Computers and Structures 89 (2011) 1286–1294

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Computers and Structures

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Critical buckling of delaminated composite plates using exact stiffness analysis

Mahdi Damghani, David Kennedy ⇑, Carol FeatherstonCardiff School of Engineering, Queen’s Buildings, The Parade, Cardiff CF24 3AA, United Kingdom

a r t i c l e i n f o

Article history:Received 14 January 2010Accepted 10 April 2011Available online 7 May 2011

Keywords:Aerospace structuresCompositesBucklingDelaminationExact stripWittrick–Williams algorithm

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.04.003

⇑ Corresponding author. Tel.: +44 29 2087 5340; faE-mail address: kennedyd@cf.ac.uk (D. Kennedy).

a b s t r a c t

The critical buckling of composite plates with through-the-length delaminations is studied using exactstiffness analysis and the Wittrick–Williams algorithm. Computational efficiency is achieved by avoidingdiscretisation into elements, making the method suitable for preliminary aircraft design. Numericalresults for longitudinal, transverse and shear loading show a transition from global to local bucklingmodes as the delamination width is increased. The critical buckling strength is dramatically reducedas the delamination is moved towards the plate surface, but is relatively insensitive to its widthwiselocation or to the edge conditions. The results are compared with finite element analysis.

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1. Introduction

Composite components are now widely used in both spacecraftand aircraft construction. Spacecraft applications include load-car-rying structural elements, fuel tanks and antennae [1], while mod-ern aircraft incorporate composites in the fuselage, wings, tailassembly, stabilisers and inner lining of cabins. Although thesematerials offer many advantages over metals in terms of specificstrength and stiffness, additional modes of failure can be encoun-tered. One of the most serious defects leading to potential failurein composite laminates is delamination, which can develop as a re-sult of low-velocity impact, fatigue load, air entrapments caused bymanufacturing processes or stress concentrations at free edges.When delamination has occurred in a composite plate, its abilityto resist compressive and shear loads will be lowered. Understand-ing the effects of delamination on buckling behaviour of suchplates is therefore necessary for the correct design and safe useof fibre-reinforced composite materials in future high performanceaircraft.

Karihaloo and Stang [2] examined, both analytically and exper-imentally, the pre- and post-buckling response of a strip delamina-tion, developing guidelines for assessing whether it poses a threatto the safe operation of a composite laminate. Riccio and Gigliotti[3] presented a fast numerical method for the simulation of delam-ination growth. Butler et al. [4] predicted the compressive fatiguelimit strain of composites containing barely visible impact damageby modelling the morphology and progression of damage.

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Lee and Park [5], Cappello and Tumino [6], and Pekbey and Say-man [7] all studied through-the-width delaminations. They consid-ered the interaction between local and global buckling behaviour,varying parameters such as the aspect ratio, width-to-thickness ra-tio and stacking sequence, as well as the size and location of thedelaminations.

Zor et al. [8], Wang et al. [9], and Hwang and Huang [10] usedfinite element analysis to investigate the effects on buckling andpostbuckling due to square and rectangular embedded delamina-tions, while Li et al. [11] solved similar problems using a semi-ana-lytical method. Wang and Lu [12] used an energy method toinvestigate the buckling behaviour of rectangular and triangular lo-cal delaminations near the surface of laminated plates undermechanical and thermal loads. Hong et al. [13] recently presenteda series solution for the closely related problems of composite lam-inates with an arbitrarily shaped cut off region.

Much of the work described above relies on finite element anal-ysis to model the critical buckling behaviour of composite lami-nates with one or more given delaminated regions. This is aversatile approach, capable of handling many combinations ofloading, boundary conditions and delamination shapes. But it doesnot provide explicit or closed-form solutions, and even withtoday’s computer hardware it can be computationally expensive.

This paper, which is an extension of an earlier conference pub-lication [14], describes an alternative approach, using exact stiff-ness analysis based on analytical solutions of the governingdifferential equations. Section 2 outlines the method of analysisand some of its applications. Section 3 introduces a through-the-length delamination model, which can be used to solve problemssimilar to those described in Refs. [5–7]. The numerical results in

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Section 4 consider the effects of delamination size and location(with respect to both depth and width) on the critical buckling loadof a composite laminate with various edge conditions and in-planeloads. The results of [14] are extended by the use of first ordershear deformation theory, by additional discussion, and by valida-tion through comparison with finite element results. Finally, Sec-tion 5 gives some conclusions and outlines how the method canbe extended to cover other types of delamination.

