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Handbook for Damage Tolerant Design

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    • Examples
      • 1. FAC Problems
      • 2. Merc Problems
        • 0. Crack Growth Prediction - Center Wing Component
        • 1. Damage Tolerance Analysis - Windshield Doubler
        • 2. Sensitivity of Crack Growth Rates to Operating Conditions
        • 3. Influences of Retardation Models on Crack Growth Predictions
      • 3. NRC Problems
      • 4. SIE Problems
      • 5. UDRI Problems

Problem Merc-1

Title:  Crack Growth Prediction of Center Wing Component with Emphasis on b-Factor Determination


To illustrate the process of applying finite element and boundary element analyses to the determination of b-factors for crack growth simulations

General Description:

This problem focuses on the determination of b-factors for crack growth analysis of a longeron in the center wing area.  Both 2-D and 3-D numerical methods are applied in the analysis.  FRANC2D/L, a 2-D finite element (FE) analysis program, is used for the b-factor determination of through cracks.  FRANC3D, a 3-D boundary element (BE) program is used to determine b-factors of a corner crack.

Topics Covered:          finite element analysis, boundary element analysis, b-factor calculation

Type of Structure:       center wing, longeron

Relevant Sections of Handbook:  Sections 2, 5, 11

Author:                     Robert D. McGinty

Company Name:  Mercer Engineering Research Center
Structures Technology Group
Warner Robins, GA  31088-7810

Contact Point:       Robert D. McGinty

                 Phone:        478-953-6800

                e-Mail:        bmcginty@merc.mercer.edu


Critical Area


This problem focuses on a critical area of an angle longeron in a center wing.  As shown in the finite element mesh below, the crack starts in, and grows across the horizontal leg of the angle longeron.  It begins as a corner crack in a fastener hole where the longeron and upper splice bar are attached. 


The dominant loading mode of the longeron is tension during banking and turning maneuvers, but due to the complex geometries and interactions of the components, the details of the stress state are more complicated than simple, uniform tension near the critical fastener.  The determination of b-factors through analytical means can, therefore, introduce significant errors in the crack growth analysis.  Numerical methods such as finite and boundary element analysis provide a powerful alternative to the analytical solutions. 

Crack Growth Phases and Modeling Approach


As shown in Figure MERC-1.2 below, crack growth behavior can be divided into phases, usually alternating between "corner crack growth" and "through crack growth".  The odd- numbered phases are corner cracks, and the even-numbered phases are through cracks.   Also, it will be assumed here that the phases of crack growth take place sequentially.  Corner crack growth starts in Phase 1 and continues until it grows through the part's thickness, at which point, through crack growth begins in Phase 2.  Finally, Phase 3 does not begin until Phase 2 growth reaches the free edge of the longeron  (Analysis has shown that the crack reaches critical length during Phase 3, therefore analysis of subsequent phases is not necessary.).


The b-factors for each phase of crack growth will be determined somewhat differently, due to the increasing complexity of each phase.  Emphasis is placed on the numerical methods used for Phases 2 and 3.


Phase 1 - Corner Crack at Fastener Hole


This represents a corner crack emanating from a loaded fastener hole that is offset from the center of the longeron leg.  AFGROW provides pre-programmed solutions of b-factors for this case.  Therefore, relatively little user expertise is required to obtain b-factors here, at least when compared to subsequent phases.  Figure MERC-1.3 shows the AFGROW input form for defining part geometry, including the crack, from which AFGROW automatically computes b-factors.


Figure MERC-1.3.  AFGROW input form for defining part geometry, including the crack, from which AFGROW automatically computes b-factors.



Phase 2 - Through Crack Growing from Fastener Hole


Part geometry and loading conditions introduce complexities into the second phase of crack growth that are not fully incorporated into existing b-factor solutions.  Therefore, numerical methods, i.e., finite elements, are used to estimate the b-factors.  Figure MERC-1.4 shows a 2-D FE model of the longeron's horizontal leg developed in FRANC2D/L.  FRANC2D/L is chosen because of its ability to automatically remesh while "growing" the crack.  The boundary conditions, tractions and fastener forces, are determined from the NASTRAN FE model shown in Figure MERC-1.1.  The crack path is also shown in Figure MERC-1.4.  Note that it passes nearby a fastener whose presence introduces additional stress concentrations that further complicate the problem. 


FRANC2D/L computes K values versus crack length as it automatically propagates the crack and remeshes.  b-factors are determined by defining a reference stress, which is usually a remote tensile value, and solving for b in Eq. (1).


where K is the stress intensity factor, a is crack length, and sref is the reference stress, in this case 6,800 psi.  Both K and b values are shown in the graph below.


Phase 3 - Corner Crack Growing from Opposite Side of Fastener Hole


In Phase 3, a corner crack is assumed to grow from the opposite side of the fastener hole after the through crack reaches a free edge in Phase 2.  But unlike Phase 1, AFGROW does not provide b-factor solutions for this situation.  It is necessary to compute them numerically as in Phase 2.  The process is substantially more involved because the problem is 3-D, however, it does begin just as the Phase 2 case.  The NASTRAN and FRANC2D/L models are developed just as before, but the Phase 2 crack is present.  The FRANC2D/L model is shown in Figure MERC-1.6 below.



The FRANC2D/L model results in Figure MERC-1.6 are used as boundary conditions for another, still more refined, numerical model.  It is a 3-D boundary element model, developed in FRANC3D, of a corner crack at the fastener hole.  This is shown in Figure MERC-1.7 below.  FRANC3D reports K values along the corner crack front, and these are used to calculate b-factors using Eq. (1) just as in Phase 2. 


Figure MERC-1.7.  FRANC3D model and principal stresses for Phase 3 corner crack
 at fastener hole.  FRANC2D/L stresses serve as boundary conditions
for the FRANC3D model


Several FRANC3D analyses are, in fact, necessary for each critical area in order to cover the possible combinations of thickness and surface lengths of the corner crack.  As an example, Figure MERC-1.8 illustrates nine (3x3) corner cracks of various lengths that are analyzed in order to provide sufficient b-factor information for corner crack growth studies.

Figure MERC-1.7.  FRANC3D model and principal stresses for Phase 3 corner crack at fastener hole.  FRANC2D/L stresses serve as boundary conditions for the FRANC3D model

Tables MERC-1.1 and MERC-1.2 give b-factors for various corner crack geometries.  Two tables are required to determine crack growth through the part thickness and along the part surface.  Table MERC-1.1 gives b-factors that apply to crack growth through the part's thickness.  Table MERC-1.2 corresponds to crack growth along the part's surface.  AFGROW uses this information to independently predict growth rates along the two directions.  Though independent, it is generally true that the growth rates produce a corner crack whose thickness length is approximately 50% greater than its surface length.


Table MERC-1.1.  Beta Factors – Thickness Direction


Thickness Crack Length (in)




Surface Crack
Lengths (in)













Table MERC-1.2.  Beta Factors – Surface Direction


Thickness Crack Length (in)




Surface Crack
Lengths (in)














Predicting Crack Growth


AFGROW is used to predict crack growth in the longeron using the b-factors presented here.  The crack is assumed to start as a 0.05" radius corner crack at the fastener hole and grow according to the discussion above.  Only surface crack length is plotted.  Details of the stress spectra and material da/dN data are not presented.  Note that Willenborg retardation is applied.