February 2012 

NASA/TM–2012-217336 

Simplified Models for the Study of Postbuckled 

Hat-Stiffened Composite Panels 

Riccardo Vescovini 

Politecnico di Milano, Milano, Italy 

Carlos G. Dávila 

Langley Research Center, Hampton, Virginia 

Chiara Bisagni 

Politecnico di Milano, Milano, Italy 
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February 2012 

NASA/TM–2012-217336 

Simplified Models for the Study of Postbuckled 

Hat-Stiffened Composite Panels 

Riccardo Vescovini 

Politecnico di Milano, Milano, Italy 

Carlos G. Dávila 

Langley Research Center, Hampton, Virginia 

Chiara Bisagni 

Politecnico di Milano, Milano, Italy 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 
1 

Simplified Models for the Study of Postbuckled 

Hat-Stiffened Composite Panels 

R. Vescovini1, C.G. Dávila2, C. Bisagni1 

1

Politecnico di Milano, Department of Aerospace Engineering, Milano, Italy 

2

NASA Langley Research Center, Hampton, Virginia, USA 

Abstract 

The postbuckling response and failure of multistringer stiffened panels is 

analyzed using models with three levels of approximation. The first 

model uses a relatively coarse mesh to capture the global postbuckling 

response of a five-stringer panel. The second model can predict the 

nonlinear response as well as the debonding and crippling failure 

mechanisms in a single stringer compression specimen (SSCS). The third 

model consists of a simplified version of the SSCS that is designed to 

minimize the computational effort. The simplified model is well-suited to 

perform sensitivity analyses for studying the phenomena that lead to 

structural collapse. In particular, the simplified model is used to obtain a 

deeper understanding of the role played by geometric and material 

modeling parameters such as mesh size, interlaminar strength, fracture 

toughness, and fracture mode mixity. Finally, a global/local damage 

analysis method is proposed in which a detailed local model is used to 

scan the global model to identify the locations that are most critical for 

damage tolerance. 

I. Introduction 

Composite stiffened panels are commonly used as components for the construction of aerospace 

structures. In most cases, stiffened panels can operate far beyond the buckling load and their collapse is 

observed deep into the postbuckling range. For this reason, the ability to predict the collapse load and the 

mechanisms involved in the failure is a crucial point for the damage tolerance assessment and the rational 

design of these structures. 

Experiments conducted on stiffened panels1,2

 and closed-box configurations3,4 represent a useful mean to 

gather insight into the complex nonlinear response governing the collapse. However, the high 

manufacturing and testing costs associated with multistringer panels strongly limits the possibility of 
2 

performing extensive experimental explorations. For this reason, the availability of analytical and 

numerical tools is fundamental to further investigate the collapse response of postbuckled composite 

panels. In recent years, several strategies have been developed for the study of the damage onset and 

propagation, including skin/stringer delaminations. An analytical approach was proposed by Taki and 

Kitagawa5

 to predict the failure load of composite stiffened panels under shear load. The methodology is 

based on a strip analysis to determine the loading conditions that would cause a disbond of the stringer 

from the skin. 

More recently, Cosentino and Weaver6

 proposed a nonlinear approach to predict crack initiation in 

composite stiffened panels. The approach is based on the use of two linked analytical models. The first 

model adopts a nonlinear formulation that provides the stress and displacement fields throughout the 

domain. The second model predicts crack initiation using a closed-form linear elastic fracture mechanics 

solution. Despite their efficiency, analytical and semi-analytical methods can be developed only by 

introducing a number of simplifying assumptions. In general, the failure of postbuckled panels involves 

several nonlinear mechanisms, including geometric as well as material nonlinearities that cannot be 

captured properly by analytical models. Finally, the inadequacy of analytical models can be exacerbated 

by the existence of different failure mechanisms, including interlaminar and intralaminar damage 

phenomena. 

Alternatively, the finite element method can be used to predict arbitrary complex postbuckling responses, 

and different analysis tools such as progressive failure methods7,8, continuum damage models9,10, virtual 

crack closure technique11, cohesive elements12,13 and X-FEM14 are available to study different aspects of 

the failure process. However, the computational requirements associated with failure analysis often render 

these methods unsuited to study even moderately large structural components. For this reason, different 

strategies have been proposed to reduce the computational cost. 

Orifici et al.15 developed a computationally efficient two-step global/local approach to identify 

interlaminar damage initiation. The approach adopts a strength-based criterion, which is applied to a 

model of the skin–stringer interface at a ply-level mesh refinement. A global/local approach has been 

proposed by Bertolini et al.16 to assess the response of stiffened panels under compression and shear. In 

this case, the VCCT technique is used to determine the energy release rates and the Benzeggagh–Kenane 

mixed-mode fracture criterion17 is applied. 

The goal of the present work is to develop simplified and computationally efficient models to predict the 

response and collapse of a stiffened panel and to perform sensitivity studies to gain deeper understanding 

of the phenomena that lead to local and global collapse and to optimize the finite element models used in 

the calculations. The optimized models at several scales of complexity are also combined in a global/local 

strategy for the collapse analysis of a multistringer panel. 

II. Modeling hat-stiffened panels with three levels of complexity 

In a previous numerical/experimental investigation18, the authors proposed the use of a Single Stringer 

Compression Specimen (SSCS) to study the response and the failure of a multistringer panel loaded in 

compression. The configuration of the SSCS is shown in Figure 1(a), and a picture of a compression test 

is shown in Figure 1(b). 
3 

(a) (b) 

Figure 1. Single Stringer Compression Specimen (SSCS): 

(a) configuration and dimensions, (b) compression test. 

The SSCS represents an intermediate level of complexity between coupon-level specimens and structural 

components while exhibiting most of the challenges that characterize the assessment of the damage 

behavior of a more complex configuration such as a multistringer panel. Advantages exist from the 

experimental point of view, due to low manufacturing and testing costs, and from the numerical point of 

view, due to the possibility of realizing refined meshes. The authors performed Abaqus19 finite element 

analyses with a user defined continuum damage model9,10 for the study of intralaminar damage 

mechanisms, and cohesive elements7,13 to account for delaminations between the skin and the stringer. 

