NASA/TP-1999-209107

Progressive Failure Analysis Methodology

for Laminated Composite Structures

David W. Sleight
Langley Research Center, Hampton, Virginia

March 1999
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NASA/TP-1999-209107

Progressive Failure Analysis Methodology

for Laminated Composite Structures

David W. Sleight
Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

March 1999
 Acknowledgments

The author would like to thank his research advisor Dr. Norman F. Knight, Jr., whose support, guidance,
understanding, and knowledge have made this work possible. A special thanks is also expressed to Dr. John Wang
for serving on my thesis committee and being a mentor for my professional development. The author would also like
to thank Ms. Tina Lotts for the countless times she has helped me with COMET.

Available from:

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS)
7121 Standard Drive 5285 Port Royal Road
Hanover, MD 21076-1320 Springfield, VA 22161-2171
(301) 621-0390 (703) 605-6000
 TABLE OF CONTENTS

TABLE OF CONTENTS..........................................................................................................................i
LIST OF TABLES .................................................................................................................................iii
LIST OF FIGURES................................................................................................................................iv
INTRODUCTION ...................................................................................................................................1
Overview of Progressive Failure........................................................................................................1
Nonlinear Analysis......................................................................................................................3
Strain/Stress Recovery ................................................................................................................4
Failure Analysis ..........................................................................................................................4
Material Degradation...................................................................................................................7
Re-establishment of Equilibrium .......................................................................................................9
Literature Review on Progressive Failure ..........................................................................................9
Objectives and Scope ...................................................................................................................... 12
PROGRESSIVE FAILURE METHODOLOGY .................................................................................... 14
Failure Detection............................................................................................................................. 14
Maximum Strain Criterion ........................................................................................................ 14
Hashin’s Criterion..................................................................................................................... 15
Christensen’s Criterion.............................................................................................................. 15
Damage Modeling ........................................................................................................................... 16
Nonlinear Analysis Solution Continuation....................................................................................... 18
NUMERICAL RESULTS ..................................................................................................................... 19
Rail-Shear Panel.............................................................................................................................. 19
Problem Statement .................................................................................................................... 20
Structural Response................................................................................................................... 22
Tension-Loaded Laminate with Hole............................................................................................... 28
Problem Statement .................................................................................................................... 28
Structural Response................................................................................................................... 28
Compression-Loaded Composite Panel ........................................................................................... 35
Problem Statement .................................................................................................................... 35
Structural Response................................................................................................................... 37
Compression-Loaded Composite Panel with Hole ........................................................................... 48
Problem Statement .................................................................................................................... 48
Structural Response................................................................................................................... 48
Composite Blade-Stiffened Panel .................................................................................................... 56
Problem Statement .................................................................................................................... 56
Structural Response................................................................................................................... 59
CONCLUSIONS AND RECOMMENDATIONS.................................................................................. 66
Conclusions..................................................................................................................................... 66
Recommendations ........................................................................................................................... 67

i
 TABLE OF CONTENTS, continued.

APPENDIX - IMPLEMENTATION INTO COMET....................................................................... 68

COMET Overview .......................................................................................................................... 68
Generic Element Processor .............................................................................................................. 69
ES5 Processor ........................................................................................................................... 71
ES31 Processor ......................................................................................................................... 71
Generic Constitutive Processor........................................................................................................ 71
Implementation of Failure and Damage Models ........................................................................ 73
Historical Material Database ..................................................................................................... 77
Nonlinear Analysis Solution Procedure ..................................................................................... 81
REFERENCES...................................................................................................................................... 82

ii
 LIST OF TABLES

Table ............................................................................................................................................. Page

Table 1. Quadratic Polynomial Failure Criteria...................................................................................6
Table 2. Options for Material Degradation for Maximum Strain Criterion ........................................ 17
Table 3. Material Degradation for Hashin’s and Christensen’s Failure Criteria ................................. 17
Table 4. Material Properties for T300/976 Material System .............................................................. 20
Table 5. Rail-Shear Problem: Effect of Mesh Size - Hashin’s Criterion. ........................................... 22
Table 6. Rail-Shear Problem: Effect of Displacement Increment Size, Hashin’s Criterion................. 23
Table 7. Rail-Shear Problem: Effect of Material Degradation Factor α, Christensen’s Criterion........ 23
Table 8. Rail-Shear Problem: Comparison of Maximum Strain Options ........................................... 25
Table 9. Rail-Shear Problem: Summary of Progressive Failure Results............................................. 25
Table 10. Material Properties for T300/1034 Material System ............................................................ 28
Table 11. Tension-Loaded Laminate with Hole: Effect of Finite Element Mesh,
Christensen’s Criterion ....................................................................................................... 31
Table 12. Tension-Loaded Laminate with Hole: Comparison of Failure Results ................................. 31
Table 13. Material Properties for C4 Panel, T300/5208 Material System ............................................ 35
Table 14. C4 Panel: Comparison of Failure Results, α = 10-20............................................................. 38
Table 15. H3/H4 panel: Comparison of Failure Results ...................................................................... 50
Table 16. Material Properties for Blade-Stiffened Panel, T300/5208 Material System ........................ 56
Table 17. Composite Blade-Stiffened Panel: Comparison of Failure Results....................................... 59
Table A-1. Constitutive Models in COMET ......................................................................................... 73
Table A-2. Constitutive Developer Interface Subroutines of GCP ......................................................... 73

iii
 LIST OF FIGURES

Figure ............................................................................................................................................. Page

Figure 1. Typical progressive failure analysis methodology. ................................................................2
Figure 2. Post-failure degradation behavior in composite laminates. ....................................................8
Figure 3. Geometry, loading, and boundary conditions of rail-shear specimen
with 48 x 8 finite element mesh. ......................................................................................... 21
Figure 4. Rail-Shear problem: Effect of displacement increment size, Hashin’s criterion.................... 24
Figure 5. Rail-shear problem: Load-deflection results, α = 10-20. ........................................................ 26
Figure 6. Rail-shear problem: Structural response at final failure, (3964 lbs.)..................................... 27
Figure 7. Geometry, loading, and boundary conditions of tension-loaded laminate with hole. ............ 29
Figure 8. Meshes for tension-loaded laminate with hole..................................................................... 30
Figure 9. Tension-loaded laminate with hole: Load-deflection results, α = 10-20. ................................ 32
Figure 10. Tension-loaded laminate with hole: Structural response at final failure (3212 lbs.).............. 33
Figure 11. Tension-loaded laminate with hole: Structural response at final failure (3261 lbs.).............. 34
Figure 12. Geometry, loading, and boundary conditions for C4 panel. ................................................. 36
Figure 13. C4 panel: Comparison of experimental and analytical linear buckling mode 1..................... 39
Figure 14. C4 panel: End-shortening results......................................................................................... 40
Figure 15. C4 panel: Out-of-plane deflection comparison. ................................................................... 41
Figure 16. C4 panel: End-shortening comparison................................................................................. 42
Figure 17. C4 panel: Photograph of failure mode from Starnes and Rouse experiment [49].................. 43
Figure 18. C4 panel: Structural response at final failure (23526 lbs.).................................................... 44
Figure 19. C4 panel: Structural response at final failure (23526 lbs.), continued. ................................. 45
Figure 20. C4 Panel: Structural response at final failure (22454 lbs.). .................................................. 46
Figure 21. C4 Panel: Structural response at final failure (22454 lbs.), continued. ................................. 47
Figure 22. Geometry, loading, and boundary conditions for H3/H4 panel. ........................................... 49
Figure 23. H3/H4 panel: Comparison of experimental and analytical linear buckling mode 1............... 51
Figure 24. H3/H4 panel: End-shortening results, α = 10-20. .................................................................. 52
Figure 25. H3/H4 panel: Out-of-plane deflection results. ..................................................................... 53
Figure 26. H3/H4 panel: Structural response at final failure (17313 lbs.).............................................. 54
Figure 27. H3/H4 panel: Structural response at final failure (17313 lbs.), continued............................. 55
Figure 28. Composite blade-stiffened panel with discontinuous stiffener. ............................................ 57
Figure 29. Finite element model of composite blade-stiffened panel. ................................................... 58
Figure 30. Composite blade-stiffened panel: End-shortening results..................................................... 61
Figure 31. Composite blade-stiffened panel: Out-of-plane deflection at hole and blade stiffener........... 62
Figure 32. Composite blade-stiffened panel: Structural response at final failure (57064 lbs.)................ 63
Figure 33. Composite blade-stiffened panel: Structural response at final failure
(57064 lbs.), continued. ...................................................................................................... 64
Figure 34. Composite blade-stiffened panel: Local region near hole..................................................... 65
Figure A-1. Graphical overview of COMET. ......................................................................................... 69
Figure A-2 GEP template...................................................................................................................... 70
Figure A-3 GCP overview. ................................................................................................................... 72
Figure A-4 Gauss quadrature points and layer-integration points........................................................... 75
Figure A-5 Progressive failure methodology in COMET....................................................................... 76
Figure A-6 Overview of the progressive failure analysis computation locations
in a composite laminate. ..................................................................................................... 78
Figure A-7 Organization of constitutive material database-1. ................................................................ 79
Figure A-8 Organization of constitutive material database-2. ................................................................ 80

iv
 ABSTRACT
A progressive failure analysis method has been developed for
predicting the failure of laminated composite structures under
geometrically nonlinear deformations. The progressive failure analysis
uses C1 shell elements based on classical lamination theory to calculate
the in-plane stresses. Several failure criteria, including the maximum
strain criterion, Hashin’s criterion, and Christensen’s criterion, are
used to predict the failure mechanisms and several options are available
to degrade the material properties after failures. The progressive
failure analysis method is implemented in the COMET finite element
analysis code and can predict the damage and response of laminated
composite structures from initial loading to final failure. The different
failure criteria and material degradation methods are compared and
assessed by performing analyses of several laminated composite
structures. Results from the progressive failure method indicate good
correlation with the existing test data except in structural applications
where interlaminar stresses are important which may cause failure
mechanisms such as debonding or delaminations.

INTRODUCTION

Composite materials have been increasingly used in aerospace and automotive applications over the
last three decades and have seen a dramatic increase in usage in non-aerospace products in the few years.
The use of composite materials is very attractive because of their outstanding strength, stiffness, and
light-weight properties. An additional advantage of using composites is the ability to tailor the stiffness
and strength to specific design loads. However, a reliable methodology for fully predicting the
performance of composite structures beyond initial localized failure has yet to be developed. Since most
composite materials exhibit brittle failure, with little or no margin of safety through ductility as offered
by many metals, the propagation of the brittle failure mechanism in composite structures must be
understood and reliable prediction analysis methods need to be available. For example, laminated
composite structures can develop local failures or exhibit local damage such as matrix cracks, fiber
breakage, fiber-matrix debonds, and delaminations under normal operating conditions which may
contribute to their failure. The ability to predict the initiation and growth of such damage is essential for
predicting the performance of composite structures and developing reliable, safe designs which exploit
the advantages offered by composite materials. Hence, the need for a reliable methodology for predicting
failure initiation and propagation in composite laminated structures is of great importance.

Overview of Progressive Failure

In recent years, the progression of damage in composite laminates has been a focus of extensive
research. Ochoa and Reddy [1] present an excellent overview of the basic steps for performing a
progressive failure analysis. A typical methodology for a progressive failure analysis is illustrated in
Figure 1. At each load step, a nonlinear analysis is performed until a converged solution is obtained
assuming no changes in the material model. Then using this equilibrium state, the stresses within each
lamina are determined from the nonlinear analysis solution. These stresses are then compared with
material allowables and used to determine failure according to certain failure criteria. If lamina failure is
detected, as indicated by a failure criterion, the lamina properties are changed according to a particular

1
degradation model. Since the initial nonlinear solution no longer corresponds to an equilibrium state,
equilibrium of the structure needs to be re-established utilizing the modified lamina properties for the
failed lamina while maintaining the current load level. This iterative process of obtaining nonlinear
equilibrium solutions each time a local material model is changed is continued until no additional lamina
failures are detected. The load step is then incremented until catastrophic failure of the structure is
detected as determined by the progressive failure methodology.

Therefore, typical progressive failure analysis methods involve five key features. First, a nonlinear
analysis capability is used to establish equilibrium. Second, an accurate stress recovery procedure is
needed in order to establish the local lamina stress state. Third, failure criteria are needed in order to
detect local lamina failure and determine the mode of failure. Fourth, material degradation or damage
models are needed in order to propagate the failure and establish new estimates for the local material
properties. Finally, a procedure to re-establish equilibrium after modifying local lamina properties is
needed. This research will focus on the last four features since nonlinear analysis procedures are already
well established.

Define initial
state

Load, Pi

Nonlinear Analysis
to Establish Equilibrium

No Yes
Converged Solution Obtained ? No
Re-establish
Yes Equilibrium ?
Predicted Final
Stress Recovery Procedure to Small
Failure Load
Compute Stresses/Strains ∆P

Stop Yes
Failure Detected ? Degrade Properties
of Material
No
Increment Load
P i = P i-1+ ∆ P

Figure 1. Typical progressive failure analysis methodology.

2
 Nonlinear Analysis

A nonlinear analysis is performed to account for the geometrically nonlinear behavior in the
progressive failure analysis. The assembled finite element equations are given by

[KT ({D})]{D} = {P} (1)

in which {D} is the displacement vector, {P} is the applied load vector, and [KT] is the assembled
tangent stiffness matrix. Composite laminates typically behave in a linear elastic manner until local
structural failures develop. After local failures within the laminate, the global structural stiffness
changes. Hence, the tangent stiffness matrix [KT] depends on the material properties as well as the
unknown displacement solution {D}. In this progressive failure analysis, a nonlinear analysis is
performed until a converged solution is obtained for a constant set of material properties. The nonlinear
analysis involves solving the linearized finite element equations for the kth iteration

[KT ](k ) {∆D} = {R}(k )

(2)
{D}(k +1) = {D}( k ) + {∆D}
where the tangent stiffness matrix [KT](k) and force imbalance vector {R}(k) are functions of the
displacements {D}(k). Solving the equilibrium equations is an iterative process where the kth step requires
computing the displacement increment {∆D} for the k + 1 load step using the kth tangent stiffness matrix.
Then the kth displacement vector {D} is updated using {∆D}. Having a new displacement solution, the
force imbalance vector {R} and possibly the tangent stiffness matrix [KT] are updated, and the process is
continued for the next iteration. The solution process is continued until convergence is achieved by
reducing the force imbalance {R}, and consequently {∆D} to within some tolerance.

Using this nonlinear solution corresponding to a given load step, the lamina stresses are determined
and used with a failure criterion to determine whether any failures have occurred during this load
increment. If no failures are detected, then the applied load is increased, and the analysis continues.
When a failure in the lamina occurs, a change in the stiffness matrix due to a localized failure is
calculated based on the material degradation model. This adjustment accounts for the material
nonlinearity associated with a progressive failure analysis embedded within a nonlinear finite element
analysis. If the load step size is too large, static equilibrium needs to be re-established by repeating the
nonlinear analysis at the current load step using the new material properties in the tangent stiffness
matrix. This process is repeated until no additional failures are detected. Alternatively, small load step
sizes can be used thereby minimizing the effect of not re-establishing equilibrium at the same load level.
This incremental iterative process is performed until a lack of convergence in the nonlinear solution
occurs.

