Composite Structures 95 (2013) 53–62

Contents lists available at SciVerse ScienceDirect

Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Progressive failure analysis of thin-walled composite structures

Diego Cárdenas a, Hugo Elizalde b, Piergiovanni Marzocca c,⇑, Frank Abdi d, Levon Minnetyan e,Oliver Probst a

a Physics Department, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Monterrey, Eugenio Garza Sada 2501 Sur, Monterrey, N.L., Mexico CP 64849, Mexicob School of Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, Campus Ciudad de México, Eugenio Garza Sada 2501 Sur, Monterrey, N.L., Mexico CP64849, Mexicoc Mechanical and Aeronautical Engineering Department, Clarkson University, Potsdam, NY, USAd Alpha Star Corporation, 5150 East Pacific Coast Highway Ste. 650, Long Beach, CA, USAe Civil and Environmental Engineering Department, Clarkson University, Potsdam, NY, USA

a r t i c l e i n f o

Article history:Available online 27 May 2012

Keywords:Progressive Failure Analysis (PFA)Damage modelFailure criteriaThin-Wall Beam (TWB)Finite Element Model (FEM)Wind turbine blade

0263-8223/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.compstruct.2012.05.022

Abbreviations: 1D, 2D, 3D, one-, two-, and threCFRP, composite fiber-reinforced polymer; DOF, definite-element model; PFA, Progressive Failure Analys⇑ Corresponding author.

E-mail address: pmarzocc@clarkson.edu (P. Marzo

a b s t r a c t

A reduced-order finite-element model suitable for Progressive Failure Analysis (PFA) of composite struc-tures under dynamic aeroelastic conditions based on a Thin-Walled Beam (TWB) formulation is pre-sented. Validation of the PFA-TWB against an integrated PFA model based on a shell formulation andimplemented in the commercial software tool GENOA is conducted for static load conditions. A helicopterblade made from composite material and previously used in literature for the discussion of damage prop-agation is used as the reference case. The failure criteria for the different layers of the composite materialused in the PFA-TWB model have been formulated in analogy with the corresponding criteria imple-mented in the shell formulation. Comparisons between the predictions of both models for progressivelyincreasing load have been conducted in terms of the cumulative overall damage volume in the thin-walled structure, the layer-resolved cumulative damage volume, as well as through spatially resolveddamage maps for both models. A strikingly similar damage topology has been found from both modelsup to load values close to final failure, in spite of the restraining assumptions of the TWB formulation.In terms of damage volume the PFA-TWB models predicts slightly higher values which can be traced backto the inevitable differences in the failure criteria formulation in the one-dimensional and the shellmodel, respectively. It is shown that a good agreement with the predictions of the shell model in termsof the cumulative damage volume is obtained if the strength values of the composite material areadjusted upwards in a uniform manner by about 10%. Considering the common safety factors usuallyapplied in the design process of composite material the agreement of the TWB and the shell model interms of damage propagation is considered excellent, allowing for the PFA-TWB to be used in systematicdesign studies.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction and dynamics [1,2]. Recent investigations [3–11] describe the cur-

Thin-walled composite beams (TWBs hereafter) are widely ap-plied in many fields of structural engineering. Helicopter and windturbine blades are examples of flexible slender structures that canbe modeled as TWB. The most critical parts of helicopters and windturbines are the rotors, which provide thrust and lift (in the case ofthe helicopter) or allow extracting power from the wind. Rotorblade design is a complex process usually involving several disci-plines such as aerodynamics and structural analysis, acoustics,

ll rights reserved.

e-dimensional, respectively;grees of freedom; FE, FEM,is; TWB, Thin-Walled Beam.

cca).

rent state-of-the-art in aeroelastic analysis, the success of whichhighly depends on the structural model’s ability to reproduce staticand dynamic behavior under complex aerodynamic flow condi-tions. To this end, Finite Element (FE) and modal models are thetwo main approaches. Due to the very extensive computational re-sources required in aeroelastic and rotor-dynamics coupled simu-lations, finite-element representations are limited in practice to 1Dlinear or non-linear beam models, with the potential of reducingthe size of the analyzed system down to a few dozens of DOF, albeitat the expense of accuracy and loss of detail due to the simplifica-tion of geometry and material layup [12]. On the other hand, modalapproaches rely on linear mode superposition in order to representthe overall structural behavior, and their accuracy depends on gen-erating sufficiently refined mode shapes for different rotor speeds[14]. To accurately evaluate the structural integrity of a rotor blade

54 D. Cárdenas et al. / Composite Structures 95 (2013) 53–62

the practitioner should perform an assessment by aeroelastic sim-ulations, in combination with failure models obtained experimen-tally or theoretically [3,15–19].

However, it is currently unfeasible to use this approach to ac-count for the interaction between the damage progression on theblade and the aeroelastic response. It is well known that the pres-ence of damage in a composite structure changes the stiffness ofthe structure [20–29], requiring a continuous update of the struc-tural properties of the model as damage progresses. Current ap-proaches for modeling composite blade dynamics in aeroelasticcodes (based on classical 1D FE-beam and modal models) use con-densed mechanical and geometrical properties; it is therefore of aparamount difficulty to track damage progression on a layer-by-layer basis. Thin-Walled Beam (TWB) models, on the other hand,provide an effective one-dimensional representation of the dynam-ics of a composite structure, while allowing recovering the layer-resolved strain and stress fields to evaluate structural damageprogression.

There is little work documented in literature on the combina-tion of beam FE models, recovery of shell strains/stresses anddamage tracking. Li [3] in her doctoral thesis uses the VABS meth-odology, developed in [13], in combination with a damage modelto optimize the structural properties of a helicopter rotor blade.She first calculates the strain/stress distributions in the blade fora typical helicopter manoeuvre and then performs an offline fati-gue damage analysis for the complete aeroelastic time series gen-erated from a multi-body dynamic software package. In the case anon-acceptable damage occurs the design is revised and newlysubjected to a failure analysis; otherwise the design is deemed tobe appropriate. To the best of the authors’ knowledge, while thisapproach allows for blade optimization in complex aeroelastic sit-uations it does not provide a platform for integrated damage pro-gression analysis, where the structural properties of the blade arecontinuously updated while damage progresses.

