Methodology for Application of 3-D Stresses to Surface Damage-Based Risk Assessment

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1 Methodology for Application of 3-D Stresses to Surface Damage-Based Risk Assessment Chris Waldhart * and Michael P. Enright Southwest Research Institute, San Antonio, TX 7838, USA Simeon H. K. Fitch Mustard Seed Software, San Antonio, TX 7859,USA A methodology is presented for application of stress results from three dimensional finite element models for use in fracture mechanics computations. It is based on the assumption that a fatigue crack propagates in a plane normal to the maximum principal stress in a region that can be idealized as a rectangular plate. The methodology, recently implemented in the DARWIN probabilistic fracture mechanics program, is demonstrated for (1) a finite width plate with a centrally located hole and () an aircraft gas turbine engine rotor disk. Computational error associated with the methodology is less than 1% compared to analytical and finite element solutions. The results can be applied to the probabilistic life prediction of components subjected to surface damage. Nomenclature X, Y, Z = global Cartesian coordinates x, y = local fracture mechanics plate coordinates xx, yy, zz, xy, xz, yz = stress tensor in Cartesian coordinate system r, θ, rθ = stress tensor for plane stress in cylindrical coordinate system 1,, 3 = principal stress (maximum to minimum) I 1, I, I 3 = stress invariants l, m, n = direction cosines N = stress acting in direction N S = applied pressure loading R = hole radius D = plate width r = radial distance from the center of a hole θ = angle from loading axis I. Introduction he presence of rare metallurgical and manufacturing anomalies in turbine disks can contribute to uncontained T aircraft engine failures 1. To address this issue, the FAA released an advisory circular recommending probabilistic damage tolerance assessment of gas turbine components and funded the Turbine Rotor Materials Design (TRMD) research program 3 and associated DARWIN 4-9 computer software. The initial focus of the TRMD project was on inherent material anomalies 10 that can occur in titanium alloys. A zone-based probabilistic framework 11 was identified to address anomaly location uncertainty and other random variables. A probabilistic fracture mechanics-based life prediction capability was developed using stress intensity factor values estimated from stress values found in D axisymmetic finite element model results. The methodology was recently extended to address surface damage-related anomalies 7, 8, 1. It is based on the assumption that a Mode I crack propagates in a plane normal to the maximum principal stress at the initial crack location in a region idealized as a rectangular plate. Stresses and temperatures in this plane are extracted from 3D * Research Engineer, Reliability and Materials Integrity, 60 Culebra Road, Member AIAA. Principal Engineer, Reliability and Materials Integrity, 60 Culebra Road, Member AIAA. Owner, Encino Way. 1

2 finite element model results for application to fracture mechanics computations. In contrast with inherent anomalies that may be present throughout a component, surface damage anomalies are found in specific features such as bolt holes or turned (machined) surfaces. Therefore, a feature-based approach is used for life prediction in which component risk is estimated as the sum of the risks of the individual features. In this paper, a methodology is presented for application of 3D stress values to surface damage-based fracture mechanics computations. The stress extraction procedure is described briefly and demonstrated for (1) a finite width plate with a centrally located hole and () an aircraft gas turbine engine rotor disk. The results can be applied to the probabilistic life prediction of components subjected to surface damage. II. Stress Extraction Methodology The procedure for extracting stress values from 3D finite element model results consists of the following steps: 1. Specify initial surface crack location in 3D finite element model (FEM).. Identify orientation of the maximum principal stress plane at crack origin. 3. Identify points of intersection between the principal stress plane and the 3D finite elements. 4. Create D finite element mesh consisting of the intersection points identified in step Specify orientation of stress gradient vector in the direction of crack propagation. 6. Extract stresses at specific locations along the vector specified in step 5. A user-friendly graphical user interface (GUI) is provided in DARWIN that allows the user to select a node on the surface of a 3D FEM as the initial crack location (step 1). The magnitude of the principal stresses ( 1,, 3 ) at this location are defined as the roots of the third order stress polynomial 13 : where 3 I I I = 0 (1) xy I 1 3 = xx yy zz () xx xy xx xz yy yz I = (3) yy xx xz xy zz xz yz zz I3 = (4) xy yy yz xz yz xx and xx, yy, zz, xy, xz, yz are the components of stress in a Cartesian coordinate system. The orientation of the vector normal to the principal stress plane (step ) is identified using the direction cosines (l, m, n) obtained from the solution of the following equations 13 : ( ) + m + n 0 l l xx 1 xy xz = (5) ( ) + n 0 xy + m yy 1 yz = (6) l xz ( ) 0 + m + n yz zz 1 = l + m + n = 1.0 (8) Step 3 (identify intersection points) is illustrated for a two-element model shown in Fig. 1. The principal stress plane ( slice plane) intersects the edges of elements 1 and at locations A, C, D, E, F, G, and A, B, and C, respectively. Stress values ( xx, yy, zz, xy, xz, yz ) at the intersection points are interpolated from values at the corner nodes of the associated three-dimensional elements. Values of stresses normal to the slice plane are computed from the stress tensor at each intersection point using the direction cosines of the slice plane: (7)