The most popular iterative schemes for the solution of nonlinear finite element equations are forms of
the Newton-Raphson procedure which is widely used because it generally converges quite rapidly.
However, one of the drawbacks of the Newton-Raphson procedure is the large amount of computational
resources needed to evaluate, assemble, and decompose the tangent stiffness matrix at each iteration. To
reduce the computational effort, a modified Newton-Raphson procedure is commonly used. The
modified Newton-Raphson procedure differs from the Newton-Raphson method in that the tangent
stiffness matrix is not updated on each iteration but periodically during the analysis, such as at the
beginning of each new load step.

3
Strain/Stress Recovery

Once the nodal values of generalized displacements have been obtained by performing the nonlinear
analysis at a particular load step, the element strains are evaluated by differentiating the displacements.
These element strains can be computed at any point in the finite element such as the center of the element
or at the Gauss points. These recovered strains are midplane strains and changes in curvature obtained by
using the strain-displacement relations for the laminate. With these values, the strains through the
laminate thickness can be determined and then the stresses can be determined based on the constitutive
relations.

Since the displacements calculated from the finite element analysis are in global coordinates, and the
failure criteria used in the laminate analysis require stresses and strains in the material (lamina)
coordinates, the strains and stresses must be transformed from global coordinates to material coordinates
for each layer of the laminate. Once the strains are transformed to the material coordinates, the lamina
constitutive equations are used to compute the in-plane stresses (σxx, σyy, σxy). The transverse stresses
(σxz, σyz, σzz) can be computed either by the lamina constitutive equations for shear deformable C0
elements or by integration of the 3-D equilibrium equations [1],

∂σ xx ∂σ xy ∂σ xz
+ + =0
∂x ∂y ∂z

∂σ xy ∂σ yy ∂σ yz
+ + =0 (3)
∂x ∂y ∂z

∂σ xz ∂σ yz ∂σ zz
+ + =0
∂x ∂y ∂z

where the body forces are neglected. The accuracy of the computed strains and stresses can be improved
by smoothing algorithms [2].

Failure Analysis

The catastrophic failure of a composite structure rarely occurs at the load corresponding to the initial
or first-ply failure. Instead, the structure ultimately fails due to the propagation or accumulation of local
failures (or damage) as the load is increased. Initial failure of a layer within the laminate of a composite
structure can be predicted by applying an appropriate failure criterion or first-ply failure theory. The
subsequent failure prediction requires an understanding of failure modes and failure propagation.

Laminated composites may fail by fiber breakage, matrix cracking, or by delamination of layers [1].
The mode of failure depends upon the loading, stacking sequence, and specimen geometry. The first
three of these failure modes depend on the constituent’s strength properties, whereas delamination may
be due to manufacturing anomalies during lay-up or curing or out-of-plane effects. In addition, failure
mechanisms such as skin-stiffener separation can be included if needed. However, this progressive
failure methodology only includes predictions for fiber breakage and matrix cracking.

Various failure criteria have been proposed in the literature [3-18]. Most failure criteria are based on
the stress state in a lamina. Ideally, a three-dimensional model is desirable for obtaining accurate stresses

4
and strains. However, due to the extensive amount of computational time required for a three-
dimensional analysis, two-dimensional failure analyses are usually performed using plate and shell finite
element models. Failure criteria are intended to predict macroscopic failures in the composite laminate
and are based on the tensile, compressive, and shear strengths of the individual lamina. If an allowable
stress limit or failure criterion within a layer is not exceeded, the material properties in the layer are not
changed and then the other layers within the laminate are checked. When a material allowable value or
failure criterion is exceeded in a given layer, the engineering material constants corresponding to that the
particular mode of failure are reduced depending on the material degradation model.

Failure criteria for composite materials are often classified into two groups: namely, non-interactive
failure criteria and interactive failure criteria. Several papers can be found which list the most commonly
used composite failure theories [3,4,5].

Non-Interactive Failure Criteria A non-interactive failure criterion is defined as one having no

interactions between the stress or strain components. These criteria, sometimes called independent
failure criteria, compare the individual stress or strain components with the corresponding material
allowable strength values. The maximum stress and maximum strain criteria belong to this category.
Both failure criteria indicate the type of failure mode. The failure surfaces for these criteria are
rectangular in stress and strain space, respectively [6].

Interactive Failure Criteria Interactive failure criteria involve interactions between stress and strain
components. Interactive failure criteria are mathematical in their formulation. Interactive failure criteria
fall into three categories: (1) polynomial theories, (2) direct-mode determining theories, and (3) strain
energy theories. The polynomial theories use a polynomial based upon the material strengths to describe
a failure surface [1]. The direct-mode determining theories are usually polynomial equations based on
the material strengths and use separate equations to describe each mode of failure. Finally, the strain
energy theories are based on local strain energy levels determined during a nonlinear analysis.

Most of the interactive failure criteria are polynomials based on curve-fitting data from composite
material tests. The most general polynomial failure criterion for composite materials is the tensor
polynomial criterion proposed by Tsai and Wu [7]. The criterion may be expressed in tensor notation as

Fi σ i + Fij σ i σ j + Fijk σ i σ j σ k ≥ 1 i, j , k = 1, K ,6 (4)

where Fi, Fij, and Fijk are components of the lamina strength tensors in the principal material axes. The
usual contracted stress notation is used except that σ4 = τ23, σ5 = τ13 and σ6 = τ12. However, the third-order
tensor Fijk is usually ignored from a practical standpoint due to the large number of material constants
required [7]. Then, the general polynomial criterion reduces to a general quadratic criterion given by

Fi σ i + Fij σ i σ j ≥ 1 i, j = 1, K ,6 (5)

or in explicit form,

F1σ1 + F2 σ 2 + F3 σ 3 + 2 F12 σ1σ 2 + 2 F13 σ1σ 3 + 2 F23 σ 2 σ 3 + F11σ12

(6)
+ F22 σ 2 2 + F33 σ 3 2 + F44 σ 4 2 + F55 σ 5 2 + F66 σ 6 2 ≥ 1.

5
The F4, F5 and F6 terms associated with σ4 and σ6, respectively, are assumed to be zero since it is assumed
that the shear strengths are the same for positive and negative shear stress. Various quadratic criteria
differ in the way that the tensor stress components are determined. Other popular quadratic failure
criteria include those by Tsai-Hill [8,9], Azzi and Tsai [10], Hoffman [11], and Chamis [12]. These
quadratic failure criteria can be represented in terms of the general Tsai-Wu quadratic criterion and are
summarized in Table 1 where X, Y, and Z are lamina strengths in the x, y, and z directions, respectively,
and R, S, and T are the shear strengths in the yz, xz, and xy planes, respectively. The subscripts T and C
in X, Y, and Z refer to the normal strengths in tension and compression. The failure surfaces for these
quadratic criteria are elliptical in shape. One of the disadvantages of these quadratic failure criteria is that
they predict the initiation of failure but say nothing about the failure mode or how the composite fails.

Table 1. Quadratic Polynomial Failure Criteria

Quadratic Polynomial Failure Criteria
* * *†
Tsai-Wu Tsai-Hill Azzi-Tsai Hoffman Chamis
F1 1
−
1 0 0 1
−
1 0
XT X C XT X C

F2 1 1 0 0 1 1
− − 0
YT YC YT YC

F3 1 1 0 0 1 1
− − 0
ZT Z C ZT Z C

F12 − 1 2 X T X CYT YC 1 1 1 1  1 1 1 1 1  K
−  + −  − −  + −  − 12
2 X 2 Y2 Z2  X2 2  X T X C YT YC ZT Z C  XY

F13 − 1 2 X T X C ZT Z C 1 1 1 1  0 1 1 1 1  K
−  + −  −  + −  − 13
2 Z2 X 2 Y2  2  X T X C Z T Z C YT YC  XZ

F23 − 1 2 YT YC Z T Z C 1 1 1 1  0 1 1 1 1  −
K 23
−  + −  −  + − 
2Y2 Z2 X 2  2  ZT Z C YT YC X T X C  YZ


F11 1 1 1 1 1
XT X C X2 X2 XT X C X2

F22 1 1 1 1 1
YT YC Y2 Y2 YT YC Y2

F33 1 1 0 1 1
ZT Z C Z2 ZT ZC Z2

F44 1 1 0 1 1
R2 R2 R2 R2

F55 1 1 0 1 1
S2 S2 S2 S2

F66 1 1 1 1 1
T2 T2 T2 T2 T2

* X, Y, and Z are either XC, YC, and ZC or XT, YT, and ZT depending upon the sign of σ1, σ2 and σ3
respectively.
† K12, K13, and K23 are the strength coefficients depending upon material.

6
 Direct-mode determining failure criteria are very useful in progressive failure analysis because they
also describe the failure mode of the composite laminate. Hashin [13,14] stated that the Tsai-Wu theory
had an intrinsic problem since it could not distinguish among the various different failure modes of the
composite material. He instead proposed a quadratic failure criterion in piecewise form based on
material strengths, where each smooth branch represents a failure mode. In unidirectional composites,
there are two primary failure modes: a fiber mode and a matrix mode subdivided into either tension or
compression failure. In the fiber mode, the lamina fails due to fiber breakage in tension or fiber buckling
in compression. In the matrix mode, failure is due to matrix cracking.

Lee [15] also proposed a direct-mode determining failure criterion. His criterion was a polynomial
equation for each mode of failure based upon the three-dimensional stress calculations. The modes of
failure determined included fiber failures, matrix failures, and delaminations. Christensen [16]
introduced a quasi-three-dimensional laminate theory which accounted for the out-of-plane stress terms.
He then developed a strain-based failure criterion which distinguished between fiber failure and fiber-
matrix interaction failure.

A nonlinear total strain energy failure criterion was developed by Sandhu [17]. This criterion is based
on the concept that the lamina fails when the sum of the ratios of energy levels (due to longitudinal,
transverse, and shear loading) to the corresponding maximum energies equals unity. A similar failure
criterion by Abu-Farsakh and Abdel-Jawad [18] was introduced based on an energy concept. However,
the failure modes could not be identified for either criterion which poses difficulties for material
degradation modeling and failure propagation.

Material Degradation

If failure is detected in a particular lamina of the composite material, the properties of that lamina
must be adjusted according to a material property degradation model. A number of post-failure material
property degradation models have been proposed for progressive failure analyses [3]. Most of these
material degradation models belong to one of three general categories: instantaneous unloading, gradual
unloading, or constant stress at ply failure [19]. Figure 2 illustrates these three categories. For the
instantaneous loading case, the material property associated with that mode of failure is degraded
instantly to zero. For the gradual unloading case, the material property associated with that mode of
failure is degraded gradually (perhaps exponentially) until it reaches zero. For the constant stress case,
the material properties associated with that mode of failure are degraded such that the material cannot
sustain additional load. The behavior of the lamina as it fails, as well as which elastic constants are
degraded, depends on the failure mode of the composite laminate.

7
 Stress constant stress

gradual unloading

instantaneous
unloading

Strain

Figure 2. Post-failure degradation behavior in composite laminates.

The Hahn-Tsai method [20] assumes that a failed lamina will support its load (load at initial lamina
failure) until total failure of the laminate occurs. This is an example of the constant stress category of
material degradation methods. In the gradual unloading model, the material elastic properties are
gradually reduced depending upon the extent of damage within a lamina until the lamina has completely
unloaded or failed. The unloading can be either linear or exponential in behavior. Petit and Waddoups’
work [21] was the first effort in material degradation by gradual unloading. Sandhu [17] also used a
degradation model by gradual unloading based on nonlinear stress-strain relations. Other gradual
unloading models are the exponential degradation model by Nahas [3] and the Weibull distribution used
by Chang [22]. Reddy and Reddy [23] used a constant degradation method in which the degraded
properties are assumed to be a constant multiple of the original properties of the undamaged material.
They divided the constant degradation methods into two types: independent and interactive. In the
independent method, it is assumed that each stress only contributes toward the degradation of the
corresponding stiffness property. In the interactive method, coupling is assumed between the normal and
shear stiffness lamina properties.

One of the most common methods used for degradation of material properties is the ply-discount
theory [19] which belongs to the instantaneous unloading category. In this method, one or more of the
elastic material properties of a lamina are set to equal zero or a small fraction of the original value once
failure is detected. As in the gradual unloading category, the degradation can be either independent or
interactive corresponding to the mode of failure. This method is described later in the paper.

8
 Re-establishment of Equilibrium

Once a lamina fails and the stiffness properties have been degraded, it is often necessary to re-
calculate the element stiffness matrices and update the tangent stiffness matrix of the model. This new
tangent stiffness matrix accounts for the local changes in material stiffness as well as any large
deformation effects associated with geometric nonlinearities. The nonlinear analysis procedure described
earlier is then used to re-establish equilibrium at the same load level for the composite structure with
localized failures. To establish equilibrium, additional iterations may be required until a new converged
solution is reached. Once obtained, checks for subsequent lamina failures are necessary. If the load steps
are restricted to be small, such a procedure may not be needed.

Literature Review on Progressive Failure

This section summarizes some of the research done in progressive failure analyses over the past two
decades. The summary discusses the type of analysis (linear or nonlinear) used in performing the
progressive failure analyses, failure criterion chosen, and prediction of progressive failure analyses
compared to experimental results.

Reddy and Pandey [6] developed a finite element procedure based on first-order, shear-deformation
theory for first-ply failure analysis of laminated composite plates subjected to in-plane and/or transverse
loads. A tensor polynomial failure criterion with failure predictions by the maximum stress, maximum
strain, Tsai-Hill, Tsai-Wu, and Hoffman failure criteria was used to predict lamina failures at the element
Gauss points. For laminates subjected to in-plane loading, all the failure criteria satisfactorily predicted
first-ply failure. However, for laminates subjected to transverse loads, the failure locations and failure
loads predicted by either the Tsai-Hill or maximum strain criteria were different than those predicted
using the other criteria.

Pandey and Reddy [24] extended their earlier work on first-ply failure of two-dimensional laminated
composites to include a progressive failure analysis capability. However, only a linear finite element
analysis was performed. Again, the same failure criteria as in the previous study were used for the
prediction of failure within the composite laminate. The elastic constants of a failed lamina were reduced
according to the dominant failure indices determined from the individual contributions of each stress
component. The individual contributions αi of each stress component to the failure index are first
determined and then the corresponding elastic properties are multiplied by a factor R(1−αi) where R is a
pre-selected reduction parameter ranging from 0 to 1. After the stiffness properties are reduced, the stress
analysis is repeated until no additional ply failures are predicted. This progressive failure analysis
method was applied to a laminated plate with a hole subjected to uniaxial tension and to a rectangular
plate subjected to a uniform transverse pressure. Comparisons with experimental results were not
provided in this reference.

