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Postbuckling behaviour of a blade-stiffened composite panel loaded in uniaxial compression
Composites: Part A 31 (2000) 459–468 www.elsevier.com/locate/compositesa
Postbuckling behaviour of a blade-stiffened composite panel loaded in uniaxial compression B.G. Falzon*, K.A. Stevens, G.O. Davies Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK Received 21 July 1999; received in revised form 14 September 1999; accepted 20 September 1999
Abstract The postbuckling behaviour of a panel with blade-stiffeners incorporating tapered flanges was experimentally investigated. A new failure mechanism was identified for this particular type of stiffener. Failure was initiated by mid-plane delamination at the free edge of the postbuckled stiffener web at a node-line. This was consistent with an interlaminar shear stress failure and was calculated from strain gauge measurements using an approximate analysis based on lamination theory and incorporating edge effects. The critical shear stress was found to agree well with the shear strength obtained from a three-point bending test of the web laminate. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Stiffened composite panel; B. Buckling
1. Introduction The dominant part of an airframe is characterised by sheet material stabilised by stiffeners for compressive loading. The use of thin skins implies that the structure is capable of supporting load in excess of the critical local buckling load of the skin. The design of metallic aircraft structures allows certain components to buckle below the design limit load although the same confidence in the design of composite airframes has yet to be realised. Accurate knowledge of a composite structure’s postbuckling behaviour is essential in order to fully exploit the potential high strength and stiffness provided by composite material. Structural improvements will become even more apparent as an increasing number of primary structural components are designed and manufactured using composite material. The failure of a postbuckled carbon–epoxy compression panel is invariably explosive and results in the complete destruction of the panel. However, the failure mechanism observed in previous experimental investigations was initiated locally at a point in the panel corresponding to either a nodal line [1–3] or an anti-nodal line [4,5] of the buckling mode. At these locations the deformations and stress resultants have maximum values—a nodal-line failure indicating that the twisting moment is the driving * Corresponding author. Tel.: 144-171-594-5116; fax: 144-171-5848120.
mechanism and an anti-nodal (or buckle crest) failure indicating a transverse bending moment driven mechanism. A program of experimental work in the Aeronautics Department at Imperial College was undertaken to investigate the postbuckling failure of a number of carbon-fibre composite compression panels with different stiffener configurations [4–6]. This work was motivated by the well-known observation that the failure load of such panels may exceed the buckling load by a factor of two or more. For this reserve of postbuckled strength to be accessible to the designer, the failure load must be amenable to calculation. That is, the failure mechanism must be understood. Previous work [4,5] has identified failure initiation mechanisms for panels which had I, J and hat-stiffeners, shown in Fig. 1. These stiffeners were characterised by their relatively high torsional stiffness arising from the torsion/bending stiffness of the stiffener caps in the I and J stiffeners and the tube-like closed-sectioned hat-stiffeners. At the anti-nodal lines in I and J stiffened panels, the buckle deformations gave rise to a moment in the stiffener which tended to pull the stiffener from the skin. The failure of each of these panel types was shown to be a consequence of the separation of the stiffeners from the skin. This initiated at an anti-nodal line, starting from a local stress concentration such as the edge of the stiffener flange, due to the bending moment discontinuity, or at the base of the stiffener web. The failure of the hat-stiffened panels was similarly characterised by the development of bending moments which,
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2. Experimental programme Nomenclature Aij Dij hk l li Mxy Nxx Nxz Q 0 ij w g xy e xx s ij t xz t xy
membrane stiffness matrix terms bending stiffness matrix terms distance from midplane to the top of layer k depth of web eigenvalue of mode i twisting moment resultant compressive loading per unit length transverse shear force per unit length transformed reduced stiffness matrix terms out-of-plane displacement experimental shear strain in microstrain experimental direct strain in microstrain 2-D plane stress terms interlaminar shear stress in-plane shear stress
depending on the curvature of the buckled crest, either initiated damage at the free edge of the flange due to the flexural stiffness discontinuity or at the inner edge of the skin-stiffener interface. In all cases the alleviating effects of flange tapering, to prevent failure at the edge of the flange, were apparent. The blade stiffeners of the panel design investigated in this paper have only low St. Venant torsional stiffness. Thus the loading action, which destroyed the I, J and hat-stiffened panels could not develop and a different failure initiation mechanism was expected, and indeed found, away from the stiffener base stress concentrations. This paper describes the test-panels and the strain gauge placement used to determine the relevant stress resultants at failure. A fractographic survey of the failure mechanism indicated a shear failure in the stiffener web. The strain gauge results were used to calculate the magnitude of the critical shear stress from lamination theory. This critical shear stress was then compared with the value obtained from a three-point bending test of the web laminate. In Refs. [4,6] the test of a component representing a small section of the test panel but exhibiting a similar failure mechanism was shown to be instructive. Consequently, two stiffener sections with attached skin were also tested in uniaxial compression. These were shown to exhibit an identical failure mode to the panels and thus gave further credence to the postulated failure mechanism.
Fig. 1. Stiffener geometry: (a) I-stiffener; (b) J-stiffener; (c) Hat-stiffener.