Despite the small size of the SSCS and the efforts devoted to maximize the computational efficiency by 

combining quasi-static and implicit steps, the numerical study of the failure of a SSCS still requires 

several days of analysis. It is clear that such lengthy analyses render the use of this approach for 

sensitivity analyses or for the analysis of multistringer panels intractable, and that an alternative 

methodology is necessary. 

The present study presents the results of an investigation into the collapse response of a multistringer 

composite panel using models characterized by the three levels of structural complexity illustrated in 

Figure 2. 

In order of decreasing complexity, the models are the complete multistringer panel, the single stringer 

compression specimen, and the simplified model, shown in Figures 2(a)-(c). The analyses in the 

following sections are conducted using Abaqus/Std19 with an implicit dynamic solution procedure. The 

models use four-noded S4R shell elements. The SSCS and simplified models also use cohesive elements 

between the skin and the stringer flange to represent skin/stringer separation. 
4 

 (a) (b) (c) 

Figure 2. Structural models: (a) multistringer panel, (b) SSCS, (c) simplified model. 

A. Multistringer panel 

The multistringer panel illustrated in Figure 2(a) has a length of 720 mm and a width of 680 mm. The 

panel is stiffened longitudinally by five hat-stringers. The height of the stringers is 30 mm, the width of 

the crown top is 15 mm, while the web is at an angle of 25° with the normal to the skin. The stringers are 

co-cured to the skin along two 15 mm-wide flanges. Both the skin and the stringers are made from 

IM7/8552 graphite-epoxy material. The skin consists of 8-ply quasi-isotropic laminate with stacking 

sequence of [45°/90°/-45°/0°]s. The stringers are composed of 7-ply laminate with a symmetric stacking 

sequence of [-45°/0°/45°/0°/45°/0°/-45°]. 

The material elastic and interlaminar properties20 are summarized in Tables 1 and 2, respectively. 

Table 1. Engineering properties – IM7/8552. 

E11 (MPa) E22(MPa) G12(MPa) Ȟ12 Į1 (1/°C) Į2 (1/°C) 

150000 9080 5290 0.32 -5.5E-6 25.8E-6 

Table 2. Interlaminar properties – IM7/8552. 

GIc (N/mm) GIIc (N/mm) ݖݖߪ) MPa) ݖݔߪ) MPa) 

0.277 0.788 50 100 

The postbuckling quasi-static response of this five stringer panel can be obtained relatively quickly as 

long as the material nonlinearities are not taken into account. However, the mesh requirements needed for 

predicting intralaminar and interlaminar damage, together with the rather large dimensions of the panel, 

render the prediction of damage at this scale computationally impractical. 

B. Single Stringer Compression Specimen (SSCS) 

The Single Stringer Compression Specimen (SSCS) considered in this study is illustrated in Figure 2(b). 

The specimen is composed of one stringer, and extends transversally from half bay to half bay, for a total 

width of 150 mm. This structural component represents a small portion of the multistringer panel and was 

designed to have similar response characteristics as the corresponding multistringer panel. Through 

parametric studies, it was found that a SSC specimen length of 240 mm resulted in a stress distribution 

that is similar to that observed in the multistringer panel18. The reduced dimensions of this specimen 
5 

diminish the manufacturing and testing costs and permit the use of sufficiently detailed finite element 

meshes to study the interlaminar and intralaminar damage mechanisms. 

The SSCS model can be used to assess the damage tolerance of the multistringer panel, as the similarity 

of the stress distributions and of the buckled patterns determines analogous failure mechanisms. 

Nevertheless, the analysis of the SSCS model with damage requires several days of computational time18, 

so that it is inadequate to perform the sensitivity studies necessary to fully understand the collapse 

response of the panel or to evaluate the effect of different modeling parameters on the analysis results. 

Consequently, a further level of approximation is introduced in the next section. 

C. Simplified model 

A simplified model developed to reduce the computational requirement for analysis is schematically 

illustrated in Figure 2(c). As is explained in the following section, the model is obtained from an 

examination of the response of the SSCS by removing the stringer from the SSCS and by using symmetry 

along the stringer mid-plane. The smaller dimensions and the reduced number of degrees of freedom 

make the model well-suited to perform sensitivity analyses with a relatively small computational effort. 

The simplified model is also useful to gain knowledge about the effects of the different modeling 

parameters on the analysis results and to define the properties of the SSCS model used in the context of 

the global/local procedure. 

III. Development of a simplified model 

The structural response of the SSC specimen is briefly described to highlight the most important aspects 

leading to the development of a simplified structural model. The panel is loaded with an end shortening 

displacement, as illustrated in Figure 2. The ending tabs of the specimens are simulated by constraining 

the translational degrees of freedom of the corresponding nodes, while the longitudinal edges are free.

In the initial loading phase, corresponding to the prebuckling conditions, the SSCS exhibits a linear 

response. The first buckling is observed at a load of about 10 kN, and mainly regards the instability of the 

skin, which buckles in three half-waves. No buckling deformations are observed on the stringer, as can be 

observed in Figure 3(a). The drop of stiffness in the initial postbuckling range is consequently due to the 

loss of stiffness of the skin only. As the load is increased further, the deflections on the skin increase in 

magnitude and, as the load redistributes, the stringer carries an increasing portion of the applied load. 

After an applied load of approximately 25 kN, nonlinear deformations of the stringer web can be 

observed, as shown in Figure 3(b). The magnitude of the displacements increases progressively with the 

applied load. 

(a) (b) 

Figure 3. Nonlinear deformation of the SSCS (end view): 

(a) at first buckling, (b) after buckling of the stringer web. 
6 

An investigation of the response and failure of the SSCS model indicates that the delamination onset is 

governed by the postbuckling deformation on the portion of skin under the stringer. This deformation 

under the stringer is characterized by three half-waves along the longitudinal direction, as shown in 

Figure 4(a). At a load of approximately 37 kN, a mode transition is observed in which the middle buckle 

divides into two smaller half-waves, as can be seen in Figure 4(b). This postbuckled configuration, which 

is composed of four half-waves, causes an internal stress redistribution that promotes the skin/stringer 

debonding. With a small increase in the load, the most severely loaded cohesive element reaches a fully 

damaged state, which triggers an unstable propagation of the delamination. As Figure 4(c) indicates, the 

delamination initiates from the inner edge of the stringer flange. The delamination is soon followed by a 

large-scale detachment of the skin from the stringer and crippling fracture of the stringer. 