Reddy and Reddy [25] calculated and compared the first-ply failure loads obtained by using both
linear and nonlinear finite element analyses on composite plates. The finite element model was based on
first-order shear deformation theory. The maximum stress, maximum strain, Tsai-Hill, Tsai-Wu, and
Hoffman failure criteria were used for failure prediction of composite plates subject to in-plane (tensile)
loading and transverse loading. The failure loads and locations predicted by the different failure criteria
differed significantly from one other. The differences between the linear and nonlinear failure loads was
found much larger for the cases involving transverse loading than for the cases involving in-plane
(tensile) loading.

9
 Reddy and Reddy [23] then developed a three-dimensional (3-D) progressive failure algorithm for
composite laminates under axial tension. The finite element analysis used Reddy’s Layerwise Laminated
Plate Theory (LWLT) and predicted both in-plane and interlaminar stresses at the reduced integration
Gauss points. In the analysis, the Tsai-Wu failure criterion along with other various failure criteria were
compared to experimental results. The other failure criteria used included maximum stress, maximum
strain, Tsai-Hill, and Hoffman criteria. Two different types of stiffness reduction methods, an
independent type and an interactive type, were considered to study the influence on the failure loads and
strains of stiffness reduction at the Gauss points where failed plies were detected. In the independent
method, it was assumed that each stress only contributes toward the degradation of the corresponding
stiffness property. In the interactive method, coupling was assumed between the normal and shear
stiffness properties. A parametric study was performed to investigate the effect of out-of-plane material
properties, 3-D stiffness reduction methods, and boundary conditions on the failure loads and strains of a
composite laminate under axial tension. Results showed the progressive failure algorithm accurately
predicted the failure loads and strains. Also, results showed that the maximum stress and maximum
strain failure criteria tend to overpredict the failure loads while the Hoffman and Tsai-Wu failure criteria
tend to underpredict the failure loads for all laminates. Also, it was noted that the Tsai-Hill failure
criteria did not consistently follow the experimental trends.

In an earlier paper, Ochoa and Engblom [26] presented a progressive failure analysis for composite
laminates in uniaxial tension using a higher-order plate theory with shear deformable elements. Hashin’s
failure criterion [13] was used to identify fiber and matrix failures while Lee’s criterion [15] was used to
identify delaminations. Stiffness reduction of failed lamina was carried out at the Gauss points for the
entire laminate. Equilibrium was then re-established and failure detection was again checked at the same
load increment before advancing to the next load increment. Analyses were performed on a plate
subjected to uniaxial tension and to four-point bending. However, comparisons with experimental results
were not provided.

Engelstad, Reddy, and Knight [27] investigated the postbuckling response and failure prediction of
flat composite unstiffened panels loaded in axial compression using 9-node shear deformable elements.
The finite element formulation accounted for transverse shear deformation and was based on virtual
displacements using the total Lagrangian description. The Newton-Raphson method was used to solve
the nonlinear analysis. For failure prediction, the maximum stress and Tsai-Wu failure criteria were
implemented and compared with limited experimental results. The stresses for the in-plane and
transverse stress components were calculated at the element Gauss points in the plane of the element and
at the middle of each lamina in the thickness direction using the constitutive relations. If failure
occurred, a reduction in the material properties was applied to the material properties at the Gauss point
corresponding to the dominant failure mode. For example, if a lamina failed due to fiber failure, the
modulus E11 was reduced to zero or if a lamina failure due to matrix failure, the modulus E22 was reduced
to zero. Equilibrium was re-established after the material properties were degraded. Good correlation
between the experimentally obtained and analytically predicted postbuckling responses was obtained for
deflections and surface strains. The Tsai-Wu method more closely estimated the apparent failure
observed than the maximum stress method due to the interaction of the stress components in the failure
criterion.

Hwang and Sun [28] performed a failure analysis of laminated composites by using an iterative three-
dimensional finite element method. A modified form of the Tsai-Wu failure criterion was used to predict
fiber breakage and matrix cracking while quadratic interactive formulas by Lee [15] and Chang [29] were
used to identify delaminations. The progressive failure problem was solved using the modified Newton-
Raphson method for the nonlinear analysis. A post-failure reduced stiffness approach was used where

10
the lamina properties are degraded depending on the failure mode. For fiber failure, only the
corresponding constitutive terms in the stiffness matrix were set to zero. For matrix failure, only the
corresponding transverse and shear properties of the stiffness matrix were removed. If damage included
both fiber and matrix modes, then the entire stiffness matrix of the failed element was removed.
Elements with delaminations were re-modeled with a new free surface at the lamina interface. After the
stiffness properties were degraded, equilibrium was re-established iteratively by the nonlinear analysis.
The three-dimensional analytical results agree favorably with the experimental results for notched and
unnotched specimens loaded in tension. However, the analytical predictions underestimated the
experimental results for angle-plied laminates with holes.

Huang, Bouh, and Verchery [30] implemented a progressive failure analysis of composite laminates
using triangular elements which include transverse shear effects. A new method was introduced for the
calculation of the shear correction factors using a parabolic function. Several failure criteria were used to
determine first-ply failure and distinguish the failure modes into fiber breakage or buckling, matrix
cracking, and delaminations.

Chang and Chang [31] developed a progressive failure damage model for laminated composites
containing stress concentrations. The progressive failure method used a nonlinear finite element analysis
using the modified Newton-Raphson iteration scheme to calculate the state of stress in a composite plate.
A modified form of the Yamada-Sun [32] failure criterion, which accounted for shear deformation and
incorporated Sandhu’s strain energy failure criterion [17], predicted fiber breakage and fiber-matrix
shearing failures. A modified form of Hashin’s failure criterion using Sandhu’s strain energy failure
criterion to account for nonlinear shear deformation was used for the matrix failure modes. For fiber
failure or fiber-matrix shearing, the degree of property degradation was dependent upon the size of the
damage predicted by the failure criterion. The property degradation model was based on a
micromechanics approach for fiber-bundle failure. For fiber failure, both the transverse modulus E22 and
Poisson’s ratio ν12 were set to zero, and the longitudinal E11 and shear moduli G12 were reduced according
to the exponential Weibull distribution. For matrix cracking in a lamina, the transverse modulus and the
Poisson’s ratio were reduced to zero, whereas the longitudinal and shear moduli remained unchanged.
After the properties were degraded, the stresses and strains in the composite laminate were redistributed
by performing the Newton-Raphson iteration again with the updated properties until no additional
failures were found. Chang and Chang applied this progressive failure method to bolted composite joints
[33], and Chang and Lessard to a laminated composite plate containing a hole [34]. Comparisons were
made to experimental results in these studies and reasonable correlation to the data was reported.

Tolson and Zabaras [35] developed a two-dimensional finite element analysis for determining failures
in composite plates. In their finite element formulations, they developed a seven degree-of-freedom plate
element based on a higher-order shear-deformation plate theory. The in-plane stresses were calculated
from the constitutive equations, but the transverse stresses were calculated from the three-dimensional
equilibrium equations. The method gave accurate interlaminar shear stresses very similar to the three-
dimensional elasticity solution. The stress calculations were performed at the Gaussian integration
points. The stresses were then inserted into the appropriate failure criterion to determine if failure had
occurred within a lamina. The maximum stress, Lee, Hashin, Hoffman, and Tsai-Wu failure criteria were
used. Since the Hoffman and Tsai-Wu failure criteria do not determine the mode of failure, the relative
contributions of the shear stress terms, transverse direction terms, and fiber direction terms were used to
determine the failure mode. Once the failure mode was determined, the stiffness was reduced at the
Gauss points. For fiber failure at all four Gauss points, the σ1, σ5, and σ6 stress terms were set to zero by
degrading the corresponding stiffness matrix terms. If less than four Gauss points failed, then the
appropriate stiffness components were proportionally reduced by the fraction of the failed area in the

11
element. Similarly, the σ2, σ4, and σ6 stresses were reduced to zero for matrix failure at all four Gauss
points and proportional values were prescribed if failure occurred at less than the four Gauss points.
Delamination was also predicted by examining the interlaminar stresses according to a polynomial failure
equation. If delaminations occurred, then the stiffness matrix in both lamina adjacent to the delamination
would be reduced such that the σ3, σ4, and σ5 stresses would vanish. Obviously, when all the terms in the
element stiffness matrix have been reduced to zero, the element makes no further contribution to the plate
stiffness and is considered to have undergone total failure. Once the stiffness properties were reduced,
equilibrium iterations were performed until no further failures were predicted. Next, the load was
incremented and the failure process was repeated until total failure of the structure was predicted. The
results obtained from the progressive failure method were compared to experimental results from
composite laminates loaded in uniaxial tension and under a transverse pressure. The Lee criterion gave
the best results for the cases tested.

One of the first finite-element-based failure analyses of composite was performed by Lee [15]. Lee
performed a three-dimensional finite element analysis and used his own direct-mode determining failure
criterion to predict the failures. He determined the stresses at the center of each element and the stresses
at the center of the interface of each element to identify the failure. According to the modes of failure,
the stiffness matrix of the element with failures was modified. Equilibrium was then re-established to
give a new stress distribution and subsequent failure zones. The process was repeated until the ultimate
strength of the laminate was obtained. The procedure was applied to a plate with a central hole subject to
uniaxial and biaxial loadings. Comparisons with experimental results were not reported.

Coats [36,37,38] developed a nonlinear progressive failure analysis for laminated composites that
used a constitutive model describing the kinematics of matrix cracks via volume averaged internal state
variables. The evolution of the internal states variables was governed by an experimentally based
damage evolutionary relationship. The methodology was used to predict the initiation and growth of
matrix cracks and fiber fracture. Most of the residual strength predictions were within 10% of the
experimental failure loads.

Objectives and Scope

The overall objective of this research is to develop a progressive failure analysis methodology for
laminated composite structures. From the literature review, few studies have been performed on
predicting the failure of composite panels based on nonlinear analyses and comparing with experimental
data. Nonlinear analyses should be used in predicting the failure of composite structures to account for
the geometric nonlinearity deformations. The progressive failure methodology in this research is
implemented into a general purpose finite element analysis code called COMET (Computational
Mechanics Testbed) [39,40]. The progressive methodology was implemented into COMET by Pifko
[41,42] using the constitutive material models developed by Moas [43]. The methodology is then
validated by comparing analytical predictions using nonlinear progressive failure analyses with
experimental data. The progressive failure methodology is also applied to a built-up composite structure,
a component of a subsonic composite aircraft. This effort incorporates some of the failure criteria and
material degradation models discussed in the literature review into a single computational structural
mechanics framework. Specific goals of this research include:

12
 1. Establish state-of-the-art perspective on computational models for progressive failure analysis.

2. Develop and implement a progressive failure analysis methodology which accommodates various
formulations for detecting failure and degrading material properties.

3. Compare and assess different formulation methods for detecting failure and degrading material
properties.

4. Perform progressive failure analysis and compare with existing test data.

The scope of the present work is limited to C1 shell elements and neglects the effects of transverse
shear deformations. The lamina properties are assumed to behave in a linear elastic manner and any
nonlinear behavior of the in-plane shear stiffness is ignored. The results from these simulations are
compared with existing experimental and analytical results. No new experimental results are presented.
This work was in partial fulfillment for a graduate degree at Old Dominion University in 1996 [44].

13
 PROGRESSIVE FAILURE METHODOLOGY

To implement and perform the progressive failure methodology described earlier, a structural analysis
software system is needed. The framework to accomplish this task is a structural analysis software
system called COMET (Computational Mechanics Testbed) [39,40]. The Appendix presents a brief
description of COMET, its element processor, its constitutive processor, the implementation process for
the failure and damage models incorporated in the progressive failure methodology, and the nonlinear
analysis solution procedure.

The progressive failure analysis methodology uses C1 shell elements based on classical lamination
theory to calculate the in-plane stresses which ignores transverse shear stresses. The nonlinear Green-
Lagrange strain-displacement relations are used in the element formulation, and large rotations are treated
through the element-independent corotational formulation in COMET. The progressive failure
methodology implemented in COMET accommodates the maximum strain criterion [6], Christensen’s
criterion [16], and Hashin’s criterion [13,14]. When a failure is detected, the progressive failure model
classifies the mode of failure as fiber failure, matrix failure, or shear failure. Two material degradation
models are implemented including instantaneous reduction and gradual reduction of the material
properties for use with the ply-discount theory.

Failure Detection

Once the strains and stresses are known throughout the composite laminate, a failure theory is used to
detect failures for each lamina at a given load level. The failure theory should be able to predict the
failure load and also the mode of failure such as fiber failure and/or matrix failure. Three failure criteria
are considered in this implementation for a progressive failure analysis.

Maximum Strain Criterion

In the maximum strain criterion, failure is assumed to occur if any of the following conditions are
satisfied:

ε1 ≥ X εT or ε1 ≥ X εC fiber failure
ε 2 ≥ YεT or ε 2 ≥ YεC matrix failure (7)
γ12 ≥ Tε shear failure

where X εT , X εC , YεT , YεC , and Tε are the material allowable strains denoted as

X εT = critical tensile strain in fiber direction

X εC = critical compressiv e strain in fiber direction
YεT = critical tensile strain in matrix direction
YεC = critical compressiv e strain in matrix direction
Tε = critical shear strain

14
The absolute value sign on γ12 indicates that the sign of the shear strain is assumed to not affect the failure
criterion. As discussed in Section 1, the maximum strain criterion is a non-interactive failure theory in
strain space. Since the maximum strain criterion provides different conditions for failure, the mode of
failure can be identified as either fiber failure, matrix failure, or shear failure.

Hashin’s Criterion

Hashin and Rotem [14] have proposed a stress-based failure criterion that has the ability to predict the
modes of failure. In this report this failure criterion will be denoted as Hashin’s Criterion. As stated
earlier, observation of failure in unidirectional fibrous composites indicates that there are two primary
failure modes: a fiber mode in which the composite fails due to fiber breakage in tension or fiber
buckling in compression; and a matrix mode in which matrix cracking occurs. Since different failure
mechanisms occur in tension and compression, Hashin further subdivided each failure mode into a
tension and compression mode. The failure modes are summarized for the case of plane stress as
follows:

Tensile Fiber Mode, σ1 > 0

2 2
 σ1  τ 
  +  12  ≥ 1 (8)
 XT   T 

Compressive Fiber Mode, σ1 < 0

2
 σ1 
  ≥ 1 (9)
 XC 

Tensile Matrix Mode, σ 2 > 0

2 2
 σ2  τ 
  +  12  ≥ 1 (10)
 YT   T 

Compressive Matrix Mode, σ 2 < 0

σ2  Y  2   σ 2  τ 2
 C
 − 1 +  2  +  12  ≥ 1 (11)
YC  2T    2T   T 

Christensen’s Criterion

Christensen [16] introduced a quasi-three-dimensional lamination theory which accounted for the out-
of-plane stress terms. In related work, Christensen then proposed a strain-based failure criterion which
distinguished the modes of failure into either fiber failure or fiber/matrix interaction failure. The
corresponding equations for failure are as follows:

15
Fiber Failure

X εC ≤ ε 1 ≤ X εT (12)

Fiber/Matrix Failure

β ε kk + eij eij ≥ k 2 (13)

where β and k are determined from experimental failure data and eij is the deviatoric strain tensor
given by

1
eij = ε ij − δ ij ε kk (14)
3

In Christensen’s analyses, the two parameters β and k in Equation (13) are evaluated to fit failure data
for tensile and compressive failure with no shear stress.