3 T E P O 3D Element 1 Principal Stress Plane C R L B K F S G A M N 3D Element Q D I Figure 1. Points of intersection of principal stress plane and edges of 3D finite elements are used to define a new mesh in the principal stress plane. J N xx yy zz ( mn + lm ) = l + m + n + ln + yz xz xy (11) In Step 4, a new two-dimensional mesh is created based on the planar set of intersection points from Step 3. Nodal connectivities for the elements in the D mesh are identified using a Delaunay triangularization algorithm 14, 15. Using this technique, the mesh shown in Fig. can be generated from the slice intersection shown in Fig. 1. E C B F A G D Figure. Two-dimensional finite element mesh created from 3D element/principal stress plane intersection points using Delaunay triangularization. Once the D mesh has been defined, the orientation of the stress gradient vector and coordinates of the associated extraction points (steps 5 and 6) can be specified by the user, shown in Fig. 3. Values of the stresses at the extraction points are computed using shape functions associated with dimensional triangular elements. These stress values are subsequently used for probabilistic fracture mechanics-based life prediction. 3

4 E C B F A G D User-Defined Stress Gradient Direction Figure 3. Stress are extracted from the D finite element mesh at user specified locations. III. Application Example 1: Plate with Centrally Located Hole Under Uniaxial Tension The stress extraction methodology is illustrated for the isotropic plate under uniform tensile loading shown in Fig. 4. Values for the geometry, material properties, and applied stress are indicated in Table 1. This configuration can be approximated as an infinite plate with a hole under uniaxial tension where the analytical solution is given by 16 : 4 S 3 4 R 1 S R R 1 r = + + cos θ 4 (1) r r r 4 S 3 θ R 1 S R = + 1 cos θ + 4 r r (13) 4 S 3 θ R R 1 r = + sin θ 4 (14) r r where S is the applied stress, R is the hole radius, r is the distance from the center of the hole and θ is the angle from the loading axis. This solution represents the spatial variation of stress in the vicinity of a hole, however, it does not address the finite width dimensions of this example. The plate shown in Fig. 4 was modeled using the finite element method (ANSYS model with 40,000 0-noded hexahedral elements) with a single axis of symmetry, shown in Fig 5. An initial crack was placed at the intersection of the hole and the upper plate surface at a circumferential location 90 degrees from the axis of loading (point A in Fig. 4). At this location, the maximum principal stress is parallel to the θ direction (i.e., parallel to the loading axis). The finite element stress result at this location (63,380 psi) is within 1% of the analytical stress (6,704 psi) predicted using a relation that accounts for the finite width of the plate 17 : 3 R R R SD max = (15) D D D ( D R) 4

Item Variable Value Units Plate Thickness t 1.00 inch Plate Height H 0.0 inch Plate Width D 0.0 inch Hole Radius R 1.

5 Figure 4. Application Example 1: isotropic plate under uniform tensile loading. Table 1. Description of Geometry and Material Variables Associated with Application Example 1. Item Variable Value Units Plate Thickness t 1.00 inch Plate Height H 0.0 inch Plate Width D 0.0 inch Hole Radius R 1.00 inch Young s Modulus E 30.0E6 psi Poisson s Ratio ν 0.3 N/A Applied Stress S 0,000 psi Figure 5. Application Example 1: isotropic plate finite element model based on a single axis of symmetry. 5