2. Exact stiffness analysis

Many authors have presented methods for determining the crit-ical buckling loads of prismatic structures which are assembled byconnecting plates together rigidly along their longitudinal edges.The use of exact stiffnesses derived from the analytical solutionof the member stiffness equations for individual plates of theassembly avoids the discretisation approximations inherent inthe usual finite element and finite strip methods, but results intranscendental (rather than linear) eigenproblems.

Wittrick and Williams [15] devised an algorithm which guaran-tees convergence to any required accuracy on the eigenvalues (i.e.critical buckling loads or undamped natural frequencies) of thetranscendental stiffness matrix K, which relates the amplitudesof the displacements and forces at the ends of the members. Theeigenvectors give the corresponding buckling and vibration modes.Although K is of finite order, it is derived from differential equa-tions which include the distributed stiffness and mass of the mem-bers. The algorithm therefore uses the Sturm sequence property toobtain J, the number of eigenvalues exceeded at a given trial loadfactor of frequency, as the sum of:

(i) the sign count of K, i.e. the number of negative elements onthe leading diagonal of K when it has been transformed toupper triangular form by the usual form of Gauss elimina-tion without scaling or pivoting; and

(ii) an analytical count of the number of eigenvalues that wouldbe exceeded for each of the members if its ends were fullyclamped.

This algorithm was used in the computer program VIPASA [16]in which the mode of buckling or vibration is assumed to varysinusoidally in the longitudinal direction with half-wavelength k.Fig. 1 shows, for an individual plate, its Cartesian axis systemxyz, displacement amplitudes u, v and w and its basic in-planeforce system, where NL, NT and NS are, respectively, uniform longi-tudinal, transverse and shear stress resultants (i.e. forces per unitlength). Eigenvalues and modes can be obtained for any half-wave-length k specified by the user. K relates the amplitudes of the dis-placements u, v and w and w at the plate edges to the amplitudes ofthe corresponding perturbation forces.

When all plates of the plate assembly are isotropic or orthotro-pic and have NS = 0, the nodal lines are straight and perpendicularto the longitudinal (x) direction. In these cases VIPASA gives exact

Fig. 1. Force and axis system of a plate component.

solutions for plate assemblies with simply supported ends, so longas k divides exactly into the length l. When more general anisot-ropy or shear load are present the nodal lines are skewed, andhence there are spatial phase differences across the widths of theplates. The solutions therefore only approximate simply supportedends, being quite accurate for short wavelength buckling, i.e. k� l,but becoming substantial underestimates as k approaches l, i.e.they are very conservative for overall modes [17].

This limitation was overcome in the computer program VICON[18] by coupling together the VIPASA stiffness matrices for differ-ent half wavelengths k using the method of Lagrangian multipliers.VICON retains the guarantee of convergence on all required eigen-values [19], and uses constraints to represent arbitrarily locatedpoint supports, or point connections of the plate assembly to sim-ple elastic supporting structures consisting of transverse beam-col-umns. This analysis has been extended to allow point connectionsbetween two or more plate assemblies, e.g. to model riveted con-nections [20]. All such constraints are assumed to repeat at inter-vals of l to give an infinitely long plate assembly for which thebuckling mode repeats over some multiple of l. The infinitely longmodel thus represents continuity with adjacent parts of the struc-ture and also gives approximate solutions for a plate assembly offinite length l.

The design software VICONOPT [21,22] incorporates both theVIPASA and VICON forms of analysis, and also has postbucklingand optimisation capabilities. While most conventional structuralanalysis and design procedures are based on finite element analy-sis, specialist software such as VICONOPT provides faster solutionsof sufficient accuracy for the preliminary design of aerospace struc-tures, e.g. for parametric studies which investigate many alterna-tive configurations before more detailed analysis is performed onthe most promising ones.

3. Through-the-length delamination model

Refs. [5–7] considered single composite plates with a through-the-width delamination. In this paper, the delamination is mod-elled as through-the-length, as illustrated in Fig. 2, in order to sat-isfy the prismatic requirements of VICONOPT, namely that thegeometry and loading are invariant in the longitudinal (x) direc-tion. The present work can therefore be compared with these ear-lier analyses by supposing that the plate is rotated by 90�.Comparison has also been made with the results of finite elementanalyses using ABAQUS [23].