Damage Modeling

A common method for degrading the material properties in laminates with fiber or matrix failure is
the ply-discount method [18]. This method belongs to the instantaneous unloading category described in
the first section. With this method, one or more of the material properties (or constitutive components)
of a location with failures are set equal to zero or reduced to a fraction of the original values. It is
assumed that the material degradation is restricted to the ply that fails.

In the present implementation of the material degradation model, the material properties which are
degraded depend upon the failure criterion chosen. The maximum strain failure criterion has three
options for material degradation. The first two are for unidirectional composites, and the third one is for
a fabric composite. These options allow failure in one direction to be independent of other failures, or
have the failures in one direction to cause failure in other directions. In Option 1, when fiber failure is
detected, the moduli E11 and the Poisson’s ratio ν12 are degraded. Similarly for matrix failure, the moduli
E22 and the Poisson’s ratio ν12 are degraded. The Poisson’s ratio ν12 is reduced to zero if a failure occurs
in both of these options to allow the constitutive matrix for the lamina to remain symmetric. Finally for
shear failure, only the shear modulus G12 is degraded. Options 2 and 3 are similar to Option 1 but also
include an induced coupling based on heuristic arguments. For example, fiber failure often induces shear
failure. Table 2 shows the material degradation model for the maximum strain criterion for each of the
three degradation options. The other two failure criteria implemented, Hashin’s criterion and
Christensen’s criterion, both include induced shear failure for fiber and matrix failure. Table 3
summarizes the material degradation for these two failure criteria.

16
 Table 2. Options for Material Degradation for Maximum Strain Criterion
Option 1 Option 2 Option 3
Primary Failure Induced Degraded Induced Degraded Induced Degraded
Direction Additional Properties Additional Properties Additional Properties
Failure Failure Failure
Fiber Failure None E11 , υ12 Shear E11 , G12 , υ12 Shear E11 , G12 , υ12
Matrix Failure None E 22 , υ12 Shear E22 , G12 , υ12 Shear E22 , G12 , υ12
Shear Failure None G12 Matrix E22 , G12 , υ12 Fiber E11 , G12 , υ12

Table 3. Material Degradation for Hashin’s and Christensen’s Failure Criteria

Hashin’s Criterion Christensen’s Criterion
Primary Failure Induced Degraded Induced Degraded
Direction Additional Properties Additional Properties
Failure Failure
Fiber Failure Shear E11 , G12 , υ12 Shear E11 , G12 , υ12
Matrix Failure Shear E 22 , G12 , υ12 Shear E22 , G12 , υ12

In the damage modeling implemented in this study, the material properties can be slowly degraded
over subsequent nonlinear load steps or can be instantaneous reduced to zero if a failure occurs. The
properties are assumed to be a constant multiple of the original material properties of the undamaged
material if no previous failures have occurred. If failures have occurred, then the degraded properties are
assumed to be a constant multiple of the updated material properties which have been previously
degraded. The material properties which are degraded depend upon the failure mode type as already
discussed. The material properties are degraded according to

E11 new = α E11 previous

E 22 new = α E 22
previous

G12 new = α G12 where α = 10 -n (0 ≤ n ≤ 20, n = integer ) (15)

previous

υ12 new = α υ12

previous

υ 21 new = 0

Therefore, if n = 0 then the properties are not degraded if failures occur. If n = −1 , then the material
properties are degraded to 10% of the previous material properties each time a failure occurs. The
constitutive matrix for the degraded lamina remains symmetric. In the present implementation, the
material properties are automatically reduced to zero if n = 20 in which α = 10-20. Studies are performed
to determine the effect of this delayed material degradation implementation.

17
 Nonlinear Analysis Solution Continuation

After the element material properties have been degraded for a failed ply, the historical database for
the element material properties is updated for the current load step in the nonlinear analysis. The tangent
stiffness matrix in the nonlinear analysis is then recalculated for the new element material properties and
the Newton-Raphson solution procedure continues. In general, static equilibrium must also be re-
established after the material properties have been degraded by repeating the nonlinear analysis at the
current load step. However, by incrementing the nonlinear analysis by small load increment sizes,
changes in the force imbalance vector should be very small and the step of re-establishing equilibrium
may be omitted.

18
 NUMERICAL RESULTS

The progressive failure methodology described in the previous sections was successfully implemented
in COMET. Several laminated composite structures are now considered to evaluate the performance of
the progressive failure model, and simulation results are compared to their experimental results. The first
problem is a composite laminate under rail-shear loading. The second problem is a composite laminate
with a central circular hole under tension loading. The third problem is a laminated composite panel
subject to an axial compressive load. The fourth problem is a laminated composite panel with an offset
circular hole subject to an axial compressive load. The final problem is a laminated composite blade-
stiffened panel with discontinuous stiffener loaded in axial compression. Numerical results obtained
using the present progressive failure methodology are compared with other reported analytical and
experimental results.

These problems are of interest because of the available experimental data and their applicability in
aircraft structures. The first two problems are essential membrane problems with the nonlinear behavior
due to material failure. The latter problems involve combined membrane and bending behavior and
combined material degradation and geometric nonlinearities.

For the problems considered, determining the structural response involves several steps using
COMET. For the problems with compressive loadings, the first step is to perform a linear static analysis.
Using the linear static analysis, the primary equilibrium path is found which produces no out-of-plane
deflection. The next step is a linear stability analysis to find the point at which the primary path will
bifurcate to a secondary equilibrium path. Along the secondary path, the transverse deflections will
increase. The third step is forming the initial geometric imperfection which serves as a trigger for the
geometric nonlinearities. For the problems with tension or shear loadings, these first three steps are
omitted. For all problems, the next step is to perform an elastic nonlinear analysis to understand how the
structure behaves without any material failures. Finally, a progressive failure analysis which includes
combined material degradation and geometric nonlinear analysis is performed.

In all problems considered, a prescribed displacement is applied to the structure to simulate the test
conditions. The initial displacement of the nonlinear analysis is applied as an initial displacement factor
times the applied displacement. Each successive displacement in the nonlinear analysis is incremented
by a displacement increment times the applied displacement. The modified energy norm error
convergence criterion with estimates of total strain energy is used in the nonlinear analysis solution
unless otherwise stated. The specified energy norm error tolerance used for convergence in the nonlinear
analysis is set to 10-3. The COMET element used in all analyses is the ES5/ES410 which is a 4-node
quadrilateral element with five Gauss quadrature points (see Appendix). Through the thickness
integrations of the element use three integration points for each layer in the laminate. The total number
of layer-integration points in an element is 5x3xNumber-of-layers.

Rail-Shear Panel

Rail-shear fixtures are frequently used to measure ply in-plane shear strength. The ply shear strength
is defined as the ultimate shear strength which is the shear load at failure divided by the area over which
the load was applied. Note that ply shear strength determined this way corresponds to the failure strength
of the laminate rather than the ply shear strength corresponding to the first matrix cracking.

19
 Chang and Chen [45] performed a series of rail-shear tests on [0n / 90n]s clustered laminates where the
number of layers n was varied. Their results showed that as the laminate thickness increased (larger
number of n clustered plies), the in-plane shear strength decreased.

Problem Statement

The rail-shear specimen used in this analysis is a cross-ply laminate performed by Chang and Chen
[45] and reported in the analytical results by Shahid [46]. The specimen used is a 24-ply [06 / 906]s
laminate fabricated from a T300/976 graphite-epoxy composite with a ply thickness of 0.0052 inches.
The specimen is 6-inches long and 1-inch wide. In order to represent the boundary conditions of the
specimen in a rail-shear fixture, one edge of the specimen is firmly fixed, while on the other parallel
edge, deformations are only allowed parallel to the edge in the y-direction and restrained from motion in
the x-direction. Upon loading, a displacement increment is applied along the latter edge. The material
properties for the T300/976 material system are shown in Table 4.

Table 4. Material Properties for T300/976 Material System

Material Properties Value [46,47]
Longitudinal Young’s Modulus E11 20.2 msi
Transverse Young’s Modulus E22 1.41 msi
Poisson’s Ratio ν12 0.29
In-Plane Shear Modulus G12 0.81 msi
Longitudinal Tensile Strength XT 220.0 ksi
Longitudinal Compression Strength TC 231.0 ksi
Transverse Tensile Strength YT 6.46 ksi
Transverse Compression Strength YC 36.7 ksi
In-Plane Shear Strength T 6.0 ksi

A finite element model of this specimen with 48 4-node elements along the length and 8 4-node
elements along the width is shown in Figure 3. The accuracy of this spatial discretization is established
by considering a finer finite element mesh of 72 4-node elements along the length and 12 4-node
elements along the width.

20
 Applied Displacement, u

6 in. Clamped Boundary Conditions

1 in.

Figure 3. Geometry, loading, and boundary conditions of rail-shear specimen with 48 x 8 finite element mesh.

21
 Structural Response

The structural response of the rail-shear specimen is studied. Progressive failure studies are
performed on the model to evaluate the three failure criteria methods, finite element mesh refinement,
rate of material degradation, and finally displacement increment of the progressive failure analysis. For
all analyses, an initial displacement of 0.00050 inches is applied to the rail-shear specimen.

In the first progressive failure analysis study on the rail-shear specimen, the effect of mesh refinement
on the model is analyzed. Hashin’s criterion is chosen as the failure criterion. Two rail-shear models
were analyzed: one with a mesh refinement of 48 x 8 elements and the other with a refinement of 72 x 12
elements. The displacement increment is 0.00050 inches and the degradation factor α is 10-20 (properties
were set to zero). The progressive failure results are compared to the analytical ultimate failure load from
Shahid [46] and the test data from Chang and Chen [45] in Table 5. The results indicate that the mesh
refinement of 48 x 8 elements is sufficient predicting the final failure load. However, the mesh
refinement did affect the prediction of the first ply failure load.

Table 5. Rail-Shear Problem: Effect of Mesh Size - Hashin’s Criterion.

Mesh Size First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
48 x 8 elements 2406 4016 Matrix Tension
(Dominated by Shear Failure)
72 x 12 elements 2101 4008 Matrix Tension
(Dominated by Shear Failure)
Shahid’s Results [46] Unavailable 4010 Shear
Test Data [45] Unavailable 3850 Shear

Since equilibrium is not re-established after the material properties have been degraded in the current
progressive failure analysis procedure, the load increment sizes must be small for accurate results. In the
next progressive failure analysis study, the effect of the displacement increment size for the nonlinear
analysis is studied. Two edge displacement increment sizes are studied: one with a displacement
increment of 0.00025 inches and the other with a displacement increment of 0.00050 inches. Again,
Hashin’s criterion is chosen for the failure criterion and the model has a mesh refinement of 48 x 8
elements. The material degradation factor α is again chosen to be 10-20 (properties were set to zero). The
results of this study are shown in Figure 4. Table 6 summarizes the progressive failure load predictions
with the test data and Shahid’s analytical results. The analysis using a displacement increment of
0.00050 inches had a converged solution for an applied edge displacement of 0.075 inches, but failed to
get a converged solution at 0.080 inches. The analysis using a displacement increment of 0.00025 inches
also had a converged solution for an applied edge displacement of 0.075 inches, and also had a converged
solution at an additional load step at 0.0775 inches in the progressive failure analysis. The results
indicate that the displacement increment sizes of 0.00050 inches and 0.00025 inches predict nearly the
same failure load at a given applied edge displacement. Even though the analysis using the smaller
displacement increment of 0.00025 inches predicted a higher final failure load, this final failure load was
only 2.7% higher than the final failure load predicted by the larger displacement increment of 0.00050
inches. Therefore, a displacement increment size of 0.00050 inches is sufficient for accurate progressive
failure prediction. The jump in the load-deflection curve at a displacement of 0.005 inches for the
progressive failure analyses was due to a load redistribution after a large number of failures occurred.

22
The progressive failure results also differed slightly from Shahid’s results. This can be attributed to
Shahid modeling the nonlinear behavior of the shear modulus in his analysis which was not modeled in
the COMET progressive failure analysis.

Table 6. Rail-Shear Problem: Effect of Displacement Increment Size, Hashin’s Criterion

Displacement Increment First Ply Failure Load Final Failure Dominant Failure
Size (lbs.) Load (lbs.) Mode Type
0.00050 2406 4016 Matrix Tension
(Dominated by Shear Failure)
0.00025 2406 4126 Matrix Tension
(Dominated by Shear Failure)
Shahid’s Results [46] Unavailable 4010 Shear
Test Data [45] Unavailable 3850 Shear

The next study to be analyzed is the effect of the material degradation factor α on the progressive
failure analysis results. Three values of α are chosen to be studied: 10-1, 10-2, and 10-20. For this study
Christensen’s criterion is chosen for the failure criterion, the mesh refinement is 48 x 8 elements, and the
displacement increment is 0.00050 inches. Table 7 shows the progressive failure results of this study for
the ultimate failure load with the test data and Shahid’s analytical results. The results show that the
material degradation factor α has little effect on the final failure prediction in this example.

Table 7. Rail-Shear Problem: Effect of Material Degradation Factor α, Christensen’s Criterion

Material degradation First Ply Failure Load Final Failure Dominant Failure
factor, α (lbs.) Load (lbs.) Mode Type
10-1 2105 3972 Fiber-Matrix interaction
10-2 2105 3965 Fiber-Matrix interaction
-20
10 2105 3964 Fiber-Matrix interaction
Shahid’s Results [46] Unavailable 4010 Shear
Test Data [45] Unavailable 3850 Shear

23
 6000

5000

4000

Nonlinear Analysis
Shear Load, P (lbs.)

3000 Shahid's Results [45]

2000

1000

0
0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

Applied Edge Displacement, u (in.)

Figure 4. Rail-Shear problem: Effect of displacement increment size, Hashin’s criterion.

24
 In the next study, the options of the maximum strain criterion described earlier are investigated. For
this study, the displacement increment is set at 0.0005 inches and α is set at 10-20. Table 8 presents the
results from this study. Option 1 of the maximum strain criterion, which allows no induced failures,
comes closest to the test data.

Table 8. Rail-Shear Problem: Comparison of Maximum Strain Options

Option First Ply Failure Load Final Failure Dominant Failure Mode
(lbs.) Load (lbs.) Type
1 2707 4065 Shear
2 2707 4312 Shear
3 2707 4269 Shear
Shahid’s Results [46] Unavailable 4010 Shear
Test Data [45] Unavailable 3850 Shear

A summary of the results for the three failure criteria is presented in Table 9. A load-deflection curve
is included in Figure 5 which compares the three failure criteria with Shahid’s analytical results. The
load-displacement curve data with the test data was not available for comparison. All progressive failure
results agree very well with Shahid’s analytical results. Figure 6 shows the structural response of the
panel at final failure using Christensen’s criterion. The left figure indicates the percentage failure of an
element which is quantified as the percentage of ply-integration points with failures within an element
(total number of integration points with failures divided by the total number of integration points in an
element). The right figure depicts the Nxy stress resultant distribution. Dark regions indicate high values.
The progressive failure analyses of the other criteria show similar structural responses. The panel
experiences 100% failure along the edges of the panel where the shear loading is applied. Consequently,
the load carrying capability along the edges is zero.