6 60,000 DARWIN 50,000 ANSYS Analytical solution for infinite width plate Stress, psi 40,000 30,000 0,000 10, Distance from hole, in Figure 6. Comparison of DARWIN stress gradient with ANSYS and analytical solutions demonstrates the validity of the stress extraction methodology. While Eq. (15) gives an indication of the maximum stress in the plate, it does not provide an indication of the spatial variation of the stress. Spatial stress variation is computed using the analytical solution for the infinite plate with a hole (Eq. 13). A comparison of this solution with the finite element solution (Fig. 6) indicates that the stress gradients predicted using these two methods are in close agreement. The difference between these two solutions is primarily due to the plate boundary effects not included in the analytical plate solution. The plate was also modeled using the 3D stress extraction methodology recently implemented 9 in DARWIN. The orientation of the principal stress plane at the initial crack location, shown in Fig. 7, is consistent with the analytical and finite element solutions. The D finite element mesh (generated from the Delaunay triangularization of the slice plane intersection points) describing the geometry and stress extraction points on the principal stress plane is shown in Fig. 8. Initial Crack Location Maximum Principal Stress Plane Figure 7. DARWIN display of 3D finite element model and associated principal stress plane at the initial crack location. 6

$Note that x and y are local coordinates defined for fracture mechanics plate. Values for the stresses extracted at the locations indicated in Fig.$ 8 are compared with the ANSYS and analytical solution for an infinite plate with a hole in Fig. 6.

8 are compared with the ANSYS and analytical solution for an infinite plate with a hole in Fig. 6.

7 Initial Crack Location Stress Extraction Points Figure 8. D finite element mesh generated by DARWIN based on intersection of principal stress plane and 3D finite element model. Note that x and y are local coordinates defined for fracture mechanics plate. Values for the stresses extracted at the locations indicated in Fig. 8 are compared with the ANSYS and analytical solution for an infinite plate with a hole in Fig. 6. It can be observed that the stress values obtained from DARWIN are in close agreement with both the analytical and finite element solutions (DARWIN and ANSYS solutions differ by a maximum of 0.3%). IV. Application Example : Gas Turbine Engine Rotor Disk The stress extraction methodology is also illustrated for an idealized aircraft gas turbine enginerotor disk (Fig. 9). Material properties associated with the disk are indicated in Table. A finite element model was created in which two sets of external loading conditions were applied to simulate operating conditions: (1) angular velocity of,300 radians/sec about the Y axis and () uniform temperature of 600 o F. Cyclic symmetry of the model was enforced by restricting displacements normal to the symmetry planes. The disk was also modeled using the 3D stress extraction methodology in DARWIN (Figs ). The initial crack location and associated principal stress plane are shown in Fig. 10. The D finite element mesh of the principal stress plane and associated stress extraction points are shown in Figure 11. Values for the stresses extracted at these points are shown in Fig. 1 along with the stress gradient extracted directly from ANSYS. A comparison of the stress values predicted using DARWIN with the finite element analysis results indicates close agreement between the two methods (DARWIN and FEM solutions differ by a maximum of 0.5%). Figure 9. Application Example : Aircraft engine rotor disk with axially oriented bolt holes. 7

Table. Material Properties Associated with Application Example. Variable Value Units Young s Modulus 17.4E6 psi Poisson s Ratio 0.

DARWIN representation of 3D finite element model and associated principal stress plane at initial crack location.

8 Table. Material Properties Associated with Application Example. Variable Value Units Young s Modulus 17.4E6 psi Poisson s Ratio N/A Coefficient of Thermal Expansion 0.3E-5 1/ o F Density 0.417E-3 lb sec/in 4 Figure 10. DARWIN representation of 3D finite element model and associated principal stress plane at initial crack location. Initial Crack Location Stress Extraction Points Figure 11. DARWIN generated D finite element model with radially oriented stress gradient superimposed on the two-dimensional planar mesh. Note: x and y axis labels apply to local fracture mechanics plate. 8

