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