Table 9. Rail-Shear Problem: Summary of Progressive Failure Results

Failure Criterion First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
Hashin, α = 10-20 2406 4016 Matrix Tension
Christensen, α = 10-20 2105 3964 Fiber-Matrix interaction
Maximum Strain 2707 4065 Shear
α = 10-20, Option 1
Shahid’s Results [46] Unavailable 4010 Shear
Test Data [45] Unavailable 3850 Shear

25
 5000

4500

4000

3500

3000
Shear Load, P (lbs.)

2500

2000

Christensen's Criterion
1500 Hashin's Criterion
Maximum Strain Criterion, Option 1
Shahid's Results [45]
1000

500

0
0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080 0.0090
Applied Edge Displacement, u (in.)

Figure 5. Rail-shear problem: Load-deflection results, α = 10-20.

26
 Max := 100% Max:= 651.9

90 % 587.7
80 % 521.5
70 % 456.3
60 % 391.2
50 % 326.0
40 % 260.8
30 % 195.6
20 % 130.4
10 % 65.2

Min := 0 % Min:= 0.0

27

a) Percentage Failure b) Nxy Stress Distribution, lb./in.

Figure 6. Rail-shear problem: Structural response at final failure, (3964 lbs.).

Christensen’s Criterion, α = 10-20
 Tension-Loaded Laminate with Hole

Problem Statement

In order to assess the accuracy of the progressive failure methodology, a 20-ply tensile specimen
containing a centrally located circular hole is considered. The calculated results are compared to
experimental results by Chang and Chang [31] and Tan [48]. The composite laminate is 8-inches long
and 1-inch wide with a hole diameter of 0.25 inches. The thickness of each ply is 0.00515 inches, and
the laminate stacking sequence is [0 / (± 45)3 / 903 ] s . The specimen is fabricated from T300/1034-C
graphite/epoxy. A finite element mesh for the laminate with the boundary conditions is shown in Figure
7. The sides of the laminate are free, and the loaded ends are clamped. The lamina properties for this
laminate are given in Table 10.

Table 10. Material Properties for T300/1034 Material System

Material Properties Value [31]
Longitudinal Young’s Modulus E11 21.3 msi
Transverse Young’s Modulus E22 1.65 msi
Poisson’s Ratio ν12 0.30
In-Plane Shear Modulus G12 8.97 msi
Longitudinal Tensile Strength XT 251.0 ksi
Longitudinal Compression Strength XC 200.0 ksi
Transverse Tensile Strength YT 9.65 ksi
Transverse Compression Strength YC 38.9 ksi
In-Plane Shear Strength T 19.4 ksi

Structural Response

The structural response of the laminate with an open circular hole is studied. An initial displacement
of 0.0005 inches is applied to the laminate. The progressive failure analysis of the tensile specimen is
analyzed using a displacement increment of 0.001 inch and a material degradation factor of α = 10-20. A
study was performed on the effect of the finite element mesh of the laminate. The original finite element
mesh with 768 elements is shown in Figure 7. Figure 8 shows a coarse mesh with 568 elements and a
fine mesh with 1,264 elements of the laminate model. Christensen’s criterion was used for the failure
criteria in this study. The first mesh (Figure 7) had eight rings of elements around the hole. The coarse
mesh had six rings of elements around the hole. The fine mesh had 12 rings of elements round the hole.
Table 11 summarizes the results of this study. These results indicate the sensitivity of the local stress
distribution near the hole in predicting final failure. The fine mesh predicted the lowest final failure load.
However, since the difference was not very significant and because the fine mesh analysis needed more
computational time, the mesh with 768 elements is used for all other analyses. Another study is
performed to analyze the effect of the material degradation factor. However, varying the α parameter had
little effect on the progressive failure results.

28
 1 in.

0.25 in.

8 in.
29

free free

y z

Figure 7. Geometry, loading, and boundary conditions of tension-loaded laminate with hole.
30

a) Coarse Mesh Model b) Fine Mesh Model

Figure 8. Meshes for tension-loaded laminate with hole.

 Next, a progressive failure analysis is performed using Hashin’s criterion. The mesh with 768
elements is used in this analysis with the same parameters of displacement increment sizes and material
degradation factor used in the previous study. A comparison of the progressive failure results for
Hashin’s criterion is shown in Table 12 with experimental results and the results using Christensen’s
criterion. The Maximum Strain criterion was not included in this study because it does not include any
strain interaction between the failure modes. The progressive failure results agree reasonably well with
the experimental results from Chang [31]. A load-displacement curve for these analyses is shown in
Figure 9.

Figures 10 and 11 show comparisons of the structural response at final failure using Hashin’s and
Christensen’s criteria. Each figure displays a close-up view of the region near the hole. The left figure
indicates the percentage of failure within an element, and the right figure depicts the Nx stress resultant
distribution. Dark regions indicate high values. The results indicate that for this problem there is little
difference between the Hashin’s criterion and Christensen’s criterion in predicting the failure loads. The
Nx stress resultant distribution for Christensen’s criterion showed a larger high stress region around the
edge of the hole than Hashin’s criterion.

Table 11. Tension-Loaded Laminate with Hole: Effect of Finite Element Mesh, Christensen’s Criterion
Number of Elements First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
568 1520 3263 Fiber/Matrix Interaction
768 1520 3261 Fiber/Matrix Interaction
1,264 1446 3195 Fiber/Matrix Interaction

Table 12. Tension-Loaded Laminate with Hole: Comparison of Failure Results

Failure Criterion First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
Hashin’s Criterion 1520 3212 Matrix Tension
Christensen’s Criterion 1520 3261 Fiber/Matrix Interaction
Experimental Results Unavailable 3523 Unavailable
[31]

31
 4000

Christensen's Criterion
Final Failure
3500 Hashin's
Criterion
Final Failure

3000

2500
Tensile Load, P (lbs.)

2000
First Ply
Failure

1500 Hashin's Criterion

Christensen's Criterion

1000

500

0
0.00 0.01 0.02 0.03 0.04 0.05
Applied Displacement, u (in.)

Figure 9. Tension-loaded laminate with hole: Load-deflection results, α = 10-20.

.

32
 Max := 100%
Max:= 5574.9

90 % 5005.7
80 % 4436.6
70 % 3867.4
60 %
3298.3
50 % 2729.1
40 % 2160.0
30 % 1590.8
20 % 1021.7
10 % 452.5

Min := 0 % Min:= -116.6

33

a) Percentage Failure b) N x Stress Distribution, lb./in.

Figure 10. Tension-loaded laminate with hole: Structural response at final failure (3212 lbs.).
Hashin’s criterion, α = 10-20
 Max := 100% Max:= 5531.0

90 % 4966.9
80 % 4402.9
70 % 3838.8
60 %
3274.7
50 % 2710.6
40 % 2146.6
30 % 1582.5
20 % 1018.4
10 % 454.3

Min := 0 % Min:= -109.7

34

a) Percentage Failure b) Nx Stress Distribution, lb./in.

Figure 11. Tension-loaded laminate with hole: Structural response at final failure (3261 lbs.).
Christensen’s criterion, α = 10-20
 Compression-Loaded Composite Panel

Problem Statement

The next problem is a composite rectangular panel loaded in axial compression. The panel length is
20.0 inches and the width is 6.75 inches and is denoted Panel C4 in the experimental results reported by
Starnes and Rouse [49]. The panel used is a 24-ply orthotropic lay-up. The thickness of each ply is
0.00535 inches and the laminate stacking sequence is [ ±45 / 0 2 / ± 45 / 0 2 / ± 45 / 0 / 90] s . The panel is
fabricated from unidirectional Thornel 300 graphite-fiber tapes preimpregnated with 450K cure Narmco
5208 thermosetting epoxy resin. The lamina properties for this panel are given in Table 13.

Table 13. Material Properties for C4 Panel, T300/5208 Material System

Material Properties Value [49,27]
Longitudinal Young’s Modulus E11 19.0 Msi
Transverse Young’s Modulus E22 1.89 Msi
Poisson’s Ratio ν12 0.38
In-Plane Shear Modulus G12 0.93 Msi
Longitudinal Tensile Strength XT 200.0 Ksi
Longitudinal Compression Strength XC 165.0 Ksi
Transverse Tensile Strength YT 11.74 Ksi
Transverse Compression Strength YC 27.41 Ksi
In-Plane Shear Strength T 10.0 Ksi

The finite element model of this panel has 40 4-node elements along the length and 14 4-node
elements along the width as shown in Figure 12. The loaded ends of the panel are clamped by fixtures,
and the unloaded ends are simply supported by knife-edge supports to prevent the panel from buckling as
a wide column.

35
 6.75 in.
x
u = v = w = θ x = θ y = θz = 0

20 in.
w = θ y = θz = 0 w = θy = θz = 0

y
z
w = θx = θ y = θ z = 0
u

Figure 12. Geometry, loading, and boundary conditions for C4 panel.

36
 Structural Response

The structural response of the C4 panel is studied. A fringe plot of the first buckling mode from the
linear stability analysis is shown in comparison to the moiré-fringe plot from the Starnes and Rouse
experiment [49] in Figure 13. The results indicate that the first buckling mode from the analysis and the
experiment are in agreement with each other. The first buckling mode has two longitudinal half-waves
with a buckling mode line at the panel midlength. An initial geometric imperfection is formed by using
the first buckling mode shape normalized by its maximum component. This normalized mode shape is
then scaled by 5% of the panel thickness and added to the nodal coordinates. The eccentricity is added to
the initial geometry to allow efficient progress past the critical buckling point, but does not affect the
results in the postbuckling range. The initial displacement applied to the panel is 0.001 inch. A
comparison between the test results and the elastic nonlinear analysis (no damage) results is shown in
Figure 14. The elastic nonlinear analysis response without damage correlates very well with the
experimental result up to the final failure of the panel and then continues on until a maximum load level
is reached.

Progressive failure analyses using Hashin’s and Christensen’s criteria were performed on the C4 panel
using a material degradation factor of α = 10-20. Similar progressive failure results have also been
performed by Engelstad et al. [27]. A displacement increment of 0.0025 inches is used for the first 10
steps in the progressive failure analysis. Then a smaller increment of 0.001 inch is chosen for the next 5
load (steps 11-15) so the analysis could pass the buckling load. In load steps 16-40, a displacement
increment of 0.0025 inches is used for the analysis. Finally, a displacement increment of 0.001 inch is
used near the failure of the panel (steps 41 to failure).

The progressive failure results for the C4 panel are presented in Figures 15 and 16 for Hashin’s and
Christensen’s criteria. Figure 15 shows the comparison of experimental and analytical out-of-plane
deflections near a point of maximum deflection as a function of the applied load. The analytical results
correlate reasonably well to the experimental results up to the buckling load and then are lower than the
experimental results in the postbuckling regime. Figure 16 shows a comparison of analytical and
experimental end shortening results as a function of the applied load. The progressive failure results for
load-end shortening also agree with the experimental results. At some point in the progressive failure
analysis, a dramatic change in the slope of the end shortening curve indicates an inability for the panel to
support any additional load. This location is designated as the analytical failure load, and the final
experimental data point is called the test failure load. The final failure loads predicted by both criteria are
very close to each other. Hashin’s criterion was less than 3% from the test failure load and Christensen’s
criterion was 8% from the test failure load. However, the first ply failure (FPF) load of Christensen’s
criterion was much lower than Hashin’s criterion. Both failure criteria predicted slightly higher failure
loads than the results from the experiment. Starnes and Rouse reported that the test failure mode was due
to transverse shear effects near the node line in the buckle pattern as shown in Figure 17. Since the
current progressive failure analysis capability does not include a transverse shear failure mode, these
analytical results cannot capture this failure mode. Despite this, the progressive failure analysis
predictions still are in good agreement with the test results. Table 14 provides a summary of the failure
loads (first ply failure and final failure) and the dominant failure mode type for both failure criteria and
the test results. The dominant failure mode of Christensen’s criterion is fiber/matrix interaction and the
dominant mode of Hashin’s criterion is failure in matrix tension. In the experiment, the panel failed
along the nodal line due to transverse shear failure as a result of coupling of large out-of-plane deflections
and high transverse shear strains near the panel’s edges.

37
 The structural response of the C4 panel at final failure is given in Figures 18 and 19 using Hashin’s
criterion. Figures 20 and 21 show the structural response at the final failure load using Christensen’s
criterion. Figures 18 and 20 display the percentage of failures within an element, the Nx stress
distribution, and the out-of-plane deflection of selected steps in the progressive failure analysis while
Figures 19 and 21 show the Ny and Nxy stress distribution. The results show that the damage is
concentrated along the nodal line just as the test results showed. The final failure results obtained using
Hashin’s criterion in Figure 18 and Christensen’s criterion in Figure 20 also reveal the possible failure
event. The deflection pattern exhibits large deflections and high local gradients. The high in-plane
membrane stress resultants and their gradients near the buckle nodal line contribute to the failure
propagation and final failure.

Table 14. C4 Panel: Comparison of Failure Results, α = 10-20

Failure Criterion First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
Christensen’s Criterion 18615 22454 Fiber/Matrix Interaction
Hashin’s Criterion 21778 23526 Matrix Tension
Test Results [49] Unavailable 21910 Transverse Shear

38
 .00638

.00553

.00468

.00383

.00298

.00213

.00128

.000432

-.000418

-.00127
39

-.00212

-.00297

-.00382

-.00467

-.00552

-.00637

a) Fringe Plot of FEM Results, in. b) Photograph of Moiré-Fringe Pattern [49]

Figure 13. C4 panel: Comparison of experimental and analytical linear buckling mode 1.
 35000

30000

25000
Applied Load, P (lbs.)

20000

15000

10000 u

5000 Elastic Nonlinear Analysis (No Damage)

Linear Response
Experimental Results [49]

0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
End Shortening, u (in.)

Figure 14. C4 panel: End-shortening results.

40
 30000

25000

Christensen's Criterion
α = 10−20
Hashin's Criterion
Experimental Results [49]
20000
Applied Load, P (lbs.)

15000

10000

5000

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Out-of-Plane Deflection, w (in.)

Figure 15. C4 panel: Out-of-plane deflection comparison.

41
 30000

25000
Hashin's Criterion
First Ply Failure

Christensen's Criterion
First Ply Failure Test Failure
20000
Applied Load, P (lbs.)

15000

Christensen's Criterion
α = 10−20
Hashin's Criterion
10000
Experimental Results [49]

5000

0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
End Shortening, u (in.)

Figure 16. C4 panel: End-shortening comparison.