Two blade-stiffened panels were manufactured by British Aerospace (Military Division) from unidirectional prepreg T800/924C with material properties as shown in Table 1. The lay-up for this panel is given in Fig. 2. Four stiffeners were secondary bonded to the skin using a film adhesive between the tapered flanges of the stiffener and the skin. It should be noted that the stiffeners are not evenly spaced— the spacing between the central two stiffeners was twice that of the outer and adjacent stiffeners. In forming the taper in the flanges, the outer 90/0/90 plies formed a protective envelope against the initiation of interply cracking due to the transverse bending which occurs in the postbuckled panel at anti-nodal locations [4]. The panel ends were potted in an epoxy resin/fibre glass mixture and machined parallel to ensure uniform loading. The unloaded edges were not supported as the stiffeners provided adequate stiffness to ensure that local buckling occurred before an Euler-type global buckling mode. The two panels tested were loaded in uniaxial compression in a custom built hyperstiff 250 ton testing machine [7] under displacement control so that catastrophic failure in compression could be assessed. Fig. 3 shows the strain gauges initially applied to the second panel tested. For brevity, not all strain-gauge results have been included in this paper although the full set of results is available in Ref. [8]. As will be discussed in a later section dealing with the failure mode, the initiation of failure in the first panel was seen to occur in the stiffener web at mid-span, i.e. corresponding to a nodal-line of the buckled mode. At the node line, the twisting moment in the stiffener is a maximum and the transverse shear force in the web due to the compression loading is also a maximum. The strain gauges were thus positioned to capture the shear strain and the distribution of compression strain in the web. From these measurements the shear stress in the web at failure could be calculated, using an approximate analysis, and compared with an experimental value of interlaminar Table 1 Material properties for T800/924C unidirectional composite @ 60 Vf, dry Property
Value
Longitudinal tensile modulus Longitudinal compressive modulus Transverse tensile modulus Transverse compressive modulus In-plane shear modulus Poisson’s ratio Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength In-plane shear strength Ply thickness
162 GPa 145 GPa 9.2 GPa 9.5 GPa 5.0 GPa 0.3 2.7 GPa 1.65 GPa 55 Mpa 225 Mpa 100 Mpa 0.125 mm
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Fig. 2. Half-section of blade-stiffened panel.
shear strength obtained from a three-point bending test of the web laminate. It will be subsequently shown that the second panel exhibited a mode switch from four half-wavelengths to five half-wavelengths at a loading which was very near the failure load of the first panel. Such a mode switch was not observed in the first panel. An acoustic emission sensor did not record any significant increase in acoustic activity thus suggesting that no significant damage had accumulated in the panel up to the mode switch. This allowed for the unloading of the panel and mounting more strain gauges in situ at the new nodal-line locations prior to the final test to failure.
3. Results 3.1. Initial buckling An in-house finite element package, FE77 [9], was used for a linear eigenvalue analysis to determine the proximity of the first few eigenmodes. Fig. 4 shows the first two buckling mode shapes of four half-wavelengths and five
half-wavelengths, respectively. The initial buckling load of 113.84 kN agreed well with the experimentally recorded value of 110 kN for the first panel and 105 kN for the second panel. Fig. 5 shows the out-of-plane displacement as measured with an LVDT placed at a buckle crest and from which the initial buckling load could be determined. Fig. 6 shows the result from back-to-back strain gauges 2/20 located at the bottom buckle crest of the central skin bay. The change in the slope of the membrane strain (calculated by averaging the two strain gauge readings) also indicates the onset of buckling. The critical load associated with the second eigenvalue, corresponding to a five half-wavelength mode shape, was calculated at 115.96 kN. The close proximity of these two modes l2 =l1 1:0186 helps to explain the mode switch observed in the second panel. A discussion of this phenomenon may be found in Refs. [10,11]. Fig. 7(a) is a shadow Moire´ fringe pattern of the second panel soon after buckling has occurred. Because the width between the central stiffeners is double that between the edge stiffeners, the buckle deformations develop earlier in the central region and thus have a higher amplitude characterised by an increased number of Moire´ fringes. Fig. 7(b) shows the higher mode shape of five half-wavelengths
Fig. 3. Strain gauge locations.
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Fig. 4. Results of an eigenvalue analysis: (a) first buckling mode shape (113.84 kN); (b) second buckling mode shape (115.96 kN). Fig. 6. Back-to-back strain gauges 2 and 20.
which occurred at a loading of 570 kN. This switch may also be observed by the strain reversal of the gauges in Fig. 6. 3.2. Postbuckling characteristics During the testing of the first panel, since the site of failure initiation was then unknown, it was also considered relevant to monitor the transverse bending strain in the skin adjacent to the central right stiffener flange at an antinodal line and 82 mm above the centreline. This is where transverse bending is a maximum and where flange peeling would occur. In Fig. 8(a) it is noted that the bending strain reverses sign at a load of approximately 250 kN. This is because of the rotation of the stiffener due to its reduced
torsional stiffness under increased compression, illustrated in Fig. 8(b). Although this may be considered to be a mode change, it occurs gradually and does not have the dynamic characteristics of the mode switch which has been observed in other experimental work on postbuckled panels and the second panel tested. This rotation will be shown to have a significant bearing in explaining the failure mechanism associated with these panels. In testing the second panel, use was made of the observations on the first. As has been mentioned earlier, a set of strain gauges were placed at nodal lines as shown in Fig. 2 to monitor the strains in the failure region deduced from the testing of the first panel. However at a load of 570 kN a dynamic mode switch was observed to a new mode shape
Fig. 5. Out-of-plane displacement at a buckle peak.
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Fig. 7. Moire´ fringe patterns: (a) initial buckling (300 kN); (b) mode switch (570 kN).
of five half-wavelengths as shown in Fig. 7(b). To investigate nodal-line strain values at failure required a repositioning of the strain gauges at the new node lines. The panel was unloaded to allow new gauges to be mounted in situ and the panel reloaded to failure. The mode switch
Fig. 8. (a) Transverse bending strain at an anti-nodal line adjacent to the stiffener flange. (b) Postbuckled cross-section showing stiffener rotation.
during the final load cycle occurred at 548 kN which was slightly lower than the load recorded in the previous cycle. This may be attributed to inevitable matrix cracking. The relocated gauge numbers were identified by the suffix B and these were placed at corresponding locations on the new node line immediately above the centreline. Although, the actual failure node-line could not be determined a priori it was assumed that the strains measured at a particular nodeline would be comparable to those developed in adjacent node-lines. In fact, failure in the second panel was shown to traverse two adjacent node-lines one of which was straingauged. Fig. 9 shows the output from back-to-back strain gauges 9 and 12. Initially these gauges were near a nodal-line so that the bending strain (the strain difference) is small. When the mode switch occurs it can be seen that a large bending strain was induced thus indicating that this location no longer corresponded to a nodal-line. Fig. 10 shows the end-displacement of the panel measured using two LVDTs near the unloaded edges of the panel. Fig. 11 shows the shear strain results measured at the initial central node-line of the stiffener web. The strain gauge rosettes 4/5 and 6/7 were positioned back-to-back on the inner left stiffener. Below the buckling of the central skin bay, no significant shear strain was recorded. Beyond this initial buckling of the skin, shear strains were recorded in the outer stiffener (strain gauges 15/17 and 16/18) implying some stiffener rotation. An increase in strain rate (indicated by a change in slope) was observed at a loading of approximately 280 kN. This was consistent with the observed rotation of the corresponding stiffener on the first panel as shown in Fig. 8(b). It may be easily shown that the twisting moment at a node-line is a maximum and induces a corresponding maximum in-plane shear strain, which, in the case of an isotropic material, would have a linear variation through
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Fig. 9. Back-to-back strain gauges 9 and 12.