(a) (b) (c) 

Figure 4. Out-of-plane-deflections (view without stringer): 

(a) initial postbuckling at 35 kN, (b) mode jumping at 37 kN, (c) delamination onset at 39 kN. 

Based on these observations of the structural response of the SSCS, a simplified structural model that can 

reproduce the collapse response with as few degrees of freedom as possible was developed. In particular, 

the model captures the correct initial skin buckling deformation mode, the mode change under the 

stringer, and the skin/stringer delamination. 

The computational efficiency of the simplified model is a result of modeling only half of the structure and 

substituting the stringer by means of nodal constraints. In addition, intralaminar damage is not represented 

and the residual thermal strains are neglected. The SSCS model and the associated simplified model are 

shown in Figure 5.

As previously observed, the postbuckling deformation of the SSCS is nearly symmetric along the stringer. 

Using the reference frame in Figure 5(b), symmetry can be imposed by setting to zero the rotations 

around the x-axis and the translation along the transverse y-axis. The imposition of symmetry introduces a 

level of approximation due to the fact that the skin is layered with a flexurally anisotropic lay-up. The 

resulting coupling between bending and twisting causes skew in the buckling half-waves that are 

inconsistent with symmetry conditions. There is also additional error when modeling a SSCS with an 

embedded delamination. In the SSCS tested, Teflon is placed under one of the stringer flanges to create 

an embedded delamination. The simplified model represents embedded delaminations under both flanges. 
7 

(b)

Figure 5. Finite element models and boundary conditions: 

(a) SSC specimen, (b) simplified model without stringer. 

The second observation that is exploited to define a simplified model is that the deformation of the 

stringer does not contribute appreciably to the process of delamination. The mode transition of the portion 

of skin under the stringer is driven by the deflections of the outer skin, and the stringer does not need to 

be modeled explicitly to reproduce the collapse response of the structure. Consequently, in the simplified 

model the stringer is removed and replaced by simply-supported boundary conditions, as illustrated in 

Figure 5(b). The model relies on the assumption that the stringer is stiff enough to furnish an ideal 

constraint to the out-of-plane displacements of the skin. The rotation along the longitudinal axis is free, 

and consequently no bending moments are exchanged between the stringer and the skin. 

As in the SSCS, the simplified model uses cohesive elements to represent the debonding of the stringer 

flange. However, intralaminar damage of the matrix and fiber, which in the SSCS model was captured 

using a continuum damage model, is not considered in the simplified model. Intralaminar damage can be 

neglected because the SSCS analyses indicates that the collapse and crippling of the stringer is caused by 

the detachment of the stringer. By limiting material nonlinearities to the cohesive layer under one flange 

the convergence rate of the load incrementation procedure improves dramatically. 

Finally, an additional decrease in the computational cost of the simplified model is achieved by neglecting 

the residual thermal strains that result from the high-temperature cure cycle used in the manufacturing 

process. 
8 

IV. Comparison between SSCS and simplified model 

The ability of the simplified structural model to reproduce the sequence of events is assessed by 

comparing the results obtained with those from the SSCS model. A mesh size of 1.0 mm is chosen for 

both models. The total number of elements and degrees of freedom of the two models is summarized in 

TTable 3, where it can be observed that the number of degrees of freedom of the simplified model is 

approximately a third of that for the SSCS model. 

Table 3: Number of elements and degrees of freedom of the SSCS and the simplified model. 

 SSCS Simplified model 

Number of elements 84,000 30,520 

D.o.f. 451,848 158,484 

The postbuckling deformations of the SSCS and the simplified model are shown in Figure 6 and the 

ability of the simplified model to predict initial buckling, mode jumping of the skin under the stringer and 

delamination onset is evaluated. Figures 6(a) and 6(b) indicate that the first instability, characterized by 

skin buckling in three half-waves, is properly represented by the simplified model. The most pronounced 

deflections are found along the edges of the skin, and minor deflections are observed on the stringer. 

Smaller half-waves are also visible in the skin under the stringer. When the load is increased, the two 

models present a similar transition of the postbuckling deformation under the stringer, as highlighted in 

Figures 6(c) and 6(d). In particular, both the SSC specimen and the simplified model display the mode 

jumping of the mid-buckle, which determines the redistribution of the internal forces and leads to the 

delamination between the skin and the stringer flange. It should be noted that the stringer is removed in 

Figures 6(c) and 6(e) for visualization purposes only. 

The force–displacement results for the SSCS and the simplified model are compared in Figures 7(a) and 

7(b). To account for the stiffness of the stringer, which is not present in the simplified model, the fraction 

of load carried by the stringer is added ex post as: 

ܲ௧௢௧௔௟ ൌ ܲ௙௘௠ ൅

ܣܧ

ܽ

ܷ (1) 

where ܲ௧௢௧௔௟ is the total load, ܲ௙௘௠is the fraction of load carried by the skin and the stringer flange and 

available from the finite element analysis, ܣܧ is the stringer axial stiffness, ܽ is the panel length and ܷ is 

the applied displacement. The results in Figure 7(a) correspond to a pristine specimen, while the results in 

Figure 7(b) correspond to a specimen with a 20-mm pre-existing delamination between the skin and the 

stringer. In the SSCS model, the pre-existing defect is placed under the middle of one of the stringer 

flanges, as shown in Figure 1(a). Since the simplified model relies on symmetry conditions, it represents a 

specimen with pre-existing defects under both stringer flanges. 
9 

SSCS Simplified model 

(a) (b) 

(c) (d) 

(e) (f) 

Figure 6. Comparison of postbuckling deflections: 

 (a)-(b) buckling, (c)-(d) mode jumping and delamination onset, (e)-(f) delamination after collapse. 

The results of Figure 7 reveal almost identical prebuckling and postbuckling stiffnesses. Indeed, the 

postbuckling response of the SSCS specimen is governed by the skin buckling, and no significant errors 

are introduced by removing the stringer from the model. 

The collapse load obtained with the simplified model is underestimated by 8% and 15% in the case of 

nominally pristine and pre-damaged panels, respectively. 