Pawar and Ganguli [32] described an integrated damage modelobtained by combining a matrix-cracking damage model devel-oped by Gudmundson et al. [34,35], ply de-bonding/de-laminationand fiber-breakage models developed by Shahid and Chang [36],and a TWB model developed by Chandra and Chopra [37]. In theirwork the authors obtained relationships between the blade re-sponse and damage densities. A static load is applied to the rotorblade and the structural response of the blade as a function ofthe damage level is calculated. In a follow-up paper [33] Pawarand Ganguli study the effect of the damage level on the natural fre-quencies of a rotating thin-walled beam. In both papers the statedobjective of the authors lies with structural health monitoring,requiring a relationship between the global damage level and thestatic and dynamic response of a blade. The authors do not provide,however, a model capable of predicting damage progression as afunction of applied load or spatial distributions of the damage inresponse to a given load.

To fill this gap in knowledge and to provide the author’s view onhow an integrated damage progression analysis can be performed,in the present work a Progressive Failure Analysis (PFA) Thin-WallBeam (TWB) Finite Element (FE) model designed to predict theprogression of damage in slender complex structures, like rotorblades, is described. As opposed to the work reviewed above, thismodel is capable of continuously updating the structural proper-ties of the blade as damage progresses, thereby providing an inte-grated description of failure propagation. The selected TWB modelwas originally developed by Librescu and co-workers [38–41] andsubsequently discretized via FE by Vo and Lee [42–44]; an applica-tion of the Librescu–Vo–Lee model to a realistic turbine blade wasdescribed in [45]. The TWB model allows including materialanisotropy, arbitrary laminate layups and shear deformabilityand has shown to reproduce the fundamental structural behavior

of 3D shell models with reasonable accuracy [45]. Stress/strainfields for individual layers can be recovered based solely on theknowledge of the nodal displacements obtained from the effective1D finite-element model, thereby allowing for Progressive FailureAnalysis (PFA) at any layer and position of the structure. A furtheradvantage is the availability of analytical expressions for the stiff-ness of arbitrary cross-sections, making damage tracking a morestraightforward task during a displacement-based FE analysis.For purposes of validation of the PFA-TWB model described here,the failure criteria proposed by Chamis and Minnetyan [46] wereused, since these criteria are implemented in the GENOA package,a commercial software tool designed for composite failure analysisin complex structures [47]. As a case study the methodology pre-sented was applied to the helicopter blade described by Pawarand Ganguli [32,33] for the case of static loads. Application of themodel to damage progression under dynamic aeroelastic condi-tions will be reported in a follow-up paper.

2. Methodology

The integrated TWB-PFA model described in the preceding sec-tion was developed and implemented for the purposes of real-timedynamic failure analysis under realistic load conditions; however,the objective of the present paper is to present a validation ofthe model against a static model implemented in the GENOA pack-age. In this section the integrated approach is discussed in detailand an application to a reference case is provided.

2.1. Integrated Thin-Walled Beam Progressive Failure Analysis(TWB-PFA) model

The Thin-Walled Beam (TWB) model used in the present workis based on the formulations given in Refs. [38–44]; an applicationof the TWB formulation to a realistic wind turbine blade was hasbeen described in [45]. The main aspects of the TWB model willbe briefly reviewed below.

The TWB model is an effective one-dimensional representationof beam-type structures such as helicopter or wind turbine blades,capable of recovering full 3D strain and stress information basedon the knowledge of the nodal displacements of the 1D modelalone. Evidently, a certain set of conditions has to be obeyed in or-der to allow such a reconstruction of information. These conditionsare the following [43]: (1) The structure is restricted to small strainvalues, (2) the beam cross sections remain undeformed in theirown plane for all load conditions applied, (3) both the transverseshear strains c0

xz; c0yz and warping shear c0

x are uniform over thecross section, (4) the Kirchhoff–Love assumption in classical platetheory remains valid for laminated composites.

Based on these assumptions, a double integration of geometricand materials properties of the cross-section (through the wall andalong the contour) can be conducted, yielding semi-analyticalexpressions for the stiffness tensor (Eij) of arbitrary cross-sections.Since the coefficients Eij contain detailed information about thestiffness of any material point in the shell (as specified by its angu-lar and thickness location), the full deformation and stress fieldscan be recovered for post-processing purposes, such as for theapplication of failure criteria for composites. If a failure criterionis met at a given location in the structure, the corresponding elasticmoduli at this location may be degraded according to the failuremodel chosen. The stiffness tensor can then be updated and anew set of nodal displacements may be calculated.

The traditional approach is to conduct the elastic calculationsand the failure analysis separately [3,15–19]; then a progressivefailure analysis often becomes a tedious process, even in the caseof static loads. PFA for dynamic loads, on the other hand, is practi-cally impossible with this approach. The use of reduced-order

Fig. 1. PFA simulation cycle.

D. Cárdenas et al. / Composite Structures 95 (2013) 53–62 55

model such as TWB, on the other hand, allows implementing a cou-pled procedure potentially capable of analyzing the progression ofdamage under dynamic load conditions in real-time.

Reduced-order models necessarily come at the expense of accu-racy, so it is important to evaluate the validity of the assumption ofthe model (1)–(4) under the load conditions studied and identify arange of applicability of the model. One of the key assumptionswhich may be violated under severe load conditions is the hypoth-esis that the contours of the beam sections remain constant(assumption 2 of the TWB); the validity of this assumption willbe further examinated in the results section.

A Timoshenko formulation is used with seven independentvariables given by the node translations U(z), V(z), W(z) where zis longitudinal or beam axis, the angular displacement /(z) of thecross sections around the z axis, and the angular displacementswx(z), wy(z), wx(z) around the x, y, and warping directions,respectively.