the thickness. At the mode-change this node-line disappears and hence the observed drop in measured shear strain. The newly introduced strain gauges designated 4B/5B and 6B/ 7B measured a corresponding step increase in shear strain (Fig. 12) indicating that a new node-line was created in the vicinity of these gauges at the mode switch. 3.3. The failure mode The first panel failed explosively at a load of 573 kN. The failed panel is shown in Fig. 13(a) where it is seen that the damage in each stiffener occurred at mid-span, i.e. at a nodal-line in the buckled mode. The failure of the second panel, which underwent a mode switch to five half-wavelengths, occurred at 601 kN. The fracture traversed two adjacent node lines—across the left-hand stiffeners at the node-line below the mid-span and at the node-line above
Fig. 10. End displacement from two LVDTs.
Fig. 11. Shear strain measurements at the pre-mode-switch central node line of the stiffener web.
mid-span for the right-hand stiffeners as shown in Fig. 13(b). In these panel failures the stiffeners, for the most part, remained attached to the skin but mid-plane delamination at the free edge of the stiffener web was observed at the failure locations (Fig. 14). This is in contrast to the failure of the I, J and hat-stiffened panels where the failure initiation was at a skin/stiffener interface [4,5] and stiffener debonding at collapse was extensive. The stress resultants acting on an element of the stiffener web are shown in Fig. 15. Observations from the experiments suggest that a critical value of interlaminar shear stress t xz was reached in the vicinity of the mid-plane of the web, at the free edge and on a nodalline. There are two contributions to this shear stress which
Fig. 12. Shear strain measurements at the location of a newly created node line due to mode switching.
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Fig. 13. (a) Failure of 1st Panel. (b) Failure of 2nd Panel.
come from the transverse shear loading Nxz and the twisting moment Mxy. Both these stress resultants have maximum values at a nodal-line and are discussed in some detail in the following section.
per unit length: Nxz Nxx
2w 2x
1
where Nxx may be approximated as: Nxx < A11 1xx ;
where A11
4. Discussion 4.1. Mid-plane shear stress t xz due to Nxz By letting w refer to the displacement in the z-direction (see Fig. 15) it can be shown that a compressive loading on the postbuckled web will give rise to a transverse shear force
Q 011 k hk 2 hk21
k1
2 0
Q ij represents the transformed reduced stiffness components of the laminate [12], k is the layer number, with the total number of layers being n, and h the distance from the mid-plane of the laminate to the top surface of the specified layer given by the subscript k. A Laminate Composite Program (LAP) [13] was used for the required laminatebased calculations where the constitutive relation was determined by applying a load of Nxx 1000 N=mm and obtaining a strain value (4487 ms). The experimental membrane strain 1 xx at the edge of the stiffener web was determined by extrapolating the measured membrane strains of the backto-back gauges at the node-line. A value of the applied compressive loading was then determined by scaling the known applied load in the LAP program by the ratio of the experimental and computed strains. The value of 2w=2x was next estimated by making use of the relationship: Mxy < 22D66 where D66
Fig. 14. Mid-plane delamination in the stiffener web.
n X
22 w 2x2y
n 1 X Q 0 h3 2 h3k21 3 k1 66 k k
3
A value of 1.835 rad/m was obtained for 22 w=2x2y at an applied twisting moment resultant of 100 N. This, in turn, yielded a value of in-plane surface shear strain which was later used to scale the value of 2w=2x based on the in-plane
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the displacement field, due to a uniform twist, represented by: u zf y
2u 2x
v zu
w 2yu
7
u angle of twist: f(y) is a warping function which asymptotes to f y y for large y. The in-plane shear stresses which react the torque must go to zero at the free-edge resulting in a sympathetic rise in the interlaminar shear stress t xz. By the Principal of Virtual Displacements [15] it may be shown that f y
where
a2 12
Gxz 2 t Gxy
8
and t is the laminate thickness. This yields the following expression for in-plane shear strain in microstrain:
Fig. 15. Failure mechanism.