The results illustrate the ability of the simplified model to reproduce the structural response and failure of 

the SSC specimen and they validate the use of symmetry boundary conditions and the representation of 

the stringer as a simple support boundary condition. The advantage of the simplified model becomes clear 

when considering that it requires less than 3 hours of computational time on a quad-core workstation, 

compared to 7 days for the SSCS model. 
10 

(a) (b) 

Figure 7. Comparison of force-displacement curves: 

(a) nominally pristine panel, (b) pre-damaged panel. 

V. Simplified model for parameter evaluation 

The simplified model is an effective tool for performing sensitivity studies to investigate the effect of the 

various modeling parameters. These studies provide deeper insight into the structural response of 

postbuckled hat-stiffened panels, and furnish information that is useful for optimizing the models. In the 

following sections, the effects of mesh size, interlaminar strength, fracture toughness, and mode mixity 

are examined. 

A. Determination of mesh size requirements 

The selection of the optimal mesh size is one of the most important factors in determining the 

computational requirements of an analysis. Cohesive elements often dictate a model’s mesh requirements 

since the cohesive elements must be small enough to capture the gradients in the process zone. Therefore, 

cohesive elements are normally much smaller than the elements needed to represent structural response. 

The length of process zone, ݈݌, can be estimated as21: 

ߛ ൌ௣ ݈

௖ܩܧ

௭௭ߪ

ଶ (2) 

where ߛ ൎ 0.884 is a nondimensional parameter, ܧ is approximately equal to the Young’s modulus in the 

direction transverse to the fibers, ܩܿ is the mode I fracture toughness and ݖݖߪ is the interlaminar peel 

strength. For the interface properties shown in Table 2, the process zone length is ݈௣ ൎ 0.85 mm. 

Consequently, it can be expected that a mesh size smaller than 0.85 mm is necessary to represent 

delamination. This assumption is evaluated by conducting the following analyses.

The results of analyses obtained using mesh sizes equal to 0.5 mm and 1.0 mm are presented in Figures 

8(a) and 8(b). The finer mesh of Figure 8(a) is able to predict the load drop due to initiation and 

propagation of a delamination between the skin and the stringer flange. The final point of the simulation 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

Simplified model

SSCS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

Simplified model

SSCS
11 

corresponds to a delamination that has propagated across the entire width of the stringer flange. The 

predicted collapse load is 37 kN. 

The analysis of a model using a 1.0 mm element size terminates due to convergence issues at an applied 

load of about 38 kN. The predicted force-displacement result shown in Figure 8(b) does not display a load 

drop due to delamination. Consequently, a 1.0 mm mesh is too coarse to predict the delamination. 

(a) (b) 

Figure 8. Mesh size: (a) 0.5 x 0.5 mm, (b) 1.0 x 1.0 mm. 

B. Interlaminar strength 

Equation 2 shows that the length of the process zone, and consequently the maximum size of the element, 

can be increased by decreasing the interlaminar strength. Clearly, the value cannot be arbitrarily modified 

because it represents a mechanical property of the material. However, it is reasonable to reduce the 

interlaminar strength if the collapse response of the structure is not significantly affected by this value. 

The investigation of this aspect is the object of the next section. 

The study of the panel response for different values of interlaminar strength has two purposes. The first 

goal is to verify the possibility of enlarging the process zone with a reduction of ݖݖߪ without affecting the 

predicted collapse load. If this is the case, the maximum size of the element can be increased, and a 

coarser mesh can be used. The second goal is to evaluate how much the collapse of the panel is influenced 

by the interlaminar strength. 

The force-displacement curve of Figure 9 reports the comparison between two different configurations. 

The first configuration is the nominal configuration and is characterized by a mesh size of 0.5 mm and 

interlaminar strength ݖݖߪ equal to 50 MPa. The second configuration considers a reduced value of 

interlaminar strength equal to 25 MPa. According to Eq. (2), a reduction of the interlaminar strength 

determines an increase of the length of the process zone. For this reason, the second configuration is 

realized with a coarser mesh size of 1.0 mm. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

σzz=50 MPa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

σzz=50 MPa
12 

Figure 9. Force-displacement curve panels with different interlaminar strengths and mesh sizes. 

It can be observed from Figure 9 that the collapse load of the nominal configuration is 36.5 kN, whereas 

the one obtained with the coarser mesh is 35.5 kN. The drawback associated to the use of the 1.0 mm

mesh is that the collapse load is underestimated. However, the percent difference between the two results 

is below 3%. It is then retained that a mesh size of 1.0 mm with an interlaminar strength equal to 25 MPa

represents a good compromise in terms of accuracy of the predicted collapse load and required analysis 

time. For this reason, these values will be used in the next sections to perform further parametric studies. 

C. Fracture toughness and R-curve effect 

The opening Mode I interlaminar fracture toughness, GIc, measured with standards such as the ASTMD5528 represent the GIc for crack initiation. Most fracture processes exhibits a resistance-type behavior 

(R-curve), which consists of an initial monotonic increase in GIc, followed by propagation at a stabilized 

value of GIc. In double cantilever beam specimens composed of unidirectional material, the increase in 

fracture toughness with crack length is mostly attributable to fiber bridging. An even larger R-curve effect 

is obtained when measuring the fracture toughness at interfaces between different fiber orientations, such 

as is the case between the skin and the stiffener. In these laminates, the increase in fracture toughness is 

likely due to the formation of somewhat tortuous delamination paths, which could include jumping of the 

delaminations to different interfaces in the laminate. For example, Shön et al.22 determined that the 

fracture toughness between 45° plies increases from about 0.22 N/mm to about 0.9 N/mm after 15 mm of 

propagation. Pereira and de Morais23 measured the fracture toughness between a 0°-ply and plies oriented 

at 0°, 67.5° and 90°, and found that for 0°/0° interfaces GIc increases from 0.3 N/mm to 0.65 N/mm after 

approximately 35 mm of propagation, while the toughness of other interfaces tested exhibits increases 

from 0.35 N/mm to 1.2 N/mm in 30 mm of propagation. 