The plate considered has length l, width b and thickness h. It iseither simply supported or clamped on all four sides. The boundaryconditions are satisfied exactly along the edges y = 0 and y = b. TheVICON analysis option of VICONOPT is employed to couple sinusoi-dal buckling responses, by dividing the plate into longitudinalstrips and using Lagrangian multipliers to enforce the boundaryconditions at intervals of 0.05b along the ends x = 0 and x = l. Thedelamination has variable width b, is at a variable depth h1 belowthe top surface of the plate, and is either symmetrically locatedabout the mid-width (as shown in Fig. 2) or extends to the plateedge y = 0. The upper and lower portions of the delaminated regioncan deform independently and are divided into separate strips, buttheir deflections and rotations must match at the edges of this re-gion. The analysis therefore allows for both overall buckling of theplate and for local buckling of the delaminated region. However,issues of contact and penetration between the upper and lowerportions are ignored, potentially giving conservative bucklingpredictions.

The ABAQUS model is as illustrated in Fig. 3. The mesh isconstructed using SC6R linear 6-node triangular in-plane general-purpose continuum shell wedge elements which have three

Fig. 2. (a) Composite plate of length l, width b and thickness h, having a central through-the-length delamination of width b = 0.4b. (b) Cross-section of plate, showingdivision into strips to enforce the end conditions and model the delamination. Solid circles denote nodes representing the longitudinal strip boundaries.

Fig. 3. Typical mesh and boundary conditions for a laminate with a through-the-width delamination of width 50 mm.

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displacement degrees of freedom at each node and use the reducedintegration method. The delaminated region is modelled as twoseparate plates with deflections and rotations matching each otherand the surrounding perfect plates at the edges of this region. Con-tact and penetration between the upper and lower portions are ig-nored to ensure consistency with the VICONOPT model. Boundaryconditions are shown in Fig. 3 and are again consistent with theVICONOPT model.

4. Numerical results

This section gives critical buckling results, assuming first ordershear deformation theory [24], for a composite laminate of lengthl = 100 mm, width b = 100 mm and thickness h = 2 mm. Thematerial has properties E1 = 130 kN mm�2, E2 = 10 kN mm�2, G12 =G13 = G23 = 5 kN mm�2, m12 = 0.3, and is laid up in 16 plies of thick-ness 0.125 mm in the sequence [0/+45/�45/90/0/+45/�45/90]S.Through-the-length delaminations occur at various locations,having varying width b .

VICONOPT results for single mid-width delaminations, locatedat depths h/2, h/4 and h/8 are compared with finite element resultsobtained using ABAQUS [23], for the three cases of longitudinal,transverse and shear loading. Further VICONOPT results are pre-

sented in the non-dimensional form N/N0, where N is the stressresultant (i.e. NL, NT or NS) at critical buckling of the delaminatedplate and N0 is the corresponding value for the perfect, undelami-nated plate (i.e. for b = 0).

4.1. Example 1: single mid-width delamination

Initially the plate is assumed to be simply supported on all fouredges and to contain a single through-the-length delamination,symmetrically located about the mid-width and at a depthh1 = h/4 below the top surface, i.e. between the 4th and 5th pliesof the laminate. Fig. 4 shows critical buckling results for the threeload cases, as the width of the delamination increases from b = 0(i.e. no delamination) to b = b (i.e. delamination extending acrossthe full width of the plate). These plots may be regarded as a datumfor comparison with the subsequent results.

In VICONOPT, the three load cases for the perfect plate give crit-ical buckling at NL = 145.8 N mm�1, NT = 145.8 N mm�1 and NS =405.6 N mm�1, respectively, which will be used as the values ofN0 for Examples 4–6. For comparison, ABAQUS gives criticalbuckling at NL = 139.7 N mm�1, NT = 139.7 N mm�1 and NS = 390.7N mm�1. Fig. 5(a) shows contour plots of the associated criticalbuckling modes. These have the expected appearance for bucklingmodes of a simply supported plate. Both ABAQUS and VICONOPTpredict the same buckling behaviour.

Fig. 4 shows that there is a negligible difference between thecritical buckling loads obtained from finite element analysis andVICONOPT. For all the load cases as long as the delamination widthb is less than or equal to 0.2b, the buckling modes remain similar tothe global modes of Fig. 5(a), i.e. all parts of the plate deflect andthe upper and lower portions of the delaminated region deflect to-gether. The almost horizontal curves for b/b 6 0.2 indicate thatsuch narrow delaminations cause only a small degradation in theglobal buckling load.