shear strain values obtained from the strain gauge rosettes mounted on the node-line. The value for 2w=2x at the edge of the web was approximated by: 2w 22 w 4 l 2x edge 2x2y where l is the depth of the web. The 2-D plane stress form of the equilibrium equations:
sij;j 0
2 2ay e 1y a
i; j x; z
5
were used to obtain the value of interlaminar shear stress at the mid-plane of the laminate. This was achieved by plylevel summation from the outer surface of the laminate to the mid-plane. The following expression was derived for the interlaminar shear stress due to Nxx:
txzuedge 1 1:35 e26 gxy 1xx MPa
6
where g xy and g xx are the extrapolated experimental shear and direct strains, respectively, in microstrain. 4.2. Mid-plane shear stress t xz due to Mxy In Ref. [14] the extent of edge effects in a composite laminate under pure torsion was assessed and a brief treatment is given herein. The plate was assumed to be a homogeneous anisotropic material having two different shear moduli Gxy and Gxz. (Refer to Fig. 15 for axis system.) The usual St.-Venant theory was modified and Table 2 Results of the approximate analysis using strain gauge data (stress in MPa) Specimen
txz 2 Nxx
txz 2 Mxy
t xz
Panel 1 Panel 2 Component Interlaminar shear strength from three-point bending test
42.3 37.2 34.7
38.6 42.1 46.3
80.9 79.3 81.0 77.0
1xy 2zu 0 1 2 e2ay
9
from which we can write an expression for the in-plane shear stress:
txy t0xy 1 2 e2ay
10
where t0xy is the shear stress obtained from classical laminate theory. By use of the equilibrium equations in Eq. (5) it may be shown that under the specified assumptions t xz will vary parabolically. The integral arising from Eq. (5) is replaced by a summation from the free surface to the mid-plane of the laminate:
txz a
mid-plane X
t0xz Dt
11
free surface
where Dt is the ply thickness. The interlaminar shear stress due to twisting moment Mxy is therefore given by:
txzuedge 2 3:45 e23 gxy MPa
12
Thus the total mid-plane shear stress at the failure load on the free edge of the stiffener web and at a node-line was estimated by the sum of the two interlaminar shear stress components. The calculated results for the two panels tested are shown in Table 2. 4.3. Component tests In previous investigations into the mechanisms leading to the failure of stiffened panels [4,6], a separate test of a small section of the panel, under representative loads or deformations, was undertaken to provide further insight into the initiation of failure. The component in this case was a length of stiffener and its associated skin, as shown in Fig. 16, loaded in uniaxial compression. The length was chosen so that the wavelength of torsional buckling was similar to the wavelength observed in the panel. The failure of the component was identical, i.e. a mid-plane delamination of the stiffener web at a nodal-line. Although the compressive loading
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tested to failure. The panels’ central skin bays buckled in a four half-wavelength mode shape. The first panel buckled at a load of 110 kN and failed at 573 kN. The second panel buckled at 105 kN and underwent a dramatic buckling mode jump to five half-wavelengths at a loading 570 kN before failing at 601 kN. Damage was shown to initiate at the freeedge of the stiffener web at a node-line. This was in contrast with the other panels tested at the Department of Aeronautics in Imperial College (I, J and hat-stiffened panels) where stiffener debonding usually preceded collapse. An approximate analysis was used to show that the failure mechanism was an interlaminar shear stress failure arising from the combination of compressive loading on the postbuckled stiffener blade and the twisting induced at the node-line of the buckled stiffener. Fig. 16. Component test specimen.
Acknowledgements was not the same as in the panel, the combination of compression and twist produced a similar total mid-plane shear stress. The full compliment of required strain gauges was only applied to one of the two component tests and the above analysis was also applied to these results as shown in Table 2. The interlaminar shear strength of the web laminate was determined by a three-point bending test on a web section of the stiffener. At failure, the mid-plane shear stress, t xz, was calculated to be 77.0 MPa. This result agreed well with the values obtained for the two panels and stiffener component using the above analysis and provided further supporting evidence to the proposed failure mechanism. The results presented in Table 2 agree well with the interlaminar shear strength of the laminate as measured using the standard three-point bending test. This is quite remarkable in the light of the simplifying assumptions that were made in the analysis. What is also of interest is that these results suggest that both the in-plane compression and the twist in the blade-stiffener web contributed approximately equally to the overall interlaminar shear stress. The failure mechanism, observed in this study, also has potential implications for the designer. It suggests that the optimizing of stiffener flanges by tapering, to reduce the interfacial shear and peeling stresses, may be effective enough to shift the initial damage site to a different location—in this case on a node-line at the edge of the web. This provides the opportunity of introducing design improvements in this region, for example stitching, to further delay the onset of damage and the quick progression to complete failure of the structure.
5. Conclusion Two blade-stiffened postbuckling composite panels were
The authors are indebted to British Aerospace Airbus and Military Aircraft Divisions, for supporting this work under research agreement DTI/SMC/4/968 and for the manufacturing of the test panels by the Military Aircraft Division.
References [1] Starnes Jr. JH, Knight Jr. FK, Rouse M. Postbuckling behaviour of selected flat stiffened graphite epoxy panels loaded in uniaxial compression. AIAA 1988;23(8):344–52. [2] Buskell N, Davies GAO, Stevens KA. Postbuckling failure of composite panels. Proceedings of the Third International Conference on Composite Structures, Paisley, UK, 1985. p. 290. [3] Falzon BG, Steven GP. Postbuckling behaviour of hat-stiffened thinskinned carbon-fibre composite panels, AIAA/ASME/AHS/ASC 36th Structures, Structural Dynamics and Materials Conference, New Orleans, LA, USA, 1995. [4] Stevens KA, Specht S, Davies GAO. Postbuckling failure of carbon– epoxy compression panels, Proceedings of ICCM-11, Gold Coast, Australia, 1997. [5] Stevens KA, Ricci R, Davies GAO. Buckling and postbuckling of composite structures. Composites 1995;26(3):189–99. [6] Stevens KA, Davies GAO, Ricci R. Postbuckling failure of composite compression panels. 19th ICAS Conference, ICAS-94-9.8.3, Anaheim, California, 1994. p. 2975–81. [7] Singer J, Arbocz J, Weller T. Experimental methods in buckling of thin-walled structures—volume 1. Buckling experiments, New York: Wiley, 1998. [8] Falzon BG, Hitchings D, Stevens KA, Davies GAO. Failure prediction of thick structural composites, Progress Report 3, DTI CARAD Contract DTI/SMC/4/968, Imperial College of Science, Technology and Medicine, London, 1998. [9] Hitchings D. FE77 User Manual, Imperial College of Science, Technology and Medicine, 1994. [10] Falzon BG, Steven BG. Buckling mode transition in hat-stiffened composite panels loaded in uniaxial compression. Composite Structures 1998;37:253–67. [11] Bushnell D, Rankin CC, Riks E. Optimization of stiffened panels in which mode jumping is accounted for. AIAA paper 1998;971141:105–45.
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[12] Jones RM. Mechanics of composite materials, New York: McGrawHill, 1975. [13] Kretsis G. Laminate Analysis Program, LAP, Centre for Composite Materials, Imperial College of Science, Technology and Medicine, London, 1995.
[14] Davies GAO, Buskell N, Stevens KA. Edge effects in failure of compression panels, Euromech 214 Symposium on Composites, Amsterdam, 1986. [15] Davies GAO. Virtual work in structural analysis, New York: Wiley, 1982.