Given the importance of GIc in the analysis of the SSCS and the uncertainty in the R-curve effect 

associated with skin/stringer debonding, analyses were conducted to examine the effect of an R-curve on 

the predicted collapse load. The analyses were performed on nominally pristine as well as specimens with 

an implanted 20-mm delamination between the skin and the stringer. For each case, three analyses were 

conducted. The first set of analyses considers the nominal, i.e., initiation value of GIc=0.277 N/mm, which 

was shown in Table 2. The second set uses GIc=1.00 N/mm, which approximates the value for steady state 

propagation described in the previous paragraph. The third set considers an R-curve that initiates at 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

Mesh 0.5 mm, σzz=50 MPa

Mesh 1.0 mm, σzz=25 MPa
13 

GIc=0.277 N/mm and increases to 1.00 N/mm in 15 mm. The R-curve simulation is conducted by means of 

a superposition of two cohesive laws21 in which the first law represents the crack initiation value of 0.277 

N/mm, which is denoted by ܩଵ, and a second law that represents the crack bridging effects and accounts 

for the remaining 0.723 N/mm of fracture toughness, which is denoted by ܩଶ. The superposition approach 

proposed in Ref.21 provides a means to determine the strengths σ1 and σ2 for the two cohesive laws : 

ߪଵ ൌ ߪ൤ ௖ͳ െ

ʹ

͵ ൬ͳ െ

ଵܩ

ଶܩ ൅ ଵܩ

൰ ݈௖

݈௦௦

൨

ߪଶ ൌ ߪ ௖െ ߪଵ

(3) 

For a steady-state length of propagation lss=15 mm, lc=2.4 mm, and σc=25 MPa, Eq. (3) gives σ1=23 MPa, 

σ2=2 MPa. 

The force-displacement curves predicted by the six analyses are shown in Figures 10(a) and 10(b). The 

response of a nominally pristine panel is shown in Figure 10(a), whereas the results for the panel with a 

pre-existing delamination is shown in Figure 10(b). The three models are realized with a 1.0 mm mesh 

and interlaminar strength of 25 MPa. 

(a) (b)

Figure 10. Effect of fracture toughness on pristine panel and panel with initial delamination: 

(a) nominally pristine panel, (b) panel with initial delamination. 

The results indicate that increasing the value of GIc from its initiation value of 0.277 N/mm to a steadystate value of 1.00 N/mm causes an increase in the collapse load of 14% in the case of the pristine panel, 

and 7.5% for the panel with a pre-existing delamination. These relatively modest increases indicate that 

the collapse of the panel is governed more by the postbuckling deformation mode than by the fracture 

toughness. The results also indicate that the incorporation of an R-curve effect by cohesive superposition 

has a negligible effect on the collapse load of both panels when compared to the predictions made with 

the initiation value of the fracture toughness. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

GIc = 0.277 N/mm

GIc = 1.00 N/mm

R curve: 0.277 to 1.00 N/mm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

GIc = 0.277 N/mm

GIc = 1.00 N/mm

R curve: 0.277 to 1.00 N/mm
14 

D. Summary of the sensitivity analyses 

The sensitivity studies performed with the simplified model provide a deeper understanding into the 

response and collapse of postbuckled hat-stiffened panels. These analyses also help to evaluate 

improvements in the model for greater computational efficiency. It was found that a mesh size of 1.0 mm

can be used if interlaminar strength is reduced from the nominal value of 50 MPa to 25 MPa. Indeed, a 

reduction of the interlaminar strength does not affect the collapse load, which is reduced by 3%. It was 

also found that the collapse is geometrically driven and, consequently, that the interlaminar strength and 

the fracture toughness have a small influence on the collapse load. 

VI. Simplified model for structural design 

In a design environment, computational efficiency and modeling simplicity are paramount. The present 

simplified model is useful for design purposes because it requires a short analysis time and because it is 

parametric, i.e., all of its dimensions, mesh density, material parameters and loading conditions are 

defined as variable parameters in the input, so the modeling effort required for analyzing multiple 

configurations is minimal. The ability of the simplified model to provide useful design information with 

modest modeling efforts and computational cost is illustrated in the following two studies. Firstly, the 

model is used to assess the effect of different sizes of pre-existing defects on the collapse load of the 

structure. Secondly, the model is used to evaluate a stringer design concept for improved strength. 

A. Evaluation of effects of bondline defects 

The ability of stiffened composite structures to operate deep into postbuckling is limited by the potential 

of presence of defects that could cause premature detachment of the stringer. In the present sudy, the 

effect of the size of a pre-existing bondline defect between the stringer and skin is investigated. These 

defects are introduced in the model at the center of the specimen, under the stringer flange. Defect sizes 

varying between 0 and 60 mm are considered. The results are summarized in Figure 11. The mesh size is 

fixed to 1.0 mm, the interlaminar strength is 25 MPa, and the fracture toughness in mode I is 0.277 N/mm. 

The defect is modeled with a layer of cohesive elements with reduced properties consisting of an 

interlaminar strength of 2.5 MPa and a mode I fracture toughness of 0.277e-2 N/mm. These properties 

represent the extremely low adhesion of the Teflon film and the presence of the cohesive elements is also 

useful to prevent the interpenetration of the skin and stringer. 

The force-displacement curves in Figure 11(a) represent the response of the six configurations taken into 

account. It can be observed that the initial axial stiffnesses for all configurations are identical since the 

stiffness is not influenced by the dimension of a pre-existing delamination. On the other hand, the 

collapse load, which is identified by the sudden drop of load, is reduced with increasing size of the preexisting damage. The results shown in Figure 11(b) indicate that the collapse load decreases with a mostly 

linear behavior in the range of inserts sizes between 0 and 40 mm. The collapse load of the panel with the 

40 mm insert is 12.8 kN, which is 55% of the load corresponding to the pristine panel. For pre-existing 

defects larger than 40 mm, the reduction in collapse load as a function of pre-existing damage is less 

significant because the skin buckles early and the stringer carries a greater portion of the load. 
15 

(a) (b) 

Figure 11. Effect of defect size: (a) force-displacement curve, (b) collapse load versus defect size. 

The panels with 10 mm and 40 mm insert sizes are illustrated in Figures 12(a) and 12(d), respectively. The 

contour of the cohesive elements damage variable is shown for the panel with a 10 mm insert size in 

Figures 12(b) and 12(c), and for the panel with 40 mm insert size in Figures 12(e) and 12(f). The results 

are reported at the maximum load and after the maximum load for both configurations. 