Once the FE displacement field (axial direction and shear) hasbeen calculated, the deformation field is recovered as follows:

ez ¼ e0z þ ðxþ n sin hÞjy þ ðy� n cos hÞjx þ ðx� nqÞjx

csz ¼ c0xz cos hþ c0

yz sin hþ c0x r þ c

2t

� �þ nþ c

2t

� �jsz

ð1Þ

where x, y are in-plane coordinates defining the cross-section’s mid-surface, and n is the thickness coordinate measured from the mid-surface; e0

z is linear strain along the blade axis; cxz, cyz, cx are trans-verse shear strain and warping strain; x is the warping function; jx,jy, jx, jsz are curvatures in the x (flapwise bending), y (edgewisebending), x(warping) and shear (torsional) directions; t is the wallthickness while k is the St. Venant circuit shear flow. In addition, rand h are geometric variables relating the local coordinate system ofthe blade section with a global coordinate system fixed at the bladeroot. The stress field can be calculated for arbitrary materials lay-ups, where each lamina k with the composite material obeys a con-stitutive law given by:

rz

rsz

� �¼

Q �11 Q �16

Q �16 Q �66

" #ez

csz

� �ð2Þ

where the coefficients Q �ij are obtained from the lamina’s stiffnesscoefficients, as defined in a local reference frame given by the mate-rials principal axes, in the following manner: first the 3D orthotro-pic law is reduced to plane stress/strain conditions [49], andsubsequently a transformation from the local to the global coordi-nates is performed. Once the finite-element stress field has beencalculated, it must be transformed back to the local (material) coor-dinates in order to evaluate the chosen failure criteria at each layer:

r11 r12

r21 r22

� �K

¼ ½T�rz

rsz

� �K

½T�T ð3Þ

where the subscripts 1 and 2 identify the directions along andtransverse to the fiber, respectively, and [T] collects the director co-sines between the global and local coordinate systems. The localstress values calculated in this manner can then be compared withthe selected failure criteria. In the present case, the failure criteriaproposed by Chamis and Minnetyan [46] have been chosen. No fail-ure occurs while the local stresses at each lamina remain within thelimits shown below:

SL11C < r11 < SL11T

SL22C < r22 < SL22T

Ss12C < r12 < Ss12ðþÞ

ð4Þ

where SL11C and SL11T are the strength values for the compressiveand tensile stress, respectively, for the along-fiber direction. Simi-larly, SL22C and SL22T are the in-plane compressive/tensile strengthvalues for the direction perpendicular to the fibers; SS12(�) and

SS12(+), finally, are the in-plane shear strength values. The reasonfor choosing the Chamis–Minnetyan criteria over other, often moreelaborate, failure criteria lies with the fact that these criteria havebeen implemented in the GENOA package [47] designed for pro-gressive failure in a shell-based environment, thereby providing auseful reference tool against which the reduced-order model pre-sented here can be compared (see Section 2.3 for a brief introduc-tion into GENOA).

2.2. Implementation of the integrated TWB-PFA model

As stated above, for the purposes of the present work only staticload are considered. Fig. 1 shows the flowchart of this implemen-tation, as described in the following.

The general procedure consists in an iterative process where thematerials properties are progressively degraded as the load is grad-ually increased. The initial step of each loop of a static TWB-PFAsimulation consists in the definition of a nodal load vector;{f} = {f0} at the first iteration. Then, the TWB finite element modelis solved for nodal displacements, where the current materialsproperties of the blade are used; the nodal displacements arestored in the vector {u}. Subsequently, the strain and stress fieldtensors expressed in the (n,s,z) coordinate system used by theTWB formulation are obtained for all layers and all azimuthaland spanwise locations. The stress tensor is then transformed ateach layer into the corresponding materials coordinate system bymeans of a passive rotation (see Eq. (3)); this coordinate systemis understood as the one defined by the principal axes of the corre-sponding material at this layer. Once the stresses in a given layerare known, the failure criteria given by Eq. (4) can be evaluated.If no increase in damage is found, the nodal load vector {f} is in-creased by a small step Df1, and the analysis is repeated. If, onthe other hand, the increase in damaged volume is higher than apercentage pre-defined by the user, the load is reduced by Df2

(|Df2| < |Df1|), i.e. to an intermediate value between the last two.This maximum tolerable increase in damage volume for one simu-lation step can be adjusted by the user but it typically of the orderof 1%. The objective of this procedure is to avoid too large increasesin damage in one simulation step, which would lead to a loss ofresolution. If a non-zero permissible damage level is observed inthis step, then the load is maintained at a constant level for severalsimulation steps until the damage stabilizes. Only after stabiliza-tion the load is increased again. Progressive Failure Analysis(PFA) takes place whenever an acceptable (e.g. sufficiently small)

Fig. 2. Cross sectional area of the blade.

Table 1Material elastic and strength properties.

CFRP (D-spar and Skin) Balsa wood (web)

E11 (MPa) 206,000 2070E22 (MPa) 20,700 2070G12 (MPa) 8300 848V12 0.30 0.22SL11 T (MPa) 1979 13SL11 C (MPa) 1000 13SL22 T (MPa) 59 13SL22C (MPa) 223 13SL12S (MPa) 103 3

56 D. Cárdenas et al. / Composite Structures 95 (2013) 53–62

increase in damage volume occurs. In such a case, materials prop-erties are updated according to the type of damage yielded by theevaluation of the failure criteria, which in turns requires updatingthe TWB model, leading to updated nodal displacements, strain/stress fields and new damage assessment, all for the same amountof load {f}. This PFA cycle is iterated until damage stabilizes for agiven load level {f}. Simulation ends when the percentage of dam-aged volume, as pre-defined by the user, is reached in one or moreTWB elements.