As b is increased further, the modes transform into local modes,of which contour plots are shown in Fig. 5(b) for the case b = 0.7b.Here the deflections occur principally in the (thinner) upper por-tion of the delaminated region, with the (thicker) lower portionand the undelaminated regions remaining essentially undeformed.This local buckling load decreases rapidly with increasing delami-nation width, as shown by the approximately parabolic curves for

Fig. 4. Plots of critical buckling stress resultants against delamination width b, for Example 1, a simply supported composite laminate of width b and thickness h, having athrough-the-length delamination located at mid-width and a depth h/4 below the top surface. (a) Longitudinal, (b) transverse and (c) shear loading.

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b/b P 0.2, where it becomes lower than the global buckling loadand hence represents the critical buckling condition. The limitb = b corresponds to buckling of the upper portion of the delami-nated region across the whole width of the plate, which occurs ata critical load much lower than that of the perfect plate. Each ofthe critical buckling plots of Fig. 4 may be interpreted qualitativelyas the envelope of a horizontal line (corresponding to the globalmode) for b/b 6 0.2, with a decreasing parabolic curve (corre-sponding to the local mode) for b/b P 0.2. Because VICONOPT as-sumes an infinitely long structure with periodically repeatingsimple supports, rather than a simply supported plate of finitelength, its global buckling results are expected to be slightly higherthan those of finite element analysis, as can be observed in Fig. 4.

4.2. Examples 2–4: single mid-width delamination at varying depth

Example 2 is derived from Example 1 by moving the delamina-tion to mid-thickness, i.e. between the 8th and 9th plies of the lam-inate, so that h1 = h/2. Critical buckling results are shown in Fig. 6.

Here the local modes involve buckling of both upper and lowerportions of the delaminated region, which each have thicknessh/2. Delamination causes much less degradation in bucklingstrength and the transition from global to local buckling is moregradual for all three load cases.

In contrast, Figs. 7 and 8 show that as the delamination is movedtowards the top surface, so that the upper portion of the delaminat-ed region comprises first two plies and then one ply to give Exam-ples 3 and 4, respectively, only a very small delamination isrequired to trigger a rapid transition to local buckling and a cata-strophic loss of buckling strength. These results are in agreementwith those of Cappello and Tumino [6] and several other authors.For the case of delamination with depth of h1 = h/8, the sharp tran-sitions for transverse and shear loading are accompanied by signif-icant changes in the mode shape. For longitudinal loading, thetransition is more gradual and there is less change in the shape ofthe critical buckling mode. When the delamination is located at adepth of h1 = h/16, all load cases show an immediate sharp dropfrom global buckling to local buckling at the onset of delamination.

Fig. 5. VICONOPT and ABAQUS plots of critical buckling modes for Example 1, with longitudinal (NL), transverse (NT) and shear (NS) loading. (a) b = 0. (b) b = 0.7b, with theedges of the delaminated region shown as vertical lines on the ABAQUS contour plots.

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4.3. Example 5: double mid-width delamination

The original delamination at depth h/4 is next accompanied byone at depth h/2, i.e. combining the delaminations of Examples 1

and 2. Fig. 9 shows that, apart from some small differences aroundthe transition from global to local modes, the critical bucklingbehaviour is almost indistinguishable from that of Example 1, i.e.Fig. 4. This is because the second delamination does not affect

Fig. 6. Plots of critical buckling load against delamination width b, for Example 2, a simply supported composite laminate of width b and thickness h, having a through-the-length delamination located at mid-width and a depth h/2 below the top surface. (a) Longitudinal, (b) transverse and (c) shear loading.

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the global mode, where all three portions of the delaminated re-gion deflect together. Local buckling of the upper portion, whichhas thickness h/4, is also unaffected by the second delamination,which produces central and lower portions of thickness h/4 andh/2, respectively. Allowing for contact, it is surmised that localbuckling of a multiply delaminated region is likely to be dominatedby buckling of the thinner of the two portions located at the topand bottom surfaces. This is in contrast to some of the vibrationresults obtained by Oh et al. [25] for plates with a double circulardelamination, which required time integration to satisfy thedynamic contact requirements.

4.4. Example 6: single edge delamination

In Example 6, the delamination of Example 1 is moved frommid-width to one edge of the plate, so that it runs from y = 0 toy = b. The limiting cases b = 0 and b = b are clearly identical to those

of Example 1. However the results in Fig. 10 give an almost perfectmatch with Fig. 4 for all delamination widths, showing that thebehaviour of this laminate is not sensitive to the widthwise loca-tion of the delamination. As in Example 1, if the width of delami-nation is less than 0.2b, the global buckling behaviour istriggered. Again shear buckling is more sensitive to delaminationthan the other two loading types. This sensitivity is such that localbehaviour occurs when the width of delamination is 0.1b orgreater.