Panel with 10 mm insert Panel with 40 mm insert

 

(a) (b) (c) (d) (e) (f)

 Figure 12. Panels with pre-existing bondline defect: 

 (a) and (d) pre-damaged area, (b) and (e) contour of cohesive element damage variable at 

maximum load, (c) and (f) contour of cohesive element damage variable after the maximum load. 

The portion of structure on the left of the cohesive element layer is the skin under the stringer. The part on 

the right is the skin of the panel. It can be observed that different insert sizes determine different failure 

mechanisms. For the panel with a 10-mm damage of Figures 12(a) and 12(b), the separation between the 

skin and the stringer starts from the inside of the stringer to the outside, while for the panel with 40 mm

the skin/stringer separation on both sides of the stringer flange since the beginning. 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

5

10

15

20

25

30

35

40

45

Displacement [mm]

Load [kN]

a=40 mm

a=60 mm

a=30 mm

a=20 mm

no defect

a=10 mm

0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

45

Defect size, a [mm]

Collapse load [kN]
16 

B. Virtual testing of a redesigned stringer 

The use of the simplified model to perform design evaluations is illustrated in the following example. A 

concept for improving the panel collapse load can be extracted from an observation of the effect of the 

postbuckling deformation of the skin inside the stringer on the collapse mode. Consequently, it could be 

expected that the collapse load would be increased if the mode jumping of the skin under the stringer 

could be delayed. 

The transition of the postbuckled deformed shape from one configuration to another is a complex 

nonlinear phenomenon. Although no closed-form solutions or simple design guidelines exist to quickly 

establish how it is possible to avoid or delay the mode jumping of a portion of structure, it can be 

expected that the mode transition of the skin inside the stringer would be delayed by narrowing the width 

of the stringer. 

A design is proposed in which the width of the skin under the stringer is reduced from 42 mm to 26 mm

by translating the stringer flange closer to the longitudinal line of symmetry of the panel, as shown in 

Figures 13(a) and 13(b). The total width of the panel and the amount of stringer material area are kept 

constant so that a comparison could be performed between equal-weight configurations. In particular, the 

height of the stringer is increased from 30 mm to 31.5 mm, and, to maintain a constant cross-section area, 

the web angle measured from the normal to the skin is reduced from 25° to 7.5°. 

(a) (b) 

Figure 13. Panel configuration: (a) reference, (b) improved. 

A comparison of the force-displacement results predicted for the two configurations is shown in Figure 

14. The first portion of the curve shows that the prebuckling stiffness of both configurations is identical. 

Indeed, the two configurations possess the same cross-sectional area. On the other hand, the postbuckling 

response of the improved configuration presents a smaller stiffness compared to the reference 

configuration. This response is explained observing that the improved configuration is characterized by a 

wider portion of external skin, which is also the portion of structure involved in the first buckling of the 

panel and responsible for the loss of stiffness. 
17 

Figure 14. Force–displacement curves for reference and improved configurations. 

It is interesting to observe that the collapse load of the improved configuration is 52.6 kN, or 32% higher 

than the reference configuration. The applied displacement at collapse is 1.2 mm. 

The skin/stringer delamination of the improved design is illustrated in Figure 15, where the contour of the 

cohesive elements damage variable is shown. 

(a) (b)

Figure 15. Delamination on the improved configuration: (a) onset, (b) at maximum load. 

Compared to the nominal design, the improved design is characterized by a different mechanism of 

delamination onset and growth. In particular, the delamination starts from the external part, as shown in 

Figure 15(a), and propagates to the internal part of the panel, as seen in Figure 15(b). This response is 

explained by the fact that no mode jumping is observed, as the narrowness of the stringer in the new 

design delays the mode jumping to load levels above the collapse load. 

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

10

20

30

40

50

60

Displacement [mm]

Load [kN]

Reference configuration

Improved configuration
18 

VII. Global/local approach 

As was previously discussed, the SSC specimen is developed such that its postbuckling response, failure 

characteristics, and damage tolerance are similar to those of a corresponding multistringer panel subjected 

to compression loads. However, when complex loading or structural configurations must be considered, 

the assumption of periodicity becomes invalid and, consequently, the evaluation of the response and 

failure must be conducted on the full multistringer panel rather than on a smaller specimen. 

A procedure is hereby proposed for assessing the damage tolerance of a postbuckled multistringer panel. 

In particular, the procedure provides a computationally efficient method to identify the critical strength 

and the critical areas of a multistringer panel. 

A. Description of the approach 

The analysis strategy consists of two main models: a global and a local model. First, the analysis of the 

entire panel is performed using a model with a relatively coarse mesh. This global model is composed 

exclusively of shell elements and is therefore unable to capture the onset and the propagation of 

delaminations. In addition, the material response is assumed to be linear elastic. Therefore, no 

convergence difficulties occur during the quasi-static load incrementation. The goal of the global analysis 

step is to determine the displacement field of the entire multistringer panel, which is used in the second 

step as a boundary condition for the local models. 

In the second step, local analyses are performed using an SSCS-type model. The local models can be any 

small portion of the structure that the designer wants to analyze in detail. The local models are 

characterized by a high level of refinement. They are realized with shell elements, while cohesive 

elements are introduced between the skin and the stringer. The local models are loaded along the 

boundaries in displacement control using the results obtained from the global analysis. An interpolated 

displacement field based on the incremental solutions of the global model is generated automatically by 

the Abaqus Submodel19 method. The local analyses are performed with an implicit dynamic step, which is 

an effective procedure to avoid the convergence issues related to the potential onset of delaminations. 

Since the collapse of the panel is sudden and governed by the unstable growth of a delamination, the 

global and local analyses can be uncoupled until the onset of collapse. If the evolution of damage were to 

change appreciably the stiffness of the local model, then an iterative procedure to update the global model 

with the results obtained from the local analysis would be necessary. 