Postbuckling behaviour of a blade-stiffened composite panel loaded in uniaxial compression B.G. Falzon*, K.A. Stevens, G.O. Davies Department of Aeronautics, Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY, UK Received 21 July 1999; received in revised form 14 September 1999; accepted 20 September 1999
Abstract The postbuckling behaviour of a panel with blade-stiffeners incorporating tapered flanges was experimentally investigated. A new failure mechanism was identified for this particular type of stiffener. Failure was initiated by mid-plane delamination at the free edge of the postbuckled stiffener web at a node-line. This was consistent with an interlaminar shear stress failure and was calculated from strain gauge measurements using an approximate analysis based on lamination theory and incorporating edge effects. The critical shear stress was found to agree well with the shear strength obtained from a three-point bending test of the web laminate. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: Stiffened composite panel; B. Buckling
1. Introduction The dominant part of an airframe is characterised by sheet material stabilised by stiffeners for compressive loading. The use of thin skins implies that the structure is capable of supporting load in excess of the critical local buckling load of the skin. The design of metallic aircraft structures allows certain components to buckle below the design limit load although the same confidence in the design of composite airframes has yet to be realised. Accurate knowledge of a composite structure’s postbuckling behaviour is essential in order to fully exploit the potential high strength and stiffness provided by composite material. Structural improvements will become even more apparent as an increasing number of primary structural components are designed and manufactured using composite material. The failure of a postbuckled carbon–epoxy compression panel is invariably explosive and results in the complete destruction of the panel. However, the failure mechanism observed in previous experimental investigations was initiated locally at a point in the panel corresponding to either a nodal line [1–3] or an anti-nodal line [4,5] of the buckling mode. At these locations the deformations and stress resultants have maximum values—a nodal-line failure indicating that the twisting moment is the driving * Corresponding author. Tel.: 144-171-594-5116; fax: 144-171-5848120.
mechanism and an anti-nodal (or buckle crest) failure indicating a transverse bending moment driven mechanism. A program of experimental work in the Aeronautics Department at Imperial College was undertaken to investigate the postbuckling failure of a number of carbon-fibre composite compression panels with different stiffener configurations [4–6]. This work was motivated by the well-known observation that the failure load of such panels may exceed the buckling load by a factor of two or more. For this reserve of postbuckled strength to be accessible to the designer, the failure load must be amenable to calculation. That is, the failure mechanism must be understood. Previous work [4,5] has identified failure initiation mechanisms for panels which had I, J and hat-stiffeners, shown in Fig. 1. These stiffeners were characterised by their relatively high torsional stiffness arising from the torsion/bending stiffness of the stiffener caps in the I and J stiffeners and the tube-like closed-sectioned hat-stiffeners. At the anti-nodal lines in I and J stiffened panels, the buckle deformations gave rise to a moment in the stiffener which tended to pull the stiffener from the skin. The failure of each of these panel types was shown to be a consequence of the separation of the stiffeners from the skin. This initiated at an anti-nodal line, starting from a local stress concentration such as the edge of the stiffener flange, due to the bending moment discontinuity, or at the base of the stiffener web. The failure of the hat-stiffened panels was similarly characterised by the development of bending moments which,
1359-835X/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S1359-835 X( 99)00 085-8
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2. Experimental programme Nomenclature Aij Dij hk l li Mxy Nxx Nxz Q 0 ij w g xy e xx s ij t xz t xy
membrane stiffness matrix terms bending stiffness matrix terms distance from midplane to the top of layer k depth of web eigenvalue of mode i twisting moment resultant compressive loading per unit length transverse shear force per unit length transformed reduced stiffness matrix terms out-of-plane displacement experimental shear strain in microstrain experimental direct strain in microstrain 2-D plane stress terms interlaminar shear stress in-plane shear stress
depending on the curvature of the buckled crest, either initiated damage at the free edge of the flange due to the flexural stiffness discontinuity or at the inner edge of the skin-stiffener interface. In all cases the alleviating effects of flange tapering, to prevent failure at the edge of the flange, were apparent. The blade stiffeners of the panel design investigated in this paper have only low St. Venant torsional stiffness. Thus the loading action, which destroyed the I, J and hat-stiffened panels could not develop and a different failure initiation mechanism was expected, and indeed found, away from the stiffener base stress concentrations. This paper describes the test-panels and the strain gauge placement used to determine the relevant stress resultants at failure. A fractographic survey of the failure mechanism indicated a shear failure in the stiffener web. The strain gauge results were used to calculate the magnitude of the critical shear stress from lamination theory. This critical shear stress was then compared with the value obtained from a three-point bending test of the web laminate. In Refs. [4,6] the test of a component representing a small section of the test panel but exhibiting a similar failure mechanism was shown to be instructive. Consequently, two stiffener sections with attached skin were also tested in uniaxial compression. These were shown to exhibit an identical failure mode to the panels and thus gave further credence to the postulated failure mechanism.
Fig. 1. Stiffener geometry: (a) I-stiffener; (b) J-stiffener; (c) Hat-stiffener.