42
Figure 17. C4 panel: Photograph of failure mode from Starnes and Rouse experiment [49].

43
 Max:= 40.0% Failure
-548 .354
-1069 .307
-1589 .260
-2110 .212
-2631 .165
Min:= 0.0% Failure
-3151 .118
-3672 .0708
-4192 .0235
-4713 -.0237
44

-5233 -.0709
-5754 -.118
-6274 -.165

-6795 -.213
-7315 -.260
-7836 -.307
-8357 -.354

a) Percentage Failure b) Nx Stress Distribution, lb./in. c) Out-of-Plane Deflection, in.

Figure 18. C4 panel: Structural response at final failure (23526 lbs.).

Hashin’s criterion, α = 10-20
 987. 788.

855. 680.

724. 573.

592. 466.

461. 358.

329. 251.

198. 144.

66.2 36.4
45

-65.3 -70.9
-197. -178.

-328. -286.

-460. -393.

-591. -500.

-723. -608.
-854. -715.

-986. -822

d) Ny Stress Distribution, lb./in. e) Nxy Stress Distribution, lb./in.

Figure 19. C4 panel: Structural response at final failure (23526 lbs.), continued.
Hashin’s criterion, α = 10-20
 Max:= 74.2% Failure

-511. .353

-1019. .306

-1526. .259

-2033. .212

-2540. .165
Min:= 0.0% Failure
-3047. .118

-3543. .0713

-4061. .0244

-4568. -.0226
46

-5075. -.0696
-5582. -.117
-6089 -.163
-6597 -.210
-7104 -.257
-7611 -.304
-8118 -.351

a) Percentage Failure b) Nx Stress Distribution, lb./in. c) Out-of-Plane Deflection, in.

Figure 20. C4 Panel: Structural response at final failure (22454 lbs.).

Christensen’s criterion, α = 10-20
 757. 971.
655. 842.
554. 713.
453. 584.
352. 455.
250. 326.
149. 197.

47.8 68.1
-53.4 -60.8
-155. -190.
47

-256. -319.
-357. -448.
-459. -577.

-560. -706.
-661. -835.

-762. -964.

d) Ny Stress Distribution, lb./in. e) Nxy Stress Distribution, lb./in.

Figure 21. C4 Panel: Structural response at final failure (22454 lbs.), continued.
Christensen’s criterion, α = 10-20
 Compression-Loaded Composite Panel with Hole

Problem Statement

Consider next a composite panel with an offset circular hole loaded in compression. The panel used
is a 24-ply quasi-isotropic laminate. Two identical specimens were tested and denoted as Panel H3 and
Panel H4 in the experimental results reported by Starnes and Rouse [49]. Each panel length is 20.0
inches, and the width is 5.5 inches. The hole is offset from the panel center in the length direction (7.5
inches from the bottom of the panel) such that it is at or near a buckle crest. Each panel is fabricated
from unidirectional Thornel 300 graphite-fiber tapes preimpregnated with 450K cure Narmco 5208
thermosetting epoxy resin. The lamina properties for this panel are given in Table 13. The thickness of
each ply is 0.00574 inches, and the laminate stacking sequence is [±45 / 0 / 90 / ± 45 / 0 / 90 / ± 45 / 0 / 90]s .

The finite element model of this panel is shown in Figure 22. The loaded ends of the panel are
clamped by fixtures, and the unloaded ends are simply supported by knife-edge supports to prevent wide-
column buckling of the panel.

Structural Response

The H3/H4 panel configuration is analyzed to investigate analytical predictions of failure for a
specimen with a hole. Engelstad et al. [27] also performed similar progressive failure analyses on this
panel. A comparison of the first buckling mode fringe plot from the experimental and analytical results
is shown in Figure 23. Four longitudinal halfwaves can be seen for the first buckling mode shape. These
results agree with the experimental results in Ref. [49] and the analytical results by Engelstad. An
imperfection of 5% of panel thickness for mode 1 is used to initiate the nonlinear analysis and
progressive failure analysis into the postbuckling region. The eccentricity is added to the initial geometry
to allow efficient progress past the critical buckling point, but does not affect the results in the
postbuckling range. Hashin’s criterion is used for the progressive failure analysis since it includes more
failure modes than the other two criteria. An initial displacement of 0.001 inch is applied to the panel
with a displacement increment of 0.0025 inches with a material degradation factor of α = 10-20. The
displacement increment error norm is used as the convergence criterion for the progressive failure
analysis. The displacement increment error norm was used because the modified energy norm error
convergence criterion had numerical convergence difficulties.

Figure 24 contains end-shortening analytical and experimental comparisons. The only test data
available for the H3 panel is for the final failure load. Figure 24 shows good correlation between the
experimental results and the analytical results. However, the progressive failure analysis encountered
convergence problems and under-predicted the final failure loads from the tests, 8.3% for the H3 panel
and 14.4% for the H4 panel, respectively. The out-of-plane deflection comparison near the hole region is
shown in Figure 25. Experimental results were available only for the H4 panel. Again, good agreement
for the out-of-plane deflection exists between the H4 panel test results and the analytical results until the
progressive failure analysis stopped due to convergence problems. Element distortion around the hole
could be a contributing factor to the convergence problems. However, no further refinement around the
hole region was performed.

48
u =v=w=0
θx = θ y = θz = 0
x
5.50 in.

w = θ y = θz = 0 w = θy = θz = 0

Hole diameter = .75 in.

20 in.

7.5 in.

u
v=w=0
θx = θ y = θz = 0

Figure 22. Geometry, loading, and boundary conditions for H3/H4 panel.

49
 Figures 26 and 27 show the structural response of the panel at the final failure load predicted by the
progressive failure analysis. Figure 26 depicts the fringe plot of the out-of-plane deflections and the Nx
stress resultant while Figure 27 shows the Nxy stress resultant and a plot of the percentage of failures
within an element (percentage of ply-integration points with failures in an element). Four longitudinal
halfwaves develop in the postbuckled out-of-plane deflection shapes for this panel. The figures shows
that high Nx and Nxy stress resultants exist near the hole region. This initiates the failure of the panel
around the hole region as shown in the figure on the right in Figure 27. The H3 panel developed local
failures around the hole region. In the experimental results [49], both the H3 and H4 panels also
experienced failures along the nodal line away from the hole due to transverse shear mechanisms. Figure
26 shows that the a high Nx stress region is located along the nodal line at the edge of the panel.
However, no failures occurred in this region in the progressive failure analysis which neglected
transverse shear effects. Table 15 summarizes the failure mechanisms for the experimental and analytical
results.

Table 15. H3/H4 panel: Comparison of Failure Results

Failure Criterion First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
Hashin’s Criterion 15507 17313 Matrix Tension/Fiber
Compression
H3 Panel Test [49] Unavailable 18884 Unavailable
H4 Panel Test [49] Unavailable 20233 Unavailable

50
 .00491

.00410

.00328

.00247

.00166

.000842

.0000283

-.000786

-.00160

-.00241
51

-.00323

-.00404

-.00486

-.00567

-.00648

-.00730

a) Fringe Plot of Analytical Results, in. b) Photograph of Moiré-Fringe Pattern from Starnes
and Rouse Test [49]

Figure 23. H3/H4 panel: Comparison of experimental and analytical linear buckling mode 1.
 25000

H4 Panel
Test Failure

20000
PFA
Final Failure
First Ply
Failure H3 Panel
Test Failure

15000
Applied Load, P (lbs.)

10000
u

Nonlinear Analysis (No Damage)

Progressive Failure Analysis (Hashin's Criterion)
5000
H3 Panel Test Results [49]
H4 Panel Test Results [49]

0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
End Shortening, u (in.)

Figure 24. H3/H4 panel: End-shortening results, α = 10-20.

52
 25000

20000
Progressive Failure Analysis (Hashin's Criterion)
H4 Panel Test Results [49]

15000
Applied Load, P (lbs.)

10000

5000

0
0.00 0.05 0.10 0.15 0.20
Out-of-Plane Deflection, w (in.)

Figure 25. H3/H4 panel: Out-of-plane deflection results.

53
 .120 -52.0

.102 -356.

.0854 -660.

.0684 -964.

.0513 -1267.

.0342 -1571.

.0172 -1875.

.0000999 -2179.

-.0170 -2483.

-.0340 -2787.
54

-.0511 -3090.
Minimum Nx Stress
-.0682 -3394.

-.0852 Maximum Nx Stress -3698.

-.102 -4002.

-.119 -4306.

-.136 -4610.

a) Out-of-Plane Deflection, in. b) Nx Stress Distribution, lb./in.

Figure 26. H3/H4 panel: Structural response at final failure (17313 lbs.).
Hashin’s Criterion
 739.

640.
541.
441. Max: 100.0 %

342.
243.

High Nxy Stress 143.

44.1
-55.1
55

-154.
-254.
-353. Min: 0.0 %

-452.
Maximum Failure (100%)
-552.
-651.
-750.

c) Nxy Stress Distribution, lb./in. d) Percentage Failure

Figure 27. H3/H4 panel: Structural response at final failure (17313 lbs.), continued.
Hashin’s Criterion
 Composite Blade-Stiffened Panel

An interest in applying graphite-epoxy materials to aircraft primary structures has led to several
studies of postbuckling behavior and failure characteristics of graphite-epoxy structural components [50].
One study of composite stiffened panels tested a blade-stiffened panel with a discontinuous stiffener [51].
The test setup for the panel is shown in Figure 28. This panel has served as a focus problem in COMET
for identifying and resolving analysis deficiencies associated with the nonlinear global/local stress
analysis of composite structures [52]. The finite element modeling and analysis needed to predict
accurately the nonlinear response of the flat blade-stiffened panel loaded in axial compression is
described in this section.

Problem Statement

The overall length of the panel is 30 in., the overall width is 11.5 in., the stiffener spacing is 4.5 in., the
stiffener height is 1.4 in., and the hole diameter is 2 in. as shown in Figure 29. The three blade-shaped
stiffeners are identical. The loading of the panel is in uniform axial compression. The loaded ends of the
panel are clamped and the sides are free. The material system for the panel is T300/5208 graphite-epoxy
unidirectional tapes with a nominal ply thickness of 0.0055 in. The lamina properties and strain
allowables for the T300/5208 material system are shown in Table 16. The blade stiffeners are 24-ply
laminates ( [±45 / 0 20 / m 45] ) , and the panel skin is a 25-ply laminate ([±45 / 0 2 / m 45 / 03 / ± 45 / 03
/ m 45 / 03 / ± 45 / 0 2 / m 45]) .

The finite element model of the blade-stiffened panel with discontinuous stiffener shown in Figure 29
has 1,184 4-node elements and 1,264 nodes. Note that the stiffeners are modeled with four shell
elements through the height of the stiffener. First a nonlinear analysis without failure predictions is
performed on the panel. A progressive failure analysis is then performed on the blade-stiffened panel and
compared with experimental results.

Table 16. Material Properties for Blade-Stiffened Panel, T300/5208 Material System
Material Properties Value [53]
Longitudinal Young’s Modulus E11 19.0 msi
Transverse Young’s Modulus E22 1.89 msi
Poisson’s Ratio ν12 0.38
In-Plane Shear Modulus G12 0.93 msi
Longitudinal Tensile Ultimate Strain XεT 0.0110 in./in.
Longitudinal Compression Ultimate Strain XεC 0.0086 in./in.
Transverse Tensile Ultimate Strain YεT 0.0036 in./in.
Transverse Compression Ultimate Strain YεC 0.0100 in./in.
In-Plane Shear Ultimate Strain Tε 0.0150 in./in.

56
Figure 28. Composite blade-stiffened panel with discontinuous stiffener.

57
u

30.0 in.
free

free

11.5 in
clamped

Figure 29. Finite element model of composite blade-stiffened panel.

58
 Structural Response

The structural response of the composite blade-stiffened panel is studied. First, an elastic nonlinear
analysis is performed to understand the behavior of the panel without any material failures. A linear
stability analysis is not required to impose geometric imperfections on the panel because the
discontinuous stiffener introduces an eccentric loading condition. The next step performed is a
progressive failure analysis using Hashin’s criterion with a material degradation factor of α = 10-20. The
initial displacement applied to the panel is 0.005 inches. The displacement increment for the nonlinear
analysis is chosen to be 0.0025 inches. The strain allowables in Table 16 used for the progressive failure
analysis are nominal values for the T300/5208 material system since the actual strain allowables for the
panel are unavailable.

End-shortening results are shown in Figure 30 as a function of the applied compressive load. The
blade-stiffened panel was tested to failure. In the test, local failures occurred prior to overall panel failure
as evident from the end-shortening results. Good agreement between the test and analysis is shown up to
the load where local failures occurred. Table 17 summarizes the failure loads of the blade-stiffened
panel. The analytically-obtained out-of-plane deflection w at the edge of the hole and blade stiffener is
shown as a function of the applied load in Figure 31. The large out-of-plane deflections indicate that the
response is nonlinear from the onset of loading.

Table 17. Composite Blade-Stiffened Panel: Comparison of Failure Results

Failure Criterion First Ply Failure Load Final Failure Dominant Failure
(lbs.) Load (lbs.) Mode Type
Hashin’s Criterion 26599 57064 Matrix Tension
Test Results [51] 35644 40613 Unavailable

Figures 32-34 show the structural response of the blade-stiffened panel at the final failure load in the
progressive failure analysis. This is the point in which the analysis experienced convergence problems.
Figures 32 and 33 show the out-of-plane deflection fringe plot and the Nx stress resultant contours on the
deformed geometry, respectively. Figure 32 shows that large out-of-plane deflections develop in the
region around the discontinuity. The Nx distribution in Figure 33 reveals that the load is re-distributed
away from the discontinuous stiffener such that the center stiffener has essentially no Nx load at the edge
of the hole. Also, the Nx load is re-distributed to the center of the outer blade stiffeners. Figure 34 shows
a close-up view near the hole of the Nx stress distribution and the percentage of failure within an element.
A closer look at the Nx distribution around the hole in indicates that high in-plane stresses and a high
stress gradient exist near the hole. The high in-plane stresses and high stress gradient coupled with the
large out-of-plane deflections near the hole ultimately caused local failures near the hole as evident in the
figure on the right in Figure 34.

Based on these simulations, two failure scenarios can be postulated. First, as the load increases a
stress concentration develops near the hole, a large normal stress also develops as the discontinuous
stiffener tends to pull the skin laminate apart. The skin continues to delaminate possibly causing the
jumps shown on the experimental data on Figure 30 until the outer stiffeners roll over leading to free
edge failures. The second scenario is similar to the first with the outer stiffeners causing the jumps
shown on Figure 30 and then the local delaminations near the hole cause final failure. Close examination
of the test panel reveals that local delaminations and disbonds are present near the hole and edge

59
delaminations are evident at the free edges near the panel midlength. These local delaminations near the
hole were not considered in this progressive failure analysis. These delaminations are most likely the
reason why the present progressive failure analysis over-predicted the final failure load of the panel.
Also, the strains allowables for the panel are not actual measured properties, but instead are obtained
from nominal values of the material system.

60
 60000

PFA
50000 Linear Response Final Failure

Progressive Failure Analysis (α = 10-20, Hashin's criterion)

Test Results [51]

40000
Applied Load, P (lbs.)

First Ply
Failure
30000
Local Failures

20000

10000

0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
End Shortening, u (in.)