4.5. Example 7: effect of edge conditions

Finally, Example 7 repeats the analysis of Example 1, but withthe edges of the plate fully clamped. With these edge conditionsthe perfect plate buckles at stress resultants of NL = 369.2 N mm�1,NT = 369.2 N mm�1 and NS = 546.7 N mm�1, which are used as thevalues of N0 in Fig. 11. The results again show a transition from

Fig. 7. Plots of critical buckling load against delamination width b, for Example 3, a simply supported composite laminate of width b and thickness h, having a through-the-length delamination located at mid-width and a depth h/8 below the top surface. (a) Longitudinal (b), transverse and (c) shear loading.

Fig. 8. VICONOPT plots of normalised critical buckling stress resultants N/N0 against delamination width b, for Example 4, a simply supported composite laminate of width band thickness h, having a through-the-length delamination located at mid-width and a depth h/16 below the top surface.

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the global mode to the local mode as the delamination widthincreases. For all types of loading the behaviour is qualitativelysimilar, with local buckling of the delaminated portion triggered

at the width of 0.1b. It is seen that by using stiffer boundary con-ditions (i.e. switching from simple supports to clamped supports),the plate is degraded at a relatively low delamination width by a

Fig. 9. VICONOPT plots of normalised critical buckling stress resultants N/N0 against delamination width b, for Example 5, a simply supported composite laminate of width band thickness h, having through-the-length delaminations located at mid-width and depths h/4 and h/2 below the top surface.

Fig. 10. VICONOPT plots of normalised critical buckling stress resultants N/N0 against delamination width b, for Example 6, a simply supported composite laminate of width band thickness h, having a through-the-length delamination located at one edge and a depth h/4 below the top surface.

Fig. 11. VICONOPT plots of normalised critical buckling stress resultants N/N0 against delamination width b, for Example 7, a fully clamped composite laminate of width b andthickness h, having a through-the-length delamination located at mid-width and a depth h/4 below the top surface.

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sudden transition to a local mode, although the overall load bear-ing capacity has increased significantly.

5. Conclusions and further work

Exact stiffness analysis has been used for the critical bucklinganalysis of composite plates with through-the-length delamina-tions. The usual discretisation of the finite element and finite stripmethods is avoided by using analytical solutions of the memberstiffness equations, the resulting transcendental eigenproblemsbeing solved by the Wittrick–Williams algorithm. Accuracy is en-hanced by coupling sinusoidal responses to give an infinitely longplate model.

Although the present study is restricted to single plates, thesoftware used readily permits the analysis of prismatic assembliesof such plates, e.g. aircraft wing and fuselage panels. It is plannedto study the effects on such structures of single and multiple del-aminations at various locations in the skin and stringercomponents.

Comparative results of finite element analysis and VICONOPTfor longitudinal, transverse and shear loading show a good agree-ment. As delamination width is increased, there is a reduction inthe critical buckling load, accompanied by a transition from a glo-bal mode to a local mode of the delaminated portion. This transi-tion tends to be sudden for shear loading, and more gradual fortransverse and longitudinal loading. The local buckling load de-creases as the delamination is moved towards the surface of the

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plate, becoming negligibly small for wide delaminations of only afew plies.

The effect of multiple through-the-length delaminations isdominated by local buckling of the top or bottom portion of thedelaminated region. The chosen example was insensitive to thewidthwise location of the delamination, and gave qualitativelysimilar results for both simply supported and clamped edgeconditions.

In local buckling, the re-distribution of stress from the buckledto the unbuckled portions might be expected to give a substantialpostbuckling reserve of strength, and this will form the basis of fu-ture investigations. It is also planned to study the effects of arbi-trarily shaped embedded delaminations, both by extending anexisting multi-structure analysis [20] and also by employingsmeared stiffness models.

Because the method uses much smaller models than finite ele-ment analysis, solution times are very competitive. For example,the 33 critical buckling results plotted in Fig. 4 were obtained ina total of 87 s and 693 s on a 2.4 GHz PC for VICONOPT and finiteelement analysis, respectively. This method therefore provides anattractive approach for parametric studies in the preliminary de-sign of aerospace structures.

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