To
illustrate the process of using the nonlinear finite element method to evaluate the
residual strength of a built-up aircraft structure possibly containing multi-site fatigue
damage and possibly having corrosion damage.

This
problem details the process of using the finite element method to predict the residual
strength of a section of built-up fuselage structure.
The section contains a lead fatigue crack, and might also contain MSD and corrosion
damage. Stable growth of the lead crack is
simulated based on the critical crack tip opening angle (CTOA_{c}) criterion. Methods for representing rivet stiffness and for
local meshing are described. The process is
applied to a section of K/C-135 fuselage, and strength assessment curves are generated for
a number of configurations with no MSD, with MSD, and with MSD and corrosion conditions.

Overview of Problem Description

A
relatively simple builtup narrow body fuselage configuration is modeled. The example
demonstrates an analysis to predict the residual strength of a pressurized fuselage,
subjected to MSD and corrosion damage [Cope 1998]. The problem chosen for analysis is a
three stringer wide, three-frame long K/C-135 fuselage panel. The panel section has a
radius of curvature of 72 inches. It contains a lap joint at the central stringer. The lap
joint is a typical three-row configuration with 3/16 inch diameter countersunk rivets. The
other two stringers are spotwelded to the skin. The upper and lower skins are made of
0.04 inch thick, 2024T3 aluminum alloy. The stringers and frames are made of 7075T6
aluminum alloy. Frames are simply connected to the stringers by rivets. The panel
configurations are shown in Figures FAC-2.1 and FAC-2.2. The frame and stringer dimensions are shown in Figure FAC-2.3.

Figure FAC-2.1. Layout of
K/C-135 fuselage section used in the present example.

Figure FAC-2.2. Detail of
lap joint rivet spacing used in the present example.

Figure FAC-2.3. Dimensions (all in inches) of stringer
and frame used in the present example.

Computational Model

All structural components including skins, stringers, and frames
are modeled by displacement-based, four-noded or five-noded Kirchoff shell finite elements
[Rankin 1991]. The geometrical and mesh models were created using FRANC3D
[www.cfg.cornell.edu], and all finite element computations were performed with STAGS
[Rankin 1997]. Each node of a shell element has six degrees of freedom. A piecewise linear
representation is used for the uniaxial stressstrain curves for 2024T3 and 7075T6
aluminum alloys (see Figures FAC-2.4 and FAC-2.5).

Symmetric
boundary conditions are imposed on all the boundary edges to simulate a cylinderlike
fuselage structure. Pressure loading is applied on all the external skins. Uniform axial
expansion was allowed at one longitudinal end. On this boundary edge, an axial force equal
to (PR/2)L was assigned where P is the applied pressure, R is the radius of the panel, and
L is the arclength of the edge. The kinematic boundary conditions (displacements and
rotations) applied along the boundaries of this local model were extracted from a global
model of the fuselage. Both geometric and
material nonlinearities are included in the analysis. The former captures the
outofplane bulging deformation and the latter captures the active plastic zone and the
plastic wake during stable crack propagation.

The nonlinear solution algorithm consists of Newton's method. Large
rotations are included in the nonlinear solution by a corotation algorithm applied at
the element level [NourOmid 1991]. The Riks arclength path following method is used to
trace a solution past the limit points of a nonlinear response [Rankin 1997; Riks 1984].

Rivets
are modeled by elasticplastic spring elements that connect finite element nodes in the
upper and lower skins. Each rivet is modeled with six degrees-of-freedom, corresponding to
extension, shearing, bending and twisting of the rivet. The stiffness of each
degree-of-freedom is defined by prescribing a forcedeflection curve. The axial,
flexural, and torsional stiffnesses of the spring elements are computed by assuming that
the rivet behaves like a simple elastic rod with a diameter of 3/16 inch. The elastic
shear stiffness of the rivet is computed by the following empirical relation developed by
Swift [1984]:

_{}

where
*E* is the elastic modulus of the sheet material, *D* is the rivet diameter, *t*_{1} and *t*_{2} are the thicknesses of the joined
sheets, and *A* = 5.0 and *C* = 0.8 for aluminum rivets. The initial shear
yielding and ultimate shear strength of the rivets are assumed to occur at load levels of
510 lb and 725 lb, respectively. Once a rivet reaches its ultimate strength, it will break
and lose its load carrying capacity. The forcedeflection curve shown in Figure FAC-2.6 for shearing is intended to represent empirically
the net shear stiffness of a riveted sheet connection, accounting for bearing deformations
and local yielding around the rivet [Young 1997; Swift 1984].

The
critical crack tip opening angle (CTOA_{c})[Dawicke 1994] is used to characterize
elasticplastic crack growth and to predict residual strength. For details on use of this
criterion, see Problem FAC-4. The CTOA_{c} used in this example was 5.7 degrees
measured 0.04 inch behind the crack tip with a plane strain core height equal to 0.08 inch
[Dawicke 1997]. Since no experimental crack
growth data are available for this structure, this particular CTOA_{c} value is
estimated based on the 5.25 degrees found for 0.09 inch thick, 2024T3 bare material. The
plane strain core height is assumed to be twice the sheet thickness.

Figure FAC-2.4. Piecewise linear representation of
uniaxial stress-strain relationship for 2023-T3 aluminum alloy used in the present
example.

Figure FAC-2.5. Piecewise linear representation of
uniaxial stress-strain relationship for 7075-T6 aluminum alloy used in the present
example.

Figure FAC-2.6. Model for
rivet shear stiffness and strength used in the present example.