Two blade-stiffened panels were manufactured by British Aerospace (Military Division) from unidirectional prepreg T800/924C with material properties as shown in Table 1. The lay-up for this panel is given in Fig. 2. Four stiffeners were secondary bonded to the skin using a film adhesive between the tapered flanges of the stiffener and the skin. It should be noted that the stiffeners are not evenly spaced— the spacing between the central two stiffeners was twice that of the outer and adjacent stiffeners. In forming the taper in the flanges, the outer 90/0/90 plies formed a protective envelope against the initiation of interply cracking due to the transverse bending which occurs in the postbuckled panel at anti-nodal locations [4]. The panel ends were potted in an epoxy resin/fibre glass mixture and machined parallel to ensure uniform loading. The unloaded edges were not supported as the stiffeners provided adequate stiffness to ensure that local buckling occurred before an Euler-type global buckling mode. The two panels tested were loaded in uniaxial compression in a custom built hyperstiff 250 ton testing machine [7] under displacement control so that catastrophic failure in compression could be assessed. Fig. 3 shows the strain gauges initially applied to the second panel tested. For brevity, not all strain-gauge results have been included in this paper although the full set of results is available in Ref. [8]. As will be discussed in a later section dealing with the failure mode, the initiation of failure in the first panel was seen to occur in the stiffener web at mid-span, i.e. corresponding to a nodal-line of the buckled mode. At the node line, the twisting moment in the stiffener is a maximum and the transverse shear force in the web due to the compression loading is also a maximum. The strain gauges were thus positioned to capture the shear strain and the distribution of compression strain in the web. From these measurements the shear stress in the web at failure could be calculated, using an approximate analysis, and compared with an experimental value of interlaminar Table 1 Material properties for T800/924C unidirectional composite @ 60 Vf, dry Property
Value
Longitudinal tensile modulus Longitudinal compressive modulus Transverse tensile modulus Transverse compressive modulus In-plane shear modulus Poisson’s ratio Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength In-plane shear strength Ply thickness
162 GPa 145 GPa 9.2 GPa 9.5 GPa 5.0 GPa 0.3 2.7 GPa 1.65 GPa 55 Mpa 225 Mpa 100 Mpa 0.125 mm
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Fig. 2. Half-section of blade-stiffened panel.
shear strength obtained from a three-point bending test of the web laminate. It will be subsequently shown that the second panel exhibited a mode switch from four half-wavelengths to five half-wavelengths at a loading which was very near the failure load of the first panel. Such a mode switch was not observed in the first panel. An acoustic emission sensor did not record any significant increase in acoustic activity thus suggesting that no significant damage had accumulated in the panel up to the mode switch. This allowed for the unloading of the panel and mounting more strain gauges in situ at the new nodal-line locations prior to the final test to failure.
3. Results 3.1. Initial buckling An in-house finite element package, FE77 [9], was used for a linear eigenvalue analysis to determine the proximity of the first few eigenmodes. Fig. 4 shows the first two buckling mode shapes of four half-wavelengths and five
half-wavelengths, respectively. The initial buckling load of 113.84 kN agreed well with the experimentally recorded value of 110 kN for the first panel and 105 kN for the second panel. Fig. 5 shows the out-of-plane displacement as measured with an LVDT placed at a buckle crest and from which the initial buckling load could be determined. Fig. 6 shows the result from back-to-back strain gauges 2/20 located at the bottom buckle crest of the central skin bay. The change in the slope of the membrane strain (calculated by averaging the two strain gauge readings) also indicates the onset of buckling. The critical load associated with the second eigenvalue, corresponding to a five half-wavelength mode shape, was calculated at 115.96 kN. The close proximity of these two modes l2 =l1 1:0186 helps to explain the mode switch observed in the second panel. A discussion of this phenomenon may be found in Refs. [10,11]. Fig. 7(a) is a shadow Moire´ fringe pattern of the second panel soon after buckling has occurred. Because the width between the central stiffeners is double that between the edge stiffeners, the buckle deformations develop earlier in the central region and thus have a higher amplitude characterised by an increased number of Moire´ fringes. Fig. 7(b) shows the higher mode shape of five half-wavelengths
Fig. 3. Strain gauge locations.
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Fig. 4. Results of an eigenvalue analysis: (a) first buckling mode shape (113.84 kN); (b) second buckling mode shape (115.96 kN). Fig. 6. Back-to-back strain gauges 2 and 20.
which occurred at a loading of 570 kN. This switch may also be observed by the strain reversal of the gauges in Fig. 6. 3.2. Postbuckling characteristics During the testing of the first panel, since the site of failure initiation was then unknown, it was also considered relevant to monitor the transverse bending strain in the skin adjacent to the central right stiffener flange at an antinodal line and 82 mm above the centreline. This is where transverse bending is a maximum and where flange peeling would occur. In Fig. 8(a) it is noted that the bending strain reverses sign at a load of approximately 250 kN. This is because of the rotation of the stiffener due to its reduced
torsional stiffness under increased compression, illustrated in Fig. 8(b). Although this may be considered to be a mode change, it occurs gradually and does not have the dynamic characteristics of the mode switch which has been observed in other experimental work on postbuckled panels and the second panel tested. This rotation will be shown to have a significant bearing in explaining the failure mechanism associated with these panels. In testing the second panel, use was made of the observations on the first. As has been mentioned earlier, a set of strain gauges were placed at nodal lines as shown in Fig. 2 to monitor the strains in the failure region deduced from the testing of the first panel. However at a load of 570 kN a dynamic mode switch was observed to a new mode shape
Fig. 5. Out-of-plane displacement at a buckle peak.
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Fig. 7. Moire´ fringe patterns: (a) initial buckling (300 kN); (b) mode switch (570 kN).
of five half-wavelengths as shown in Fig. 7(b). To investigate nodal-line strain values at failure required a repositioning of the strain gauges at the new node lines. The panel was unloaded to allow new gauges to be mounted in situ and the panel reloaded to failure. The mode switch
Fig. 8. (a) Transverse bending strain at an anti-nodal line adjacent to the stiffener flange. (b) Postbuckled cross-section showing stiffener rotation.
during the final load cycle occurred at 548 kN which was slightly lower than the load recorded in the previous cycle. This may be attributed to inevitable matrix cracking. The relocated gauge numbers were identified by the suffix B and these were placed at corresponding locations on the new node line immediately above the centreline. Although, the actual failure node-line could not be determined a priori it was assumed that the strains measured at a particular nodeline would be comparable to those developed in adjacent node-lines. In fact, failure in the second panel was shown to traverse two adjacent node-lines one of which was straingauged. Fig. 9 shows the output from back-to-back strain gauges 9 and 12. Initially these gauges were near a nodal-line so that the bending strain (the strain difference) is small. When the mode switch occurs it can be seen that a large bending strain was induced thus indicating that this location no longer corresponded to a nodal-line. Fig. 10 shows the end-displacement of the panel measured using two LVDTs near the unloaded edges of the panel. Fig. 11 shows the shear strain results measured at the initial central node-line of the stiffener web. The strain gauge rosettes 4/5 and 6/7 were positioned back-to-back on the inner left stiffener. Below the buckling of the central skin bay, no significant shear strain was recorded. Beyond this initial buckling of the skin, shear strains were recorded in the outer stiffener (strain gauges 15/17 and 16/18) implying some stiffener rotation. An increase in strain rate (indicated by a change in slope) was observed at a loading of approximately 280 kN. This was consistent with the observed rotation of the corresponding stiffener on the first panel as shown in Fig. 8(b). It may be easily shown that the twisting moment at a node-line is a maximum and induces a corresponding maximum in-plane shear strain, which, in the case of an isotropic material, would have a linear variation through
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Fig. 9. Back-to-back strain gauges 9 and 12.