9 DARWIN ANSYS 140,000 10, ,000 Stress, psi 80,000 60,000 40,000 0, Distance From Crack, in Figure 1. Stress gradient values extracted using DARWIN are in close agreement with ANSYS finite element results for Application Example. V. Summary A methodology was presented for application of stress values from 3D finite element model results for use in fracture mechanics computations. It is based on the assumption that Mode I cracks propagate in the maximum principal stress plane. The methodology was demonstrated for a crack at a hole in an isotropic plate and an aircraft gas turbine engine rotor disk. Computational error associated with the methodology is less than 1% when compared to analytical and finite element solutions. The results can be applied to the probabilistic life prediction of components subjected to surface damage. VI. Acknowledgments This work was supported by the Federal Aviation Administration under Cooperative Agreement 95-G-041 and Grant 99-G-016. The authors wish to thank Joe Wilson (FAA Technical Center project manager) and Tim Mouzakis (FAA Engine and Propeller Directorate) for their continued support. The ongoing contributions of the Industry Steering Committee (Darryl Lehmann, Pratt & Whitney; Jon Tschopp, General Electric; Ahsan Jameel, Honeywell; Jon Dubke, Rolls-Royce) are also gratefully acknowledged. VII. References 1 National Transportation Safety Board, Aircraft Accident Report - United Airlines Flight 3 McDonnell Douglas DC Sioux Gateway Airport, Sioux City, Iowa, July 19, 1989, NTSB/AAR-90/06, Washington, DC., Federal Aviation Administration, Advisory Circular - Damage Tolerance for High Energy Turbine Engine Rotors, AC , U.S. Department of Transportation, Washington, DC., Leverant, G.R., "Turbine Rotor Material Design - Final Report," Federal Aviation Administration, DOT/ FAA/ AR-00/64, Washington, DC, Leverant, G.R., Littlefield, D.L., McClung, R.C., Millwater, H.R., Wu, Y-T, A probabilistic approach to aircraft turbine material design, Proceedings of the 4nd ASME International Gas Turbine & Aeroengine Technical Congress, Paper 97-GT-, McClung, R.C., Enright, M.P., Millwater, H.R., Leverant, G.R., and Hudak, S.J, A software framework for probabilistic fatigue life assessment, Probabilistic Aspects of Life Prediction, ASTM STP 1450, ASTM International, West Conshohocken, PA, 003 (in press). 9

10 6 Leverant, G.R., McClung, R.C., Millwater, H.R., and Enright, M.P, A new tool for design and certification of aircraft turbine rotors, Journal of Engineering for Gas Turbines and Power, ASME, Vol. 16, No. 1, 003, pp McClung, R.C., Enright, M.P., Lee, Y-D., Huyse, L., and Fitch, S.H.K, Efficient fracture design for complex turbine engine components, Proceedings of the 49th ASME International Gas Turbine & Aeroengine Technical Congress, Vienna, Austria, June 14-17, Enright, M.P., Huyse, L., McClung, R.C., and Millwater. H.R, Probabilistic methodology for life prediction of aircraft turbine rotors, Proceedings, 9th Biennial ASCE Aerospace Division International Conference on Engineering, Construction and Operations in Challenging Environments (Earth & Space 004), R.B. Malla & A. Maji, Eds., ASCE, Houston, TX, March 7-10, 004, pp DARWIN, Design Assessment of Reliability with INspection, Software Package, Ver. 5.1, Southwest Research Institute, San Antonio, TX, Aerospace Industries Association Rotor Integrity Subcommittee, The development of anomaly distributions for aircraft engine titanium disk alloys, Proceedings, 38th Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, April 7-10, 1997, pp Wu, Y.T., Enright, M.P., and Millwater, H.R., Probabilistic methods for design assessment of reliability with inspection, AIAA Journal, Vol.40, No. 5, 00, pp Enright, M.P., Lee, Y-D, McClung, R.C., Huyse, L., Leverant, G.R., Millwater, H.R., and Fitch, S.K, Probabilistic surface damage tolerance assessment of aircraft turbine rotors, Proceedings of the 48th ASME International Gas Turbine & Aeroengine Technical Congress, Atlanta, GA, June 16-19, Boresi, A P., Schmidt, A P., and Sidebottom, O. M., Advanced Mechanics of Materials, John Wiley & Sons, Inc., de Berg, M., van Krevald, M., Overmars, M., and Schwarzkopf, O., Computational Geometry Algorithms and Applications, Spinger, Heckbert, P. S. ed., Graphics Gems IV, Morgan Kaufmann, Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill Book Company, Young, W. C., Roark s Formulas for Stress and Strain, McGraw-Hill Book Company,

Summary of New DARWIN TM Capabilities for Surface Damage Tolerance Assessment and Related Enhancements

Summary of New DARWIN TM Capabilities for Surface Damage Tolerance Assessment and Related Enhancements Mike Enright, Luc Huyse, Yi-Der Lee, Graham Chell, Craig McClung, Gerald Leverant Southwest Research