2.3. Finite-element shell model (reference model)

The GENOA [47] package was used a reference tool for compar-isons with the TWB-PFA model described in this work. GENOA com-bines a commercial Finite Element Analysis (FEA) package withproviding Progressive Failure Analysis (PFA) on various scales[47]. GENOAs approach for PFA is based on the Composite Durabil-ity Structural Analysis computer program, CODSTRAN [46],originally developed at NASA Glenn Research Center for polymer–matrix composite structures, and significantly extended in capabil-ities by the Alpha Star Corporation. It consists of three modules:composite mechanics, finite-element (FE) analysis, and damageprogression modeling. The ICAN composite mechanics module[48] is recalled before and after each FE analysis. The module com-putes the composite properties from the fiber and matrix constitu-ent characteristics and the composite layup. Then, the FE moduleaccepts the composite properties at each node and performs theanalysis at each load increment, computing generalized nodalforces and deformations. This information is subsequently suppliedto the ICAN module which evaluates the amount and nature of localdamage, if any, in all plies of the composite laminate. Individual plyfailure modes are assessed by ICAN using failure criteria associatedwith the negative and positive limits of the six ply-stress compo-nents in the material directions. No failure occurs as longs as thestress values remain within the limits exhibited by the followingequation:

SL11C <r11 < SL11T Along-fiber ðlongitudinalÞtensile=compressive strengthSL22C <r22 < SL22t In-plane tensile=compressive strengthSL33C <r33 < SL33T Through-the-thickness tensile=compressive strengthSS12ð�Þ <r12 < SS12ðþÞ In-plane shear strengthSS23ð�Þ <r23 < SS23ðþÞ 1st through-the-thickness shear strengthSS13ð�Þ <r13 < SS13ðþÞ 2nd through-the-thickness shear strength

ð5Þwhere r and S are ply stress and strength in directions defined bynumerical indexes (1,2,3) according to usual tensorial notation,respectively. Directions 11, 22, 33 are given in relation to fiber ori-entation: along, in-plane transversal and through-the-thicknesstransversal, while 12, 23, 13 represent in-plane, first and secondout-of-plane shear directions. Strengths are calculated by ICANbased on constituent fiber and matrix strengths and micromechan-ics equations [48]. In addition to failure criteria based on the stresslimits described above, a modified distortion energy (MDE) failurecriterion that takes into account combined stresses can also be con-sidered [46]. For the current analysis, however, this failure criterionwas not considered as to facilitate comparison with the TWB-PFAmodel.

2.4. Description of case study

Both the reduced-order TWB-PFA model presented in this workand the GENOA package were used to conduct a comparative studyfor the damage prediction in a structural model of a helicopterblade as described by Pawar and Ganguli [32–33]. The bladesexternal geometry is based on the NACA 0012 airfoil with a con-stant 305 mm chord and a two-cell cross sectional area, as shown

in Fig. 2; the total blade length is 5080 mm. The D-spar and skinsections are divided at 35% of the chord. The blade reported in[32–33] does not specify the use of a web; however, in this workboth webless and webbed cases were simulated in order to facili-tate extending results to wind turbine blades. For the latter case,the web consisted of a 2 mm width solid cross-section the centre-line of which is located at 35% of the chord, running throughout theentire length of the blade. Table 1 lists the elastic and strengthmaterials properties for Carbon Fiber Reinforced Polymer (CFRP,used for the D-spar and Skin sections) and Balsa (used for theweb) used in the blade. Table 2 details the material layup for theD-spar and Skin sections, specifying the thickness of each layerfrom the outmost to the innermost.

Based on the databases corresponding to Tables 1 and 2, two FEmodels were generated: Model 1 consists of 13,600 4-node shellelements, 13,467 nodes and 80,802 degrees of freedom (DOFs)and was subjected to progressive failure analysis using GENOA.Model 2 consists of 100 TWB elements, 101 nodes and 707 DOFs.Both models were subject to equivalent boundary conditions (fixedat the root end) and loading (tip force applied in the out-of-planedirection). To assess the extent of cross-section deformation inthe Shell model and therefore the range of applicability of theTWB-PFA model, two versions of model 1 (webless and webbed)were constructed.

3. Results and discussion

Fig. 3 illustrates the overall blade tip response during PFA. It canbe observed that the three models (shell webless, webbed andTWB) have identical flexural stiffness for small loading, the regionwhere no or only negligible damage would be expected. It shouldbe noted that the reduced-order (TWB) model provides an excel-lent reproduction of the predictions of the far more detailed shellmodel. For higher loadings, the webless shell model exhibits a ra-pid decrease in flexural stiffness, explained by the collapse of theblade’s cross section, with a deviation from the other models ofabout 10% at a tip load of 1000 N and a rapidly growing discrep-ancy for higher loads. This reduction in flexural stiffness can betraced back to a collapse of the cross-sectional area of the webless

Table 2Material layup for the airfoil profile, listed from the outmost to the innermost layer.

# Layer D-Spar Skin

Thickness(mm)

Fiber angle(�)

Thickness(mm)

Fiber angle(�)

1 0.762 0 0.381 02 0.508 45 0.508 453 0.508 �45 0.508 �454 1.016 90 0.254 905 0.508 45 0.508 456 0.508 �45 0.508 �457 0.762 0 0.381 0

Fig. 4. Evolution of the blade cross-section at 450 mm from the root end as afunction of the applied load.

D. Cárdenas et al. / Composite Structures 95 (2013) 53–62 57

blade, as illustrated in Fig. 4. In this figure the cross sectionmeasured at 450 mm from the root end, the location where themost notorious change in profile area occurs, has been plottedfor different values of the tip load. Interestingly, a reduction incross-sectional area can be observed for load values well before adifference in flexural response becomes conspicuous, with theonset of the cross-sectional collapse occurring for load values aslow as 200 N. The webbed shell model, on the other hand, main-tains its cross-sectional area up to load values well into the regionwhere structural damage occurs, as shown below. For a load valueof 1000 N the reduction of the cross section is barely 5%, and stillfor 1500 N the decrease is limited to about 10%. It is evident fromthese findings that the load carrying capabilities of the blade arewidely extended by the use of the web, in accordance with goodengineering practices in other fields, such as wind turbine bladeengineering [50]. Apart from providing higher load carrying capac-ity, the use of the web ensures the approximate constancy of thecross-sectional area and justifies the use of the TWB model usedin this work. After these introductory findings, the core of the re-sult section will concentrate on the comparison of the TWB andthe webbed shell model.