B. Global and local models 

The multistringer panel considered for the present study is shown in Figure 16. The panel has dimensions 

of 680 x 720 mm and it is longitudinally stiffened by five hat stringers. The area investigated using 

detailed local analyses consists of the central stringer. The local zone extends along the transverse 

direction to include half portion of the bay. The two external portions close to the loaded boundaries are 

excluded to avoid local effects. The resulting area of interest has a length 580 mm and a width equal to 

152 mm. For efficiency, the local analysis is performed by dividing the area of interest into five smaller 

local models. In particular, the local models were selected to be 116-mm long, as illustrated in Figure 

16(a) and identified by the numbers 1 to 5. In addition, the four overlapping models numbered 6-9 in 

Figure 16(b) were used to avoid potential boundary effects in the local models in cases where 

delaminations were to initiate near the edges of the local models. The overlap models are offset by 58 mm

in the longitudinal direction with respect to models 1 to 5. 
19 

(a) (b) 

Figure 16. Local models and panel geometry: (a) local models 1 to 5, (b) overlap local models 6 to 9. 

C. Results 

The global/local procedure is conducted using a global model composed of shell elements with a typical 

mesh size of 5 x 5 mm. The total number of degrees of freedom is 171,390. Based on the analysis 

presented in the previous sections, the local model uses a typical mesh size of 1.0 x 1.0 mm. The 

corresponding number of degrees of freedom for each local model is 185,328.

The results of the global/local procedure are summarized in Figures 17 and 18. The out-of-plane 

deflection contours on the skin are shown on the global model for an applied end displacement of 2.50 

mm, which corresponds to an applied strain of 0.35%. The hat-stringers are removed for visualization of 

the response in the portions of skin under the stringers. The results of the local models are reported in 

terms of the damage state of the cohesive elements at the corresponding maximum load. The critical areas 

are the ones exhibiting fully damaged cohesive elements, and the corresponding longitudinal positions on 

the global panel are identified by the dotted lines. The applied displacements at the failure of the local 

models are reported on the right columns of the figures. 

The results indicate that delaminations can initiate as a consequence of two different postbuckling 

deformations modes. The first and most common delamination mode initiates on the inside edge of the 

stiffener flange and is induced by the splitting of a half-wave of the skin under the stringer. It can be 

observed in Figure 17, for instance, that the buckles for model 1 in the internal portion of the skin tend to 

pull away the skin from the stringer, and therefore represent a failure mechanism which is mainly 

governed by mode I. This delamination event was also observed in the SSCS and simplified models. 
20 

Failure 

displacement

2.97 mm

2.82 mm

2.90 mm

2.46 mm

Figure 17. Results of global analysis and local models 1 to 5. 

Failure 

displacement

2.87 mm

2.59 mm

2.47 mm

2.40 mm

Figure 18. Results of global analysis and local models 6 to 9. 

A second type of delamination can initiate at skin inflection points along the stringer flanges. This second 

mode of delamination has a stronger mode II component and it propagates at higher applied loads than the 

first mode of delamination. 

It can also be noted from Figure 17 that displacement for collapse of the local model 2 (2.90 mm) is much 

higher than that of the local model 1 (2.46 mm), even though the modes of deformation appear to be 

similar. This difference can be explained by the fact that the delaminations predicted by model 2 are close 

to the boundaries of the local model. The influence of the boundary conditions is confirmed by the 

comparison with the local models 6 and 7, which cover the critical areas predicted by the models 1 and 2. 

However, the longitudinal position of the models 6 and 7 reduces the influence of the boundaries and 
21 

provides a more accurate description of the delamination onset and propagation. Indeed, the models 6 and 

7 collapse at lower load levels that correspond to applied displacements of 2.40 mm and 2.47 mm, 

respectively, which are similar to that of model 1. 

The critical displacements for models 3 and 4 are higher than those of models 1 and 2 and, therefore, they 

do not represent critical areas of a pristine stringer. However, it is interesting to observe that a different 

failure mechanism is exhibited in these local models compared to models 1 and 2. This behavior is 

clarified by the representation of the local model 4, which is reported in Figure 19. The contours of the 

out-of-plane displacements and the damage state of the cohesive elements are reported at the peak load, 

which corresponds to an applied displacement of 2.97 mm. 

Figure 19. Results on model 4. 

The deflected surface of Figure 19 displays two nodal lines in the skin outside the stringer, as highlighted 

by the two dotted lines. It can be observed that two delaminations develop at the nodal lines, and the 

failure mechanism is characterized by a significant mode II component. 

The models 8 and 9 collapse at displacements of 2.59 and 2.87 mm, respectively, in the same mode Idominated delamination as models 1 and 2, 6 and 7. 

It is interesting to observe that the applied displacement corresponding to the collapse load of the local 

models increases from the bottom to the top of the panel. This effect is ascribed to the skewness of the 

buckled surface, which is due to the bending/twisting coupling induced by the lay-up stacking sequence. 

It is interesting to observe the skew angle of the buckling pattern at different longitudinal positions. The 

skew angle is the angle of a line connecting the buckle crests of four adjacent bays. It can be observed in 

Figure 17 that the line passing through the crests of the lower section is approximately horizontal at the 

lower section, while it is characterized by a not zero skew angle at the upper section of the panel. 

A comparison of the critical loads for local models 1 to 9 indicates that the weakest location along the 

central stringer is contained within local model 6. The critical displacement predicted by the global/local 

approach is 2.40 mm, which corresponds to a 0.33% strain. It is interesting to compare this result with the 

failure strain obtained from the SSCS analysis, and whose results are reported in Figure 7. The SSCS fails 

for an applied displacement of 0.89 mm, which corresponds to a 0.37% strain. The difference between the 

failure strain obtained with the global/local approach and the SSCS is 10%, so revealing the ability of the 

SSCS to reproduce the failure response of the multi-stiffened panel with a satisfactory level of accuracy. 

In terms of computational times, the global analysis requires approximately 2 hours on a desktop 

workstation, while the local analysis requires approximately 3 hours. Consequently, the overall procedure 

can be carried out in a total time of 29 hours. 
22 

D. Mesh sensitivity 

A comparison is presented between the local model 6, shown in Figure 20(a), and a model realized with a 

finer mesh of 0.5 mm and interlaminar strength equal to 50 MPa, reported in Figure 20(b). The goal is to 

verify that even the collapse load of the multistringer panel is not influenced by the interlaminar strength, 

and the use of the 1.0 mm mesh and ݖݖߪ equal to 25 MPa is appropriate. 