Figure 30. Composite blade-stiffened panel: End-shortening results.

61
 90000

Linear Response
80000 Progressive Failure Analysis (α = 10-20, Hashin's criterion)

70000

60000
Applied Load, P (lbs.)

50000

40000

30000

20000

10000

0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Out-of-Plane Deflection, w (in.)

Figure 31. Composite blade-stiffened panel: Out-of-plane deflection at hole and blade stiffener.

62
 .117

.0868

.0572

.0275

-.00220

-.0319

-.0616

-.0912

-.121
63

-.151

-.180

-.210

-.240

-.269

-.299

-.329

Figure 32. Composite blade-stiffened panel: Structural response at final failure (57064 lbs.).
Out-of-plane deflection fringe plot, in.
 3397.

2744.

2092.

1440.

788.

136.

-516.

-1168.
64

-1820.

-2472.

-3124.

-3776.

-4428.

-5080.

-5732.

-6384

Figure 33. Composite blade-stiffened panel: Structural response at final failure (57064 lbs.), continued.
Deformed geometry shape with Nx stress distribution, lb./in.
 Center Stiffener
Failures

3397.
2744.
2092.
1440.
788. Max:= 100%

136.
-516.
-1168.
65

-1820.
-2472.
Min:= 0.0%
-3124.
-3776
-4428
-5080
Nx Stress Distribution, lb./in. x Percentage Failure
-5732
-6384
y

Figure 34. Composite blade-stiffened panel: Local region near hole.

 CONCLUSIONS AND RECOMMENDATIONS

Conclusions

A state-of-the-art perspective on computational models for progressive failure analysis on laminated

composite structures has been presented. A progressive failure analysis model has been developed for
predicting the nonlinear response and failure of laminated composite structures from initial loading to
final failure. This progressive failure methodology is based on Pifko’s approach and has been developed,
extended, and successfully implemented in COMET. The progressive failure analyses use C1 plate and
shell elements based on classical lamination theory to calculate the in-plane stresses. Several failure
criteria, including the maximum strain criterion, Hashin’s criterion, and Christensen’s criterion, are used
to predict the failure mechanisms and several options are available to degrade the material properties after
failures. These different formulation methods are compared and assessed by performing analyses on
several laminated composite structures.

The first laminated composite structure to be analyzed was a rail-shear specimen. Studies were
performed to test the effect of the displacement increment size in the nonlinear analysis procedure. The
study showed that a displacement increment of 0.00050 inches is sufficiently small for accurate
progressive failure predictions. The next study tested the material degradation factor, α, to check its
effect on the failure load prediction. The study showed that the rate of material degradation had little
effect on the failure prediction. Progressive failure analyses were then performed on the rail-shear
specimen to compare the maximum strain criterion, Hashin’s criterion, and Christensen’s criterion. The
progressive failure results showed that all criteria compared very well with the results from Shahid.

The next composite structure analyzed was a tension-loaded laminate with a centrally located hole.
Analyses were performed to study the effect of the mesh size on the progressive failure predictions. The
study for the tension-loaded laminate showed that the mesh size had a small effect on the prediction of the
progressive failure loads. Another study was performed to again test the material degradation factor on the
progressive failure predictions. Again, varying the material degradation factor had little effect on the
progressive failure results. In later analyses, the material degradation factor α was set to 10-20 which
essentially zeroed out the material properties at the element integration points with failures. Progressive
failure analyses were then performed to compare Hashin’s criterion and Christensen’s criterion to the
experimental results. The results indicated that for this problem there was little difference between
Hashin’s criterion and Christensen’s criterion in predicting the failure loads and failure distribution.

The next problem analyzed was a composite rectangular panel loaded in axial compression. A linear
buckling analysis was performed and the results were compared to the experimental results. The first
buckling mode shape from the analysis was in good agreement with that from the experiment. An initial
geometric imperfection using the first buckling mode shape was then applied to the panel. The load-end
shortening curve from the analysis correlated well to the experimental results up to the failure of the panel.
Progressive failure analyses using Hashin’s and Christensen’s criteria were performed on the panel.
Hashin’s criterion was less than 3% from the test failure load while Christensen’s criterion was less than
8% from test final failure load. Both failure criteria predicted slightly higher failure loads than the results
from the experiment. Christensen’s criterion predicted first ply failure at a much lower load than the load
predicted by Hashin’s criterion. The failure mode reported in the test was due to transverse shear effects
along the node line in the buckle pattern. The progressive failure analysis results showed that the damage

66
was concentrated along the node line just as the test results showed. Despite the progressive failure
analysis not including the transverse shear failure mode, the progressive failure analysis predictions still
agreed well with the test results.

A compression-loaded composite panel with an offset hole was also analyzed and compared to the test
results for two panels. Again, a linear buckling analysis was performed to impose geometric
imperfections on the panel model. The first buckling mode shape predicted by the analysis agreed with
the test results. Hashin’s criterion was used for the nonlinear progressive failure analysis of the panel
because it provided more failure modes than the other failure criteria. There was good correlation in the
load end-shortening curve for the analytical and experimental results. However, the progressive failure
analysis final failure load predictions under-predicted the failure load from the test by 8.3% for the H3
panel and 14.4 % for the H4 panel. In the experiment, both panels experienced failures along the nodal
lines away from the hole due to transverse shear mechanisms. In addition, the H3 test panel developed
local failures around the hole region. The progressive failure results also showed that the panel
experienced local failures near the hole region due to shear stress failure.

The final problem analyzed was a composite blade-stiffened panel with a discontinuous stiffener
loaded in axial compression. A progressive failure analyses using Hashin’s criterion was performed on
the blade-stiffened panel. The progressive failure analysis and test results showed good correlation for
the load end-shortening results up to the load where local failures occurred. The progressive failure
results under-predicted first ply failure and severely over-estimated the panel’s final failure. The
progressive failure results predicted failures around the hole region at the stiffener discontinuity. The
final failure of the experiment showed that local delaminations and disbonds were present near the hole
and edge delaminations were present near the panel midlength. The delaminations are most likely why
the progressive failure analysis results did not compare well to the test results since delamination failure
modes are not included in the progressive failure methodology.

Recommendations

Because of the complexity in developing a progressive failure model capable of predicting all types of
failures, there are several recommendations that should be made to enhance the current progressive
failure methodology. The first recommendation would be to extend the current progressive failure model
to include failure criteria to predict failure interlaminar mechanisms. One option to accomplish this using
the current progressive failure model would be to calculate the interlaminar stresses (σ xz , σ yz , σ zz ) by
integration of the 3-D equilibrium equations. A more efficient option would be to modify the current
progressive failure model so that it would use the C0 elements which already account for these
interlaminar stresses. Once the interlaminar stresses were known, other failure mechanisms to predict
debonding or delaminations could added. However, for accurate interlaminar stresses even for C0
elements, integration of the 3-D equilibrium equations will be necessary. The second recommendation
would be to modify the degradation model such that an integration point with failures would still have a
small stiffness which could eliminate singularities in the stiffness matrix caused by the material
degradation model. Finally, the progressive failure analysis model should be modified to re-establish
static equilibrium after material properties have been degraded. This could be accomplished by repeating
the nonlinear analysis at the current load step until a converged solution exists. Such a capability would
permit the use of arbitrary step sizes during the nonlinear analysis and provide for an automatic step size
control rather than fixed step size.

67
 APPENDIX

IMPLEMENTATION INTO COMET

COMET Overview

At NASA Langley Research Center, a research effort is being directed towards developing advanced
structural analysis methods and identifying the requirements for next generation structural analysis
software. This activity has developed into a computational framework to aid in the definition of these
requirements and to serve as a “proving ground” for new methods on complex structural application
problems. This framework has yielded COMET, a structural analysis software system, which was
developed jointly between NASA Langley Research Center and Lockheed Palo Alto Research
Laboratory. COMET is a modular, extendible, machine-independent, architecturally-simple, multi-level
software system enabling researchers to implement their formulations as generically as possible.
COMET utilizes a high-level command language and data manager that allows the coupling of
independent FORTRAN processors together such that specific structural analysis functions may be
performed. Because of these features, it has become an extremely powerful tool to researchers and
developers in the field of computational mechanics. COMET’s capabilities include linear and nonlinear
stress analyses of large-scale built-up structures, transient dynamic analyses, and eigenvalue analyses.

A graphical overview of COMET is shown in Figure A-1. No single processor (FORTRAN program)
controls all aspects of the analysis in COMET. Instead, the steps for an analysis are performed by a
number of independent analysis application processors and high-level, command-language procedures.
Processors and procedures communicate with one another by exchanging named data objects in a global
computational database managed directly by a data manager called GAL (Global Access Library) [54].
Execution of the application processors is controlled by the user with an interactive, high-level, command
language called CLAMP (Command Language for Applied Mechanics Processors) [55] which is
processed by the command language interpreter CLIP (Command Language Interface Program).

The progressive failure analysis methodology developed as part of this research exploits features of
COMET and is enabled by the design of this computational framework. Specifically, four main
functions are defined. First, the capability to develop and modify application processors for model
generation and analysis enhanced this research. Second, the generic element processor (GEP) provided
element data and results through the computational database which represented the element developer’s
best strategy for stress and strain recovery. Third, the generic constitutive processor (GCP) provided an
effective mechanism for implementing different failure models and archiving constitutive data through
the primary database and through auxiliary historical database for nonlinear, path-dependent constitutive
models. Finally, the procedure library provided a springboard for developing a progressive failure
analysis strategy and its assessment. COMET allows a researcher to focus on their main area of
concentration and to benefit from the breakthroughs of other researchers. Additional details of the GEP
and GCP are provided next.

68
 COMET (CSM Testbed)

Global Computational Database

CLAMP Structural Generic Constitutive

Analysis Element Processors
Processors Processor

Element Solids LAU

Assemblers

Equation Shells LAUB

Solvers

Beams
Eigenvalue Beams GCP
Solvers
Historical
Pre/Post Constitutive
Processors Database

Figure A-1. Graphical overview of COMET.

Generic Element Processor

COMET offers an interface called the GEP (Generic Element Processor) [56] which allows finite element
developers to implement their finite element formulations quickly and efficiently. The GEP acts as a
generic template (see Figure A-2) for implementing a multitude of structural-element (ES*) processors.
Each ES* processor performs all element operations for all elements implemented within the processor
such as element definition, stiffness, force, and mass matrix generation, and various other pre-processing
and post-processing functions such as stress recovery. The computation of the stresses and/or element
strains is primarily a post-processing command. These element quantities may be computed at element
integration points, at element centroids, or at element nodes by extrapolation from the integration points.
The GEP is designed to implement virtually all types of elements, both standard elements (1-D, 2-D, and
3-D element types) and non-standard elements which do not fit within the mold of standard elements.
All element processors share the same standard generic software interface (called the ES shell routines) to
COMET which ensures that all element processors understand the same command-language directives
and create and access the database in the same way. Because of the ES shell routines, element users can

69
access all ES processors in the same manner and element developers can implement new elements
regardless of their complex internal formulations.

*CALL ES ( FUNCTION = 'FORM STIFFNESS/MATL' , ES_PROC = ES* )

PROCEDURE ES (. . .)

[XQT ES*

PROCESSOR ESi

GENERIC ELEMENT PROCESSOR SOFTWARE "SHELL" G C D

L O A
STANDARD "SHELL/KERNEL" INTERFACE ROUTINES O M T
G B P A
C ... A U B
ES_K ES_M ES_F
P L T A
A S
T E
I
STIFFNESS MASS FORCE ... O
N
ELEMENT DEVELOPER'S "KERNEL" ROUTINES A
L

Figure A-2. GEP template.

For the progressive failure analysis, the GEP software shell also interacts with the generic constitutive
processor (GCP). For a linear elastic analysis, the material is assumed to be a Hookean material and
constant throughout the analysis. For nonlinear material problems, the material model is not constant
and perhaps not even continuous as in the case of brittle failures. In these cases, the element processor
must compute the stress and strain resultants for later use in the through-the-thickness calculations of the
stress and strain. Then the GCP constitutive kernel routines are called to evaluate failure models and to
assess damage models prior to computing new constitutive terms for the element.

The progressive failure methodology implemented into COMET currently uses only the C1 (slope-
continuous) elements for the analysis. Two C1 shell elements are available for use in COMET: a 4-node
quadrilateral shell element named E410 in processor ES5 and a 3-node triangular element named TP2L in
processor ES31. These elements are implemented into two ES processors in the GEP and are described
below.

70
 ES5 Processor

Processor ES5 [57] contains a displacement-based 4-node quadrilateral shell element originally
developed for the STAGS code [58]. The E410 element is a C1 (slope-continuous) finite element based
on the Kirchhoff-Love shell hypothesis (normals stay normal, no transverse-shear deformation). The
element has 3 translational and 2 rotational degrees of freedom per node. The element also has a
“drilling” rotational stiffness which eliminates the need to suppress the drilling degree of freedom. For
geometrically nonlinear analysis problems, the E410 element includes the full nonlinear Green-Lagrange
strains.

ES31 Processor

Processor ES31 contains a discrete-Kirchhoff triangular element, known as the DKT element. The 3-
node DKT element implemented into COMET is referred to as the TP2L element. This element was
developed by Garnet, Crouzet-Pascal, and Pifko [59] and implemented in COMET by Pifko and Crouzet-
Pascal [41]. The element has 5 degrees of freedom per node, 3 translational and 2 rotational degrees of
freedom and has an artificial drilling term. For geometrically nonlinear analysis problems, the TP2L
element includes the full nonlinear Green-Lagrange strains.

Generic Constitutive Processor

Each structural element requires the evaluation of various constitutive functions including evaluation
of element constitutive matrix, determination of tangent moduli (for the tangent stiffness matrix), and
evaluation of failure criteria (for failure and material degradation). The constitutive modeling capabilities
of COMET are centered towards the analysis of laminated composite structures. Element developers and
structural analysts have access to constitutive models for 1-D beam elements, 2-D plate and shell
elements, as well as 3-D solid elements. Processor LAU is a laminate analysis utility for calculating the
constitutive relations for 2-D and 3-D isotropic, orthotropic, anisotropic, and laminated structures. For
2D structures, processor LAU can use classical lamination theory or traditional first-order, shear
deformation theory. Processor LAU is limited to performing elastic structural analyses. Processor
LAUB is an extension of processor LAU to calculate the constitutive relations for 1-D beam elements.
Both LAU and LAUB are described in Ref. 40.

To enhance the constitutive modeling capabilities of COMET, the Generic Constitutive Processor
(GCP) was developed by Lockheed Palo Alto Research Laboratory [60]. GCP allows researchers to
implement new constitutive models, failure models, or damage models into COMET conveniently. The
GCP replaces COMET’s current elastic constitutive capability, as implemented in both LAU and LAUB.
The GCP is similar to the GEP in that various constitutive models may be implemented in COMET and
accessed using other independent processors within the COMET framework. Once the midplane strains
and curvatures are known from the element processor, the through-the-thickness in-plane strains and
corresponding stresses may be calculated and used to evaluate selected failure criteria using features of
the GCP.