In a first level of analysis the total damage volume of both theTWB and the shell models was calculated and plotted against theload value (Fig. 5). Total damage volume is defined as the sum ofdamaged TWB or shell elements, respectively, resolved both bylayer and span/contour-wise location. It is conspicuous that theprediction of the onset of damage is very similar in the TWB andwebbed shell model, with an initial damage occurring at a loadof about 1250 N in the case of the TWB model and 1300 N forthe shell model. This corresponds to an error of about 3.8% in theprediction of the load where initial damage occurs, which is a fairlyreasonable agreement. It should be noted that the TWB errors onthe safe (conservative) side. Consistent with the slightly earlier on-set of damage in the TWB model is the larger rate of increase withload which amounts to about 1.5% per 1000 N of load for the TWBin the linear region of Fig. 5 versus 1.2% per 1000 N of load for the

Fig. 3. Out-of-plane blade tip displacement versus applied load.

(webbed) shell model. While this difference in slope is far moresubstantial (25%) than the difference in the damage onset values,the subsequent layer-resolved damage analysis (below) will dem-onstrate that this discrepancy is actually much smaller if moreappropriate error metrics are used. For the moment it can be statedthat the general trend predicted by the TWB and the webbed shellmodel beyond initial failure is very similar, including the change ofslope occurring at about 2000 N. Not unexpectedly, the weblessmodel shows an onset of damage at about 1050 N, well beforethe TWB and the webbed shell model. At this load value thecross-section at 450 mm from the root end (Fig. 4) can be seen tobe partially collapsed (at about 75% of its original area), leadingto significant loss of the section’s structural properties. In thefollowing, the comparative study will not further consider thewebless model.

As stated earlier, the main motivation for using the TWB-PFAmodel described in this work is its capability of providing de-tailed damage maps for different load values, thereby allowingthe use of the model as a structural design tool, identifying andmodifying critical areas in a sequence of rapid design cycles. Inthe following only the webbed shell and the TWB will be com-pared in detail. As described above, the damage maps are con-structed from the nodal displacements of the low-DOF TWBmodel based on the analytical expressions (Eq. (1)) between thenodal displacements and local strains at arbitrary locations alongthe perimeter of a given contour and through the wall. In the fol-lowing only the webbed shell and the TWB will be compared indetail. Only layers 4 (90�) and 7 (0�) of the thin-walled structurepresent damage at any of the load values studied. This finding isconsistently predicted by both the TWB and the shell model. Inorder to compare the predictions of both models in a detailedmanner, a damage map for layer 4 has been plotted for four dif-ferent load values for both the TWB and the shell model; the re-sults are shown in Fig. 6. Damaged cells are shown in red (dark)on top of the backdrop of a three-dimensional visualization of the

Fig. 5. Percentage of damage volume vs. load for the three blade models.

(a) (b)

Fig. 6. Damage progression of layer 4 (90�). (a) Plot showing location of damagedcells in layer 4 for different values of the external load as predicted by the GENOAmodel. (b) Corresponding plot built from the results obtained with the TWB-PFAmodel. Note that different scales have been used for visualization the deflection ofthe beam in the GENOA model (a) and the TWB-PFA model (b).

58 D. Cárdenas et al. / Composite Structures 95 (2013) 53–62

blade, where undamaged cells are displayed in light blue. Pleasenote that due to the different visualization tools used the scalesof the blade displacements are different in the TWB and the shellpart, respectively. Please also refer to Fig. 3 showing that theblade displacements of both models are essentially identical foridentical loads. It should also be noted that due to the algorithmicnature of the load variation in both PFA tools the load value se-quences corresponding the two models in Fig. 6 are not identical,although reasonably similar.

It is conspicuous from Fig. 6 that the damage maps for corre-sponding loads at layer 4 are strikingly similar between the two

(a)

(b)

Fig. 7. Top view of the damage progression of layer 4 (90� fiber orientation). (a) Damagedamage map obtained with the TWB-PFA model.

models. Most importantly, both models predict the occurrenceof damage under tensile stress only. Secondly, in both cases thedamage zone is a tongue-like structure which propagates fromthe root zone towards the spanwise direction. While the damageregion at the lowest load value shown in Fig. 6 is a little larger inthe TWB model than the corresponding region in the shell model,the increase in size is practically identical in both cases. Takingthe total length of the damage size as one possible metric, a linearincrease as a function of load can be observed for load valueshigher than about 1600 N (the three highest load cases for bothmodels). This increase is 1.84 mm per applied N of load for bothmodels. If the first load point is included (leading to a somewhatless linear relationship), then the incremental length of the dam-age zone is 2.2 mm/N for the shell model and 2.3 mm/N for theTWB model, still only a 5.5% difference. The striking similarityof the topology of the damage zones for the two models is furtherillustrated by a plane map for the four load cases studied (Fig. 7).The similarities include the contour-wise location of the damagezone which is centered at about �60 mm for both models. As ex-pected from its more detailed structure, the shell model showsmore irregular damage patterns, but the overall topology is verysimilar.

Damage in layer 4 is caused on the tension side of the blade andoccurs because of the 90� orientation of the fibers with respect tothe blade axis, leading to a situation where much of the load hasto be carried by the matrix. Layer 7, on the other hand, fails onthe compressed side of the blade, which can be explained by thefact that the stresses are highest in this layer because of the 0� fiberorientation and the strength for compression loads is relatively lowcompared to its tensile strength.

Additional insight into the progression of damage at layer 4 canbe obtained by studying the cumulative damage along the blade, asshown in Fig. 8. Three curves have been plotted in the sequence offour subplots, each corresponding to one load situation: the predic-

map calculated from GENOA for different values of the tip load. (b) Corresponding

Fig. 9. Damage progression of layer 7 (0�) as shown as damage maps for the twoload cases where damage was observed. Left: Predictions of the shell model. Right:Predictions of the TWB-PFA model.