the thickness. At the mode-change this node-line disappears and hence the observed drop in measured shear strain. The newly introduced strain gauges designated 4B/5B and 6B/ 7B measured a corresponding step increase in shear strain (Fig. 12) indicating that a new node-line was created in the vicinity of these gauges at the mode switch. 3.3. The failure mode The first panel failed explosively at a load of 573 kN. The failed panel is shown in Fig. 13(a) where it is seen that the damage in each stiffener occurred at mid-span, i.e. at a nodal-line in the buckled mode. The failure of the second panel, which underwent a mode switch to five half-wavelengths, occurred at 601 kN. The fracture traversed two adjacent node lines—across the left-hand stiffeners at the node-line below the mid-span and at the node-line above
Fig. 10. End displacement from two LVDTs.
Fig. 11. Shear strain measurements at the pre-mode-switch central node line of the stiffener web.
mid-span for the right-hand stiffeners as shown in Fig. 13(b). In these panel failures the stiffeners, for the most part, remained attached to the skin but mid-plane delamination at the free edge of the stiffener web was observed at the failure locations (Fig. 14). This is in contrast to the failure of the I, J and hat-stiffened panels where the failure initiation was at a skin/stiffener interface [4,5] and stiffener debonding at collapse was extensive. The stress resultants acting on an element of the stiffener web are shown in Fig. 15. Observations from the experiments suggest that a critical value of interlaminar shear stress t xz was reached in the vicinity of the mid-plane of the web, at the free edge and on a nodalline. There are two contributions to this shear stress which
Fig. 12. Shear strain measurements at the location of a newly created node line due to mode switching.
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Fig. 13. (a) Failure of 1st Panel. (b) Failure of 2nd Panel.
come from the transverse shear loading Nxz and the twisting moment Mxy. Both these stress resultants have maximum values at a nodal-line and are discussed in some detail in the following section.
per unit length: Nxz Nxx
2w 2x
1
where Nxx may be approximated as: Nxx < A11 1xx ;
where A11
4. Discussion 4.1. Mid-plane shear stress t xz due to Nxz By letting w refer to the displacement in the z-direction (see Fig. 15) it can be shown that a compressive loading on the postbuckled web will give rise to a transverse shear force
Q 011 k hk 2 hk21
k1
2 0
Q ij represents the transformed reduced stiffness components of the laminate [12], k is the layer number, with the total number of layers being n, and h the distance from the mid-plane of the laminate to the top surface of the specified layer given by the subscript k. A Laminate Composite Program (LAP) [13] was used for the required laminatebased calculations where the constitutive relation was determined by applying a load of Nxx 1000 N=mm and obtaining a strain value (4487 ms). The experimental membrane strain 1 xx at the edge of the stiffener web was determined by extrapolating the measured membrane strains of the backto-back gauges at the node-line. A value of the applied compressive loading was then determined by scaling the known applied load in the LAP program by the ratio of the experimental and computed strains. The value of 2w=2x was next estimated by making use of the relationship: Mxy < 22D66 where D66
Fig. 14. Mid-plane delamination in the stiffener web.
n X
22 w 2x2y
n 1 X Q 0 h3 2 h3k21 3 k1 66 k k
3
A value of 1.835 rad/m was obtained for 22 w=2x2y at an applied twisting moment resultant of 100 N. This, in turn, yielded a value of in-plane surface shear strain which was later used to scale the value of 2w=2x based on the in-plane
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the displacement field, due to a uniform twist, represented by: u zf y
2u 2x
v zu
w 2yu
7
u angle of twist: f(y) is a warping function which asymptotes to f y y for large y. The in-plane shear stresses which react the torque must go to zero at the free-edge resulting in a sympathetic rise in the interlaminar shear stress t xz. By the Principal of Virtual Displacements [15] it may be shown that f y
where
a2 12
Gxz 2 t Gxy
8
and t is the laminate thickness. This yields the following expression for in-plane shear strain in microstrain:
Fig. 15. Failure mechanism.