The GCP architecture contains five major functional components as shown in Figure A-3. The
combination of the generic constitutive interface, GCP inner “shell”, constitutive developer interface,
historical database, and the constitutive kernel provide links with the element processor for efficient
element constitutive functions. The GCP outer “shell” provides a common user interface to the GCP by
processing commands for input of material/fabrication data, interacting with the database, and directing

71
the flow of computational procedures. The GCP outer “shell” also incorporates the nonlinear analysis
algorithm for stand-alone testing of constitutive models, material failure criteria, and material damage
models. The GCP inner “shell” performs through-the-thickness integration for composite laminates,
interpolates state-dependent material properties, performs transformations from element-to-material
coordinate systems, calls the constitutive kernel routines, and performs database management functions
of the constitutive historical data, point stress/strain quantities, and material tangent stiffnesses. In order
to provide the capability for performing stand-alone constitutive analyses and analyses involving the
element processors, the GCP utilizes a generic constitutive interface which provides a flexible, efficient,
computational link to each individual element processor.

G C D
L O A
GCP Outer Shell O M T
B P A
A U B
L T A
A S
Generic Constitutive Interface (GCI)
T E
I
O
GCP Inner Shell N
A
G L
E Constitutive Developer Interface
P
STRESS STIFFNESS DAMPING INITIALIZE

H C D
I O A
S N T
T S A
O T B
Constitutive Kernels R I A
I T S
C U E
A T
L I
V
E

Figure A-3. GCP overview.

72
 Implementation of Failure and Damage Models

Implementation of new constitutive models into the GCP is accomplished using the Constitutive
Developer’s Interface in the GCP as indicated in Figure A-3. This interface allows developers of new
constitutive models to include constitutive kernel routines such as failure detection models or material
damage models easily through a set of FORTRAN subroutine entry points with standardized argument
lists. This capability and its implementation template is very similar to the GEP described earlier.

GCP kernel subroutines are denoted by the name of CSiX where the index i indicates the ith
constitutive model, and the selected constitutive function to be performed is represented by X in the
GCP’s FORTRAN material library. Table A-1 summarizes the types of constitutive models existing in
COMET wherein at least the minimal capabilities are provided to form the constitutive matrix for
different element types. Table A-2 summarizes the functions of the CSiX predefined constitutive
subroutine entry points in GCP where X denotes a specific constitutive function.

Two tasks must be performed by the constitutive developer to implement a new constitutive model
within the GCP. First, constitutive kernel routines must be provided to perform the necessary and
desired constitutive functions at a material point. Secondly, the selected subroutines within the
constitutive developer interface must be modified to read the material property input for the new
constitutive model and also to perform any constitutive related postprocessing.

Table A-1. Constitutive Models in COMET

Constitutive Model i Constitutive Model Description
CS1X Linear Elastic Isotropic Model
CS2X Linear Elastic Orthotropic Model
CS3X Mechanical Sublayer Plasticity Model
CS4X Linear Elastic-Brittle Orthotropic Model

Table A-2. Constitutive Developer Interface Subroutines of GCP

Subroutines Constitutive Function Description
CSiX
CSiV Material property input verification.
CSiI Initialization of constitutive model.
CSiS Point stress calculation subroutine.
CSiC Constitutive matrix calculation subroutine.
CSiM Returns mass density at material-point.
CSiD Returns material damping matrix at material-point.

73
 One set of constitutive interface subroutines initially and implemented into COMET through the GCP
developed by Moas [43] were enhanced by Pifko of Grumman Aerospace Corporation [41,42]. These
subroutines are identified as the CS4 constitutive model. The CS4 constitutive model is for 2D linearly
elastic, brittle, orthotropic laminates. The constitutive kernel subroutines used for the progressive failure
analysis are therefore called: CS4V, CS4I, CS4S, and CS4C. The CS4V subroutine performs the
verification of the material property input for the progressive failure constitutive model. The laminate
material properties, lamina orientations, and strain allowables for failure assessment are read when the
CS4V subroutine is called. The CS4I subroutine entry point performs the initialization required for the
constitutive model by setting flags for the historical material database and initializing that database if
necessary. The CS4S subroutine recovers the element stresses from the element strains determined in the
element processor (GEP). The CS4C subroutine is modified to perform failure detection and material
model degradation following Pifko’s strategy.

In the progressive failure strategy implemented by Pifko, the stresses are recovered for each element
(IEL = 1,2, K , NEL) in the finite element model after a converged nonlinear solution is obtained. This is
accomplished by calling the generic ES processor shell to compute the element stresses (ES procedure
command “FORM STRESS”) of a given element type within a given ES processor. The ES* element
processor (ES5 or ES31) for the specific element type is then executed to initiate the stress recovery
process. For each Gauss quadrature point of an element (IQP = 1, 2, K , NQP ) , the middle surface strains
and curvatures, {ε0} and {κ}, are calculated in the plane of the element. Then the GCP shell is called to
read in the material properties for each layer in the laminate (NL = 1,2, K , NLAYER) . If no previous
failures have occurred in the analysis, then the original elastic material properties are used. These
properties are stored in the array mpd. However, if previous failures have occurred, then the properties
from the previous load step, stored in array oldhmd, which have been previously degraded, are read into
the computational database. For each integration point through the thickness of a layer
(LIP = 1,2, K , NLIP ) , the point strains are calculated according to

 ε x   ε 0x   κ x 
   0   
 ε y  =  ε y  + z κ y  (A-1)
γ   0   κ 
 xy  γ xy   xy 

at the integration points associated with Simpson’s integration rule. Numerical integration is needed to
calculate the point strains because the lamina properties are not constant through the thickness if the
material properties have been degraded due to lamina failures. Once the point strains for the integration
point within the layer have been calculated, the point stresses in the material reference frame are
calculated by the constitutive relations, {σ} = [Q]{ε} .

Next the failure criteria are evaluated for failures using the point strains or stresses at the integration
points within the layer. If failures are detected, then the material properties for that integration point are
degraded according to the damage model. The updated material properties for each layer-integration
point are then saved in an array called newhmd. This progressive failure analysis process is continued for
each layer-integration point within each layer at each Gauss quadrature point for each element in the
finite element model. Figures A-4 (a) and (b) show the number of Gauss quadrature points for the
ES5/E410 and ES31/TP2L elements, respectively. Figure A-4 (c) illustrates a general NLAYER laminate
and possible layer-integration points through the thickness of each layer. The computational procedure
of the progressive failure methodology just described is presented in Figure A-5 for each element.

74
 Gauss Quadrature
Points

NQP=5 NQP=3

(a) ES5/ES410 element (b) ES31/TP2L element

•
•
•
•
•
NL
AY Layer-integration
ER
• points through each layer
• (LIP=1,NLIP)
2
1

N-Layered
Laminate
(NL=1,NLAYER)
(c) Layer-integration points

Figure A-4. Gauss quadrature points and layer-integration points.

75
*CALL STRESS

For each element

CALL ES (Function = ‘FORM STRESS’) (IEL=1,2,…,NEL)

[XQT ES* (ES5 or ES31)

For each Gauss integration point (IQP=1,2,…,NQP)
Compute{ε0} and {κ} at each Gauss point
in the plane of the element

CALL GCP shell

{ε0} and {κ} from Global Computational Database

For each layer in laminate (NL=1,2,…,NLAYER)
For each layer-integration point (LIP=1,2,…,NLIP)

• Read in material properties (E11, E22 ,G12 ,u 12 , etc.)

(array oldhmd for previous failures or mpd for no failures)
from Historical Constitutive Database
• Compute point strains {ε} = {ε0} + z{κ}

• Compute point stresses {σ} = [Q]{ε}

• Evaluate Failure Criteria

- Christensen
- Hashin
- Maximum Strain

• Damage Model
- Degrade properties

Next layer-integration point

Next layer
*CALL UPDATE_GCP
Update historical constitutive database (array newhmd )
Next Gauss integration point

Next element

Figure A-5. Progressive failure methodology using COMET.

76
 The constitutive processor CS4 is used as the starting point of this research. The main contribution to
this processor is in subroutine CS4C and involves the implementation of additional failure theories and
material degradation models. The CS4 processor now accommodates the maximum strain criterion,
Christensen’s criterion, and the Hashin criterion as described in the Progressive Failure Analysis
Methodology section. The failure criteria desired for the analysis is selected by the analyst when the
CS4V subroutine is called. When a failure is detected, a failure flag classifies the type of failure as fiber
failure, matrix failure, or shear failure. Two material degradation models are implemented including
instantaneous reduction and gradual reduction for use with the ply-discount approach which are also
described in earlier in the paper. The material properties which are degraded when a failure is detected
depend upon the type of failure as described above. The degraded material properties for the element
with failures are updated in the historical material database.

In this progressive failure methodology, historical material information for every layer-integration
point needs to be stored. This is illustrated in Figure A-6. The failure analysis must loop over all
elements (IEL = 1, 2, K , NEL) . For each Gauss point (IQP = 1, 2, K , NQP ) in each element, a detailed
assessment is performed through the thickness of the laminate, layer by layer
(ILAYER = 1,2, K , NLAYER ) , using a numerical integration method with multiple layer-integration
points (LIP = 1,2, K , NLIP ) . New updated material data is then computed and stored for each point in
the historical material database which is discussed in detail next.

Historical Material Database

The historical material database is independent of the global computational database and contains all
information related to the failure history of each element and its material properties. A schematic of the
structure of the historical material database is shown in Figures A-7 and A-8. This database is updated if
new failures are detected after a converged solution in the nonlinear analysis. If new failures are
detected, then the degraded material properties become the new material properties (array newhmd) which
are used in the next load step of the nonlinear analysis. Initially the historical data has only NEL items
(one item for each element) in the HISTDIR record, and these items are all zeros indicating that no
failures have occurred at any point in any of the elements. If a failure is detected in an element, then the
item for that element with failures becomes nonzero, and it defines the data location of the block in the
HISTPTR$ record. The HISTPTR$ record is NEF items (number of elements with failures) in length,
and each item is a pointer which points (gives the data location) to the element pointer data in the
HISTPTR record. Thus, after a failure in the IELth element, the IELth item in the HISTDIR record is
checked. The value in that item points to the element record pointer in the HISRPTR$ record. This item
points to the first item of the element pointer data contained in the HISTPTR record. Thus, each element
block in HISTPTR has a total length of NQP + (NQP × NLAYER) × (NLIP × 2 + 1) items. Within
this element pointer block in the HISTPTR record, the first NQP items are pointers for each Gauss
quadrature point to the layer pointers. The next NQP × NLAYER items are the pointers to the layer-
integration-point pointers for each Gauss quadrature point. Each of these NLIP × 2 layer-integration-
point pointers specifies the number of items for each constitutive historical data block and the pointer to
the data location in the HISTDATA record for the LIPth layer-integration point with the NLth layer at the
IQPth Gauss quadrature point for the IELth element. For the constitutive model implemented here, nine
items per block are updated. These items include the material properties and the failure flag types.
These failure flag types are defined as lfail(1) for fiber failure, lfail(2) for matrix failure, and lfail(3) for
shear or transverse failure.

77
 th
IQP Gauss
Quadrature Point
th
Element
IEL

NLth Layer
of Laminate

•
•
•
•
•

th
LIP Layer-Integration Point

Updated Material Data

E11, E22 ,G12 ,υ12 , etc.

Figure A-6. Overview of the progressive failure analysis computation locations in a composite laminate.

78
 Element pointers in HISTDIR record
1 2 3 • • • • • • NEL

NEL items in length

Contains data location
in HISTPTR$ record

HISTPTR$ record

NEF items in length

Contains element pointer data in HISTPTR record

HISTPTR record

NQP + (NQP x NLAYER ) x (NLIP x 2 +1) items per element block

HISTDATA record

Constitutive historical data 9 items per layer-integration point

for the C1 elements considered

Figure A-7. Organization of constitutive material database - 1.

79
 HISTDIR record
IEL= 1 IEL= 2 ••• IEL= NEL

1 2 NEL

= 0, No failure 1 item per element

≠ 0, Pointer to element pointer
data in HISTPTR$

(a) Element pointers in HISTDIR

HISTPTR$ record

1 item per element with failures, NEF

Element pointer data in HISTPTR

Pointer to first layer pointer Pointer to first layer-integration point

for the IGPth Gauss quadrature point pointer for the NLth layer

IGP IGP IGP = NL = NL = NL = LIP LIP LIP =

••• ••• •••
=1 =2 NQP 1 2 NLAYER =1 =2 NLIP

HISTPTR record
Pointer to first item of constitutive
data for the LIPth layer-integration point

(b) One block of element pointer data in HISTPTR for each element

HISTDATA

E11 E22 G12 G13 G23 u12 lfail(1) lfail(2) lfail(3)

9 items in order

(c) Constitutive data in HISTDATA at each layer-integration point pointer

Figure A-8. Organization of constitutive material database - 2.

80
 Nonlinear Analysis Solution Procedure

The nonlinear analysis solution procedure described in Progressive Failure Analysis Methodology
section is implemented as a procedure in COMET named NL_STATIC_1. This procedure performs a
nonlinear analysis using a modified Newton-Raphson algorithm with corotational and an arc-length
control strategy for either applied force or applied displacement problems. The procedure uses a global
load-stepping algorithm for advancing the nonlinear analysis solution during a static analysis. The
implementation in NL_STATIC_1 involves a linearized version of the quadratic arc-length constraint
equation. At the beginning of each “arc-length” step, a new tangent stiffness matrix is formed and
factored. This tangent stiffness matrix is used for all iterations at this load step. Hence, the nonlinear
analysis procedure implemented in COMET may be viewed as a modified Newton-Raphson algorithm
with simultaneous iteration on the generalized displacements and the load factor. This procedure is
modified such that once failure occurs, the analysis is converted into a load or displacement-controlled
procedure since the equilibrium iteration is not performed at constant load. Instead, the strategy is to use
small load increments.

81
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85
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Progressive Failure Analysis Methodology for Laminated Composite
Structures 538-13-12-04

6. AUTHOR(S)
David W. Sleight

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13. ABSTRACT (Maximum 200 words)
A progressive failure analysis method has been developed for predicting the failure of laminated composite
structures under geometrically nonlinear deformations. The progressive failure analysis uses C1 shell elements
based on classical lamination theory to calculate the in-plane stresses. Several failure criteria, including the
maximum strain criterion, Hashin’s criterion, and Christensen’s criterion, are used to predict the failure
mechanisms and several options are available to degrade the material properties after failures. The progressive
failure analysis method is implemented in the COMET finite element analysis code and can predict the damage
and response of laminated composite structures from initial loading to final failure. The different failure criteria
and material degradation methods are compared and assessed by performing analyses of several laminated
composite structures. Results from the progressive failure method indicate good correlation with the existing
test data except in structural applications where interlaminar stresses are important which may cause failure
mechanisms such as debonding or delaminations.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Composites, Failure Prediction, Progressive Failure, Material Degradation 94
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