D. Cárdenas et al. / Composite Structures 95 (2013) 53–62 59

tions directly obtained from both the TWB and the webbed shellmodel, respectively, and an additional TWB model output curvewith adjusted failure strength values. The adjustment factor wasequal for all strength values at a given load, but was allowed tovary among load values. In all load cases a relatively small adjust-ment of the strength values used in conjunction with Eq. (4) is suf-ficient to reproduce the predictions of the more detailed (shell)model. The criterion use to fit the cumulative damage curves wasthe requirement that the total damage along the blade for a givenload value should be identical. In the case of the lowest load valuethis was achieved by increasing the strength values used in theTWB-PFA model by 6.5%, whereas in the higher load cases the in-crease was 9.5%, 12%, and 14.5%, respectively. A still fairly accept-able fit is achieved if a uniform increase of 10% is used throughout.Even in the cases of the best fits for each load case the TWB-PFAcan still be seen to slightly overpredict the damage at the root sec-tion, but the error is typically only of the order of a few percent.Moreover, the error is conservative in all cases, i.e. a designer rely-ing on the TWB-PFA model will always err on the conservativeside.

Fig. 8. Cumulative damage volume as a function of the spanwise coordinate forlayer 4 (90� fiber orientation) and four load cases. Continuous curves: Predictions ofthe shell model. Dashed curves: Predictions of the unadjusted TWB-PFA model.Fine-dashed curve: Predictions of the TWB-PFA model with adjusted strengthvalues.

While the agreement in the detailed predictions of the damageprogression between the TWB-PFA and the GENOA (shell) model isnot perfect, it is still surprisingly good, particularly if the hugereduction in degrees of freedom (DOF) is considered. Whereasthe shell model needs 80,202 DOFs to accurately represent thegeometry and materials layup of the blade, the reduced-ordermodel uses only 707 DOFs, a mere 0.88% of the DOFs of the shellmodel. Consequently, the TWB model is generally more rigid andless capable of accommodating the externally imposed load bystraining its internal DOFs.

Evidently, the TWB-PFA model cannot account for the interac-tion between a web and the blade shell such as in the case of thewebbed shell model used here in conjunction with GENOA. Asshown in Figs. 9 and 10 for the case of the 1800 N load case a nee-dle-like longitudinal damage feature can be seen to emerge in theshell model which is not predicted by the TWB model. This featurecan be attributed to the presence of the web in the shell model,leading to a localized concentration of stress in the interface withthe web. Apart from this feature, the agreement between the dam-age topologies predicted by both models is strikingly similar.Regarding the total damage volume at the two load values wheredamage arises (about 1800 and 2000 N), the agreement betweenthe predictions between the TWB and GENOA is remarkably good(about 0.025% at 1800 N and 0.3% at 2000 N for both models,Fig. 11).

It should be emphasized that layer 7 represents only a smallfraction of the total damage in the blade (0% at the lowest two loadvalues, about 0.025% of the total volume versus about 1% in layer 4for the 1800 N load case, and about 0.3% versus 1.2% in layer 4 forthe 2000 N load case), so discrepancies in this layer are of nopractical concern at the design stage. Even so, the TWB correctlypredicts the onset of damage, the total damage volume and(roughly) even the shape of the damage zone, with the exceptionof the needle-like feature caused by the presence of the web. Thisis quite an achievement for a reduced-order model. For forensicanalyses, a more detailed model may be indispensible, but at thedesign stage the TWB-PFA model may be good enough.

A few remarks on computational economy: setting up the shellmodel took about 3 person-h, while the corresponding task for theTWB model was estimated at 0.5 person-h. Execution time wassome 4 h for the shell model, compared to 5 min for the reduced-order model. This difference in set-up and computation time islikely to increase for more complex structures than the helicopterblade studied in the present work.

Fig. 10. Top view of the damage progression of layer 7 (0� fiber orientation). Upper graph: Predictions of the shell model. Lower graph: Predictions of the TWB-PFA model.

Fig. 11. Cumulative damage volume as a function of the spanwise coordinate forlayer 7 (0�). Continuous curve: Predictions of the shell model. Dashed curve:Predictions of the TWB-PFA model.

60 D. Cárdenas et al. / Composite Structures 95 (2013) 53–62

4. Summary and conclusions

In the present work, a reduced-order Thin-Wall Beam (TWB)Progressive Failure Analysis (PFA) model has been presented anda detailed validation of its predictions for a helicopter blade subjectto static tip loads against a more elaborate shell model, built in thecommercial PFA software suite GENOA, has been conducted. TheTWB-PFA represents an equivalent one-dimensional model witha dramatically reduced number of degrees of freedom (DOF) com-pared to the detailed shell model. In spite of being a one-dimen-

sional representation of the blade, the TWB-PFA model is capableof generating detailed maps of the progressively occurring damagein response to increasing external loads by taking advantage of theanalytical relationships between the node displacements and thelongitudinal and shear strains which, together with the constitu-tive laws for each lamina of the composite material, allowingapplying failure criteria to the blade resolved both by layer andby contour-/spanwise position.

Different levels of analysis were carried out in course of thestudy. At the first level the total damage volume as a function ofthe external load was calculated for both approaches; a similar va-lue of the onset of damage was predicted by the shell (GENOA) andthe TWB-PFA model if the shell model of the blade was equippedwith an internal web, thereby avoiding a collapse of the cross sec-tions near the root and ensuring the validity of one of the keyassumptions of the TWB model used. The damage-onset valuewas predicted with an accuracy of about 5.5%, indicating a promis-ing initial evidence for the suitability of the reduced-order model.While the damage after the onset progresses at a rate about 25%higher in the TWB-PFA model (1.5% of damage per 1000 N of loadvs. 1.2%/1000 N for the shell model), this difference can be substan-tially reduced by a small adjustment of the strength values usedwith the failure criteria for the TWB-PFA model, as discussedbelow.