Six different crack configurations with various lengths of lead and
MSD cracks are studied. The initial configurations prior to crack growth are:

1.
A
7.14 inch lead crack,

2.
A
7.14 inch lead crack with 0.025 inch MSD cracks emanating from both sides of a fastener
hole,

3.
A
7.14 inch lead crack with 0.046 inch MSD cracks emanating from both sides of a fastener
hole,

4.
A
10 inch lead crack,

5.
A
10 inch lead crack with 0.025 inch MSD cracks emanating from both sides of a fastener
hole, and

6.
A
10 inch lead crack with 0.046 inch MSD cracks emanating from both sides of a fastener
hole.

The
lead crack is located symmetrically about the central frame line. The MSD pattern is
symmetric about the lead crack at the 3 rivets in front of the lead crack. The lead and
MSD cracks are located along the upper rivet row in the upper skin of the joint. The crack
configurations with a 10 inch initial lead crack are shown in Figure
FAC-2.7. Since rivet holes are not modeled explicitly in the finite element model, a
small crack with a length equal to the rivet diameter plus the MSD length is used to model
the MSD crack. The finite element mesh for the model is shown in Figures
FAC-2.8. Figure FAC-2.9 shows details of the near-tip mesh
pattern with the 0.04 inch crack tip elements used there.
In addition to the effects of MSD, material thinning due to corrosion damage is
also studied. The effect of material thinning is modeled by a uniform reduction in
thickness of the upper skin at the lap joint in the two center bays.

Figure FAC-2.7. Crack
configurations for 10 inch lead crack and MSD.

Figure FAC-2.8. Overall
finite element mesh for present example.

Figure FAC-2.9. Typical
near-tip finite element meshing for present example.

Computational Results

Figure FAC-2.10 shows the predicted results of the operating
pressure loading versus the total crack extension for all the cases conducted in this
study. The predicted residual strengths summarized in Figure
FAC-2.11 indicate:

· The
MSD cracks significantly reduce the residual strength of the fuselage panel. A 21.8 to
28.0% loss of residual strength due to the presence of small MSD is observed.

· A
10% uniform thickness degradation due to corrosion damage reduces the residual strength by
3.4 to 9.0%. The coupling of MSD and corrosion damage leads to the most severe damage
scenario.

· In
general, increasing the lead and MSD crack lengths reduces the residual strengths.
However, for the cases with a 10 inch initial lead crack, residual strength seems to be
relatively insensitive to the MSD crack sizes.

The
deformed structure at residual strength for the case with a 10 inch initial lead crack but
without MSD and corrosion damage is shown in Figure FAC-2.12.
Out-of-plane bulging is observed in the skin crack edges. Because of the stiffness of the
stringer, the bulging at the lower crack edge is much smaller than the opposing edge. The
unsymmetric, outofplane bulging thus leads to an antisymmetric bending deformation
field at the crack tips [Potyondy 1995].

Figure FAC-2.10. Predicted operating pressure versus
total crack extension for the present example: (a) 7.14 inch initial lead crack, (b) 7.14
inch initial lead crack with corrosion damage, (c) 10 inch initial lead crack, and (d) 10
inch initial lead crack with corrosion damage.

Figure FAC-2.11. Predicted residual strength versus
initial lead crack length for present example.

Figure FAC-2.12. Typical deformed shape of the present
example (pressure = 15.3 psi, magnification factor = 5.0).

__References__

Cope,
D., West, D., Luzar, J., Miller, G. Corrosion Damage Assessment Framework:
Corrosion/Fatigue Effects on Structural Integrity. Technical Report D500130081, The
Boeing Defense and Space Group, 1998.

Dawicke,
D., Piascik, R. and Newman, J. Prediction of Stable Tearing and Fracture of a 2000Series
Aluminum Alloy Plate Using a CTOA Criterion. In *Fracture
Mechanics: 27th Volume, ASTM STP 1296*, Philadelphia, pp 90-104, 1997.

Dawicke,
D.and Sutton, M. CTOA and Cracktunneling Measurements in Thin Sheet 2024T3 Aluminum
Alloy. *Experimental Mechanics*, Volume 34, pp
357-368,1994.

NourOmid,
B. and Rankin, C. Finite Rotation Analysis and Consistent Linearization Using Projectors*. Computer Methods in Applied Mechanics and
Engineering*, Volume 93, pp 353-384, 1991.

D.
O. Potyondy, P. A. Wawrzynek, and A. R. Ingraffea. Discrete Crack Growth Analysis
Methodology for Through Cracks in Pressurized Fuselage Structures. *International Journal for Numerical Methods in
Engineering*, Volume 38, pp 1611-1633,1995.

Rankin,
C. and Brogan, F. The Computational Structural Mechanics Testbed Structural Element
Processor ES5: STAGS Shell Element, 1991. NASA CR4358.

Rankin,
C., Brogan, F., Loden, W. and Cabiness, H. STAGS User Manual Version 2.4. Lockheed Martin
Missiles & Space Co., Inc., Advanced Technology Center, 1997.

Riks,
E. Some Computational Aspects of the Stability Analysis of Nonlinear Structures. *Computer Methods in Applied Mechanics and Engineering*,
Volume 47, pp 219-259, 1984.

T.
Swift. Fracture Analysis of Stiffened Structure. In *Damage
Tolerance of Metallic Structures: Analysis Methods and Application, ASTM STP 842*,
Philadelphia, pp 69-107, 1984.

R.
D. Young, C. A. Rose, C. G. D'avila, and J. H. Starnes, Jr. Crack Growth and Residual
Strength Characteristics of Selected Flat Stiffened Aluminum Panels. *Proceedings of the First Joint DoD/FAA/NASA Conference
on Aging Aircraft, *Ogden, Utah, July 1997.