(a) (b) 

Figure 20. Critical local model: (a) interlaminar strength 25 MPa, mesh 1.0 x 1.0 mm, (b) 

interlaminar strength 50 MPa, mesh 0.5 x 0.5 mm. 

The comparison between Figures 20(a) and 20(b) reveals no differences in terms of delamination onset 

and propagation mechanism. Indeed, the contour of the damage variables of the cohesive elements of the 

two models is very similar. A slight difference is observed in terms of critical applied displacement, 

which increases from 2.40 mm to 2.54 mm. 

The force displacement-curve of the multistringer panel is presented in Figure 21, together with the 

collapse loads calculated with the two different meshes. 

Figure 21. Comparison of predicted collapse loads using different mesh size. 

0 1 2 3 4 5 6 0

50

100

150

200

250

300

350

Displacement [mm]

Load [kN]

σzz = 25 MPa

Mesh 1.0x1.0 mm

σzz = 50 MPa

Mesh 0.5x0.5 mm
23 

The collapse load obtained with the 1.0 mm mesh is equal to 190.6 kN, while the more refined mesh of 0.5 

mm determines a value of 198.0 kN. The two loads differ by less than 4%, showing that the 1.0 mm mesh 

with reduced interlaminar strength properties (25 MPa) allows an accurate prediction of the collapse laod. 

VIII. Conclusions 

A study of the response of postbuckled panels using models of different levels of approximation was 

conducted. A fast parametric finite element model was developed to gain insight into the structural 

postbuckling response, the initiation and propagation of delaminations, and their effect on the collapse 

load. The reduced computational time for this simplified model was also useful to conduct sensitivity 

studies on the effects of mesh refinement, material properties, defect dimensions, and stringer 

configuration. The results of the analyses indicate that the delamination-induced collapse of the stiffened 

panel investigated is geometrically driven. A postbuckling mode transition of the portion of skin between 

the two stringer webs was found to induce a mode I delamination of the stringer. The influence of 

different bond defect sizes on the collapse load was examined. In addition, the understanding of the 

mechanism of collapse extracted from the analyses was also used to develop an improved stringer design 

that constrains the mode transition for a higher collapse load. The computational efficiency of the 

simplified model developed in this study illustrates the ease with which such a parametric model can be 

used in a design environment. 

Finally, a global/local methodology was presented to identify the weakest spots of the multistringer panel 

configuration loaded in compression. The procedure makes use of a relatively coarse global mesh of the 

panel in conjunction with a local model that relies on cohesive elements to capture the delamination 

failure modes. The multi-level methodology represents an efficient strategy to identify the weak spots and 

to predict the collapse load of postbuckled stiffened panels subjected to arbitrary loading conditions. 

References 

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Panels under Combined Shear-Axial Compression,” Journal of Aircraft, Vol. 45, No. 2, 2008, pp. 

402–413. 

2. Cordisco, P., Bisagni, C., “Cyclic Buckling Tests under Combined Loading on Predamaged 

Composite Stiffened Boxes,” AIAA Journal, Vol. 49, No. 8, 2011, pp. 1795-1807. 

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Post-Buckled CFRP Stringer-Stiffened Panels,” Composites Part A: Applied Science and 

Manufacturing, Vol. 34, No. 7, 2003, pp. 623–633. 

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pp. 150–161. 
24 

5. Taki, T., Kitagawa, T., “Post-Buckling Strength of Composite Stiffened Panel under Shear Load,” 1st 

AIAA Aircraft Engineering, Technology, and Operations Congress, AIAA Paper 95-393, September 

1995. 

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Skin/Stringer Composite Assemblies,” AIAA Journal, Vol. 46, No. 5, 2008, pp. 1144-1159. 

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Panels with and without Cutouts,” NASA/CR-2001-211223, ICASE Report No. 2001-27, 2001. 

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Structures Using Shells and Cohesive Elements,” Journal of Aircraft, Vol. 45, No. 2, 2008, pp. 663-

672. 

14. Moës, N., Dolbow, J., Belytschko, T., “A Finite Element Method for Crack Growth without 

Remeshing,” International Journal for Numerical Methods in Engineering, Vol. 46, No. 1, 1999, pp. 

131-150. 

15. Orifici, A.C., Thomson, R.S., Degenhardt, R., Bisagni, C., Bayandor, J., “A Finite Element 

Methodology for Analysing Degradation and Collapse in Postbuckling Composite Aerospace 

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16. Bertolini, J., Castanié, B., Barrau, J.J., Navarro, J.P., Petiot, C. “Multi-Level Experimental and 

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25 

20. Camanho, P.P., Maimí, P., Dávila, C.G., “Prediction of Size Effects in Notched Laminates Using 

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REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Simplified Models for the Study of Postbuckled Hat-Stiffened Composite 

Panels

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Vescovini, Riccardo; Davila, Carlos G.; Bisagni, Chiara

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20118

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 24

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

The postbuckling response and failure of multistringer stiffened panels is analyzed using models with three levels of 

approximation. The first model uses a relatively coarse mesh to capture the global postbuckling response of a five-stringer 

panel. The second model can predict the nonlinear response as well as the debonding and crippling failure mechanisms in a 

single stringer compression specimen (SSCS). The third model consists of a simplified version of the SSCS that is designed to 

minimize the computational effort. The simplified model is well-suited to perform sensitivity analyses for studying the 

phenomena that lead to structural collapse. In particular, the simplified model is used to obtain a deeper understanding of the

role played by geometric and material modeling parameters such as mesh size, inter-laminar strength, fracture toughness, and 

fracture mode mixity. Finally, a global/local damage analysis method is proposed in which a detailed local model is used to 

scan the global model to identify the locations that are most critical for damage tolerance. 

15. SUBJECT TERMS

Cohesive elements; Composites; Delamination; Fracture; Material characterization

18. NUMBER

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30

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(443) 757-5802

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5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

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5f. WORK UNIT NUMBER

284848.02.04.07.01

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NASA/TM-2012-217336

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1. REPORT DATE (DD-MM-YYYY)

01- 02 - 2012