In a second level of analysis the spatial distribution of the dam-age was studied for different load levels. Both in the TWB-PFA andthe GENOA model damage is predicted in layers 4 and 7 only, andthe topology of the damage maps is strikingly similar for bothmodels. In the case of layer 4, where most of the damage occurs,the shape and the contour-wise location of the damage zone pre-dicted by the reduced-order model is almost identical to the pre-dictions of the more complex model, although the total length ofthe failure zone is somewhat larger for all load cases, which canbe traced back to an almost constant offset between the two pre-dictions, since the rate of growth of the length of the failure region

D. Cárdenas et al. / Composite Structures 95 (2013) 53–62 61

is almost identical for both models. In the case of layer 7, wherevery little damage occurs and no damage is observed until fairlyhigh load values (about 1800 N of tip load for both models) theshape of the damage zone is well predicted by the TWB-PFA modelexcept for a needle-like damage feature which can be traced backto the presence of the web in the shell model.

In a third level of analysis the layer-resolved cumulative damagevolume as a function of the spanwise coordinate has been calcu-lated for both models as a function of the applied tip load and thepredictions have been compared. Consistently with the findings de-scribed above, the cumulative damage volume found from theTWB-PFA model is somewhat higher than in the GENOA model. Apossible explanation for this behavior lies with the locally stifferstructure of the reduced-order model given its much smaller num-ber of degrees of freedom (DOFs) and the correspondingly smallerability of distributing the strain energy among the DOFs. Another is-sue refers to the appropriateness of the failure criteria, which mayhave to be modified for an effective reduced-order model. In orderto explore this line of thought the strength values used in the TWB-PFA model were increased by a constant factor and the simulationswere re-run for layer 4 and all load cases, matching the total cumu-lative damage volume. It was possible to quite accurately reproducethe cumulative damage curves of the GENOA model by a relativelysmall adjustment in materials strength, with the adjustment factorranging from 1.065 (lowest load case) to 1.14 (highest load case); aconstant adjustment factor of 1.1 (10% higher) still yielded accept-able results. The increase in the value of the adjustment factor from1.065 to 1.14 can be explained by the fact that the damage progres-sively weakens the contour of the affected sections, thereby allow-ing for a progressive reduction of the section stiffness due to thedecrease in cross-sectional area. As discussed above, this behavioris correctly modeled by the shell but not the TWB model.

In conclusion, the TWB-PFA model presented in this work hasshown its capability of correctly predicting both the onset andthe propagation of damage in the composite blade studied forthe case of static loads, providing reliable information on the loca-tion of the damage zone (resolved both by layer and the contour-and spanwise location of the damage). A small overprediction ofthe damage compared to the more detailed shell model can be cor-rected by a small upward adjustment of the (equivalent) strengthvalues to be used in the model. In all cases, the reduced-order mod-el proved to be more conservative that the shell model, so it shouldbe possible to safely use it for design purposes, especially if a seriesof iteration with quick turnaround times is desired. While the finaldesign of a rotor blade will still remain confined to the domain ofhigher resolution models such as shell and volume element mod-els, the TWB-PFA approach is a promising tool for pre-design andaeroelastic design simulation, where the use of high-resolutionmodels would be prohibitive.

Acknowledgements

Financial support from Tecnológico de Monterrey, includingfunds obtained through the Research Chair for Wind Energy (Con-tract No. CAT158), are gratefully acknowledged. One of the authors(PM) acknowledges support from the New York State Energy Re-search and Development Authority (NYSERDA) Project No. 18812,and the National Science Foundation, Grant No. CMMI-1031036,for partially funding this research project. Finally, the support ofAlpha Star Corporation, provided in term of software and training,is greatly appreciated.

References

[1] Hau E. Wind turbines: fundamentals, technologies, application,economics. Berlin: Springer-Verlag; 2006.

[2] Sobieszczanski-Sobieski J, Haftka, R. Multidisciplinary aerospace designoptimization: Survey of recent developments. In: American institute ofaeronautics and astronautics, proceedings of the 34th aerospace sciencesmeeting and exhibit, Reno; 15–16 January, 1996. p. 1–33.

[3] Li L. Structural design of composite rotor blades with consideration ofmanufacturability, durability and manufacturing uncertainties. Doctoralthesis in school of aerospace engineering. Georgia Institute of Technology;2008.

[4] Shahverdi H, Nobari AS, Behbahani-Nejad M, Haddadpour H. Aeroelasticanalysis of helicopter rotor blade in hover using an efficient reduced-orderaerodynamic model. J Fluids Struct 2009;25(8):1243–57.

[5] Friedmann P, Glaz B, Palacios R. A moderate deflection composite helicopterrotor blade model with an improved cross-sectional analysis. Int J Solids Struct2009;46(10):2186–200.

[6] Murugan S, Chowdhury R, Adhikari S, Friswell MI. Helicopter aeroelasticanalysis with spatially uncertain rotor blade properties. Aerosp Sci Technol2011. http://dx.doi.org/10.1016/j.ast.2011.02.004.

[7] Hansen MOL, Sorensen JN, Voutsinas S, Sorensen N, Madsen HA. State of the artin wind turbine aerodynamics and aeroelasticity. Prog Aerosp Sci2006;42(4):285–330.

[8] Jonkman JM, Sclavounos PD. Development of fully coupled aeroelastic andhydrodynamic models for offshore wind turbines. In: ASME, proceedings of thewind energy symposium, Reno; 10–12 January, 2006.

[9] Ahlstrom A. Aeroelastic simulation of wind turbine dynamics. Doctoral thesisin structural mechanics. KTH, Sweden; 2005.

[10] Buhl ML, Manjock A. A comparison of wind turbine aeroelastic codes used forcertification. In: 44th AIAA aerospace sciences meeting and exhibit, Reno,Nevada; 9–12 January, 2006.

[11] Passon P, KÜhn M, Butterfield D, Jonkman J, Camp T, Larsen TJ. Benchmarkexercise of aero-elastic offshore wind turbine codes. In: Proceedings of EAWEconference, Lyngby, Denmark; 28–31 August, 2007.