shear strain values obtained from the strain gauge rosettes mounted on the node-line. The value for 2w=2x at the edge of the web was approximated by: 2w 22 w 4 l 2x edge 2x2y where l is the depth of the web. The 2-D plane stress form of the equilibrium equations:
sij;j 0
2 2ay e 1y a
i; j x; z
5
were used to obtain the value of interlaminar shear stress at the mid-plane of the laminate. This was achieved by plylevel summation from the outer surface of the laminate to the mid-plane. The following expression was derived for the interlaminar shear stress due to Nxx:
txzuedge 1 1:35 e26 gxy 1xx MPa
6
where g xy and g xx are the extrapolated experimental shear and direct strains, respectively, in microstrain. 4.2. Mid-plane shear stress t xz due to Mxy In Ref. [14] the extent of edge effects in a composite laminate under pure torsion was assessed and a brief treatment is given herein. The plate was assumed to be a homogeneous anisotropic material having two different shear moduli Gxy and Gxz. (Refer to Fig. 15 for axis system.) The usual St.-Venant theory was modified and Table 2 Results of the approximate analysis using strain gauge data (stress in MPa) Specimen
txz 2 Nxx
txz 2 Mxy
t xz
Panel 1 Panel 2 Component Interlaminar shear strength from three-point bending test
42.3 37.2 34.7
38.6 42.1 46.3
80.9 79.3 81.0 77.0
1xy 2zu 0 1 2 e2ay
9
from which we can write an expression for the in-plane shear stress:
txy t0xy 1 2 e2ay
10
where t0xy is the shear stress obtained from classical laminate theory. By use of the equilibrium equations in Eq. (5) it may be shown that under the specified assumptions t xz will vary parabolically. The integral arising from Eq. (5) is replaced by a summation from the free surface to the mid-plane of the laminate:
txz a
mid-plane X
t0xz Dt
11
free surface
where Dt is the ply thickness. The interlaminar shear stress due to twisting moment Mxy is therefore given by:
txzuedge 2 3:45 e23 gxy MPa
12
Thus the total mid-plane shear stress at the failure load on the free edge of the stiffener web and at a node-line was estimated by the sum of the two interlaminar shear stress components. The calculated results for the two panels tested are shown in Table 2. 4.3. Component tests In previous investigations into the mechanisms leading to the failure of stiffened panels [4,6], a separate test of a small section of the panel, under representative loads or deformations, was undertaken to provide further insight into the initiation of failure. The component in this case was a length of stiffener and its associated skin, as shown in Fig. 16, loaded in uniaxial compression. The length was chosen so that the wavelength of torsional buckling was similar to the wavelength observed in the panel. The failure of the component was identical, i.e. a mid-plane delamination of the stiffener web at a nodal-line. Although the compressive loading
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tested to failure. The panels’ central skin bays buckled in a four half-wavelength mode shape. The first panel buckled at a load of 110 kN and failed at 573 kN. The second panel buckled at 105 kN and underwent a dramatic buckling mode jump to five half-wavelengths at a loading 570 kN before failing at 601 kN. Damage was shown to initiate at the freeedge of the stiffener web at a node-line. This was in contrast with the other panels tested at the Department of Aeronautics in Imperial College (I, J and hat-stiffened panels) where stiffener debonding usually preceded collapse. An approximate analysis was used to show that the failure mechanism was an interlaminar shear stress failure arising from the combination of compressive loading on the postbuckled stiffener blade and the twisting induced at the node-line of the buckled stiffener. Fig. 16. Component test specimen.
Acknowledgements was not the same as in the panel, the combination of compression and twist produced a similar total mid-plane shear stress. The full compliment of required strain gauges was only applied to one of the two component tests and the above analysis was also applied to these results as shown in Table 2. The interlaminar shear strength of the web laminate was determined by a three-point bending test on a web section of the stiffener. At failure, the mid-plane shear stress, t xz, was calculated to be 77.0 MPa. This result agreed well with the values obtained for the two panels and stiffener component using the above analysis and provided further supporting evidence to the proposed failure mechanism. The results presented in Table 2 agree well with the interlaminar shear strength of the laminate as measured using the standard three-point bending test. This is quite remarkable in the light of the simplifying assumptions that were made in the analysis. What is also of interest is that these results suggest that both the in-plane compression and the twist in the blade-stiffener web contributed approximately equally to the overall interlaminar shear stress. The failure mechanism, observed in this study, also has potential implications for the designer. It suggests that the optimizing of stiffener flanges by tapering, to reduce the interfacial shear and peeling stresses, may be effective enough to shift the initial damage site to a different location—in this case on a node-line at the edge of the web. This provides the opportunity of introducing design improvements in this region, for example stitching, to further delay the onset of damage and the quick progression to complete failure of the structure.
5. Conclusion Two blade-stiffened postbuckling composite panels were
The authors are indebted to British Aerospace Airbus and Military Aircraft Divisions, for supporting this work under research agreement DTI/SMC/4/968 and for the manufacturing of the test panels by the Military Aircraft Division.
References [1] Starnes Jr. JH, Knight Jr. FK, Rouse M. Postbuckling behaviour of selected flat stiffened graphite epoxy panels loaded in uniaxial compression. AIAA 1988;23(8):344–52. [2] Buskell N, Davies GAO, Stevens KA. Postbuckling failure of composite panels. Proceedings of the Third International Conference on Composite Structures, Paisley, UK, 1985. p. 290. [3] Falzon BG, Steven GP. Postbuckling behaviour of hat-stiffened thinskinned carbon-fibre composite panels, AIAA/ASME/AHS/ASC 36th Structures, Structural Dynamics and Materials Conference, New Orleans, LA, USA, 1995. [4] Stevens KA, Specht S, Davies GAO. Postbuckling failure of carbon– epoxy compression panels, Proceedings of ICCM-11, Gold Coast, Australia, 1997. [5] Stevens KA, Ricci R, Davies GAO. Buckling and postbuckling of composite structures. Composites 1995;26(3):189–99. [6] Stevens KA, Davies GAO, Ricci R. Postbuckling failure of composite compression panels. 19th ICAS Conference, ICAS-94-9.8.3, Anaheim, California, 1994. p. 2975–81. [7] Singer J, Arbocz J, Weller T. Experimental methods in buckling of thin-walled structures—volume 1. Buckling experiments, New York: Wiley, 1998. [8] Falzon BG, Hitchings D, Stevens KA, Davies GAO. Failure prediction of thick structural composites, Progress Report 3, DTI CARAD Contract DTI/SMC/4/968, Imperial College of Science, Technology and Medicine, London, 1998. [9] Hitchings D. FE77 User Manual, Imperial College of Science, Technology and Medicine, 1994. [10] Falzon BG, Steven BG. Buckling mode transition in hat-stiffened composite panels loaded in uniaxial compression. Composite Structures 1998;37:253–67. [11] Bushnell D, Rankin CC, Riks E. Optimization of stiffened panels in which mode jumping is accounted for. AIAA paper 1998;971141:105–45.
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[12] Jones RM. Mechanics of composite materials, New York: McGrawHill, 1975. [13] Kretsis G. Laminate Analysis Program, LAP, Centre for Composite Materials, Imperial College of Science, Technology and Medicine, London, 1995.
[14] Davies GAO, Buskell N, Stevens KA. Edge effects in failure of compression panels, Euromech 214 Symposium on Composites, Amsterdam, 1986. [15] Davies GAO. Virtual work in structural analysis, New York: Wiley, 1982.