[12] Volovoi V, Hodges DH, Cesnik CES, Popescu B. Assessment of beam modellingmethods for rotor blade applications. Math Comput Model 2001;33:1099–112.

[13] Hodges D, Atilgan AR, Cesnik CES, Fulton MV. On a simplified strain energyfunction for geometrically nonlinear behaviour of anisotropic beams. ComposEng 1992;2(5):513–26.

[14] Jonkman JM, Marshall LB. FAST user’s guide. Technical, report NREL/EL-500-38230; August 2005.

[15] Noda M, Flay RGJ. A simulation model for wind turbine blade fatigue loads. JWind Eng Ind Aerodynam 1999;83(1–3):527–40.

[16] White D. New method for dual-axis fatigue testing of large wind turbineblades using resonance excitation and spectral loading. Technical, reportNREL/TP-500-35268; April 2004.

[17] Nijssen RPL. Fatigue life prediction and strength degradation of wind turbinerotor blade composites. Doctoral thesis in design and production of compositestructures. Delft University of Technology; 2006.

[18] Epaarachchia JA, Clausen PD. The development of a fatigue loading spectrumfor small wind turbine blades. J Wind Eng Ind Aerodynam 2006;94(4):207–23.

[19] Marin JC, Barroso A, Parıs F, Canas J. Study of fatigue damage in wind turbineblades. Eng Fail Anal 2009;16(2):656–68.

[20] DimarogonaS AD. Vibration of cracked structures: a state of the art review. EngFract Mech 1996;55(5):831–57.

[21] Gadelra RM. The effect of delamination on the natural frequencies of alaminated composite beam. J Sound Vib 1996;197(3):283–92.

[22] Yan YJ, Chen L, Wu ZY, Yam LH. Development in vibration-based structuraldamage detection technique. Mech Syst Signal Process 2007;21(5):2198–211.

[23] Zou Y, Tong L, Steven GP. Vibration-based model-dependent damage(delamination) identification and health monitoring for compositestructures. J Sound Vib 2000;230(2):357–78.

[24] Dilena M, Morassi A. Identification of crack location in vibrating beams fromchanges in node positions. J Sound Vib 2002;255(5):915–30.

[25] Nahvi H, Jabbari M. Crack detection in beams using experimental modal dataand finite element model. Int J Mech Sci 2005;2005(47):1477–97.

[26] Alvandi A, Cremona C. Assessment of vibration-based damage identificationtechniques. J Sound Vib 2006;47(10):179–202.

[27] Hu H, Wang BT, Lee CH, Su JS. Damage detection of surface cracks in compositelaminates using modal analysis and strain energy method. Compos Struct2006;74(4):399–405.

[28] Karthikeyan M, Tiwari R, Talukdar S. Crack localization and sizing in a beambased on the free and forced response measurements. Mech Syst SignalProcess 2007;21(3):1362–85.

[29] Shih HW, Thambiratnam DP, Chan THT. Vibration based structural damagedetection in flexural members using multi-criteria approach. J Sound Vib2009;323(3–5):645–61.

[32] Pawar PM, Ganguli R. Modeling progressive damage accumulation in thinwalled composite beams for rotor blade applications. Compos Sci Technol2006;66(13):2337–49.

[33] Pawar PM, Ganguli R. On the effect of progressive damage on compositehelicopter rotor system behavior. Compos Struct 2007;78(3):410–23.

[34] Adolfsson E, Gudmundson P. Matrix crack induced stiffness reduction in [(hm/90n/+hp/�hq)s]m composite laminates. Compos Eng 1995;5(1):107–23.

[35] Gudmundson P, Zang W. An analytic model for thermoelastic properties ofcomposite laminates containing transverse matrix cracks. Int J Solids Struct1993;30(23):3211–31.

[36] Shahid IS, Chang FK. An accumulative damage model for tensile and shearfailure of laminated composite plates. J Compos Mater 1995;29:926–81.

62 D. Cárdenas et al. / Composite Structures 95 (2013) 53–62

[37] Chandra R, Chopra I. Structural response of composite beams and blades withelastic couplings. Compos Eng 1992;2(5–6):347–74.

[38] Song O, Librescu L, Jeong NH. Static response of thin-walled composite I-beamsloaded at their free-end cross-section: analytical solution. Compos Struct2001;52(1):55–65.

[39] Librescu L, Na S. Active vibration control of doubly tapered thin-walled beamsusing piezoelectric actuation. Thin-Walled Struct 2001;39(1):65–82.

[40] Qin Z, Librescu L. On a shear-deformable theory of anisotropic thin-walledbeams: further contribution and validations. Compos Struct2002;56(1):345–58.

[41] Librescu L, Song O. Thin-walled composite beams: theory andapplication. Springer; 2006.

[42] Vo T, Lee J. Flexural torsional behavior of thin walled closed-section compositebox beams. Eng Struct 2007;29(8):1774–82.

[43] Lee J, Vo T. Flexural–torsional behavior of thin walled composite box beamsusing shear-deformable beam theory. Eng Struct 2008;30(7):1958–68.

[44] Vo T, Lee J. Free vibration of thin-walled composite box beams. Compos Struct2008;90(2):11–20.

[45] Cardenas D, Elizalde H, Marzocca P, Probst O. Numerical validation of a finiteelement thin-walled beam model of a composite wind turbine blade. WindEnergy 2011. http://dx.doi.org/10.1002/we.462.

[46] Chamis CC, Minnetyan L. Defect/damage tolerance of pressurized fibercomposite shells. Compos Struct 2001;51(2):159–68.

[47] http://www.ascgenoa.com/main/.[48] Murthy PLN, Chamis CC. ICAN: integrated composite analyzer user’s and

programmer’s manual. Technical paper NASA TP-2515. National Aeronauticsand Space Administration, Washington, DC; 1985.

[49] Tuttle M. Structural analysis of polymeric composite materials. NewYork: Marcel Dekke; 2004.

[50] Lindenburg C, Winkel GD. State of the art of rotor blade buckling tools.Technical report ECN-C-05-054. Energy Center of the Netherlands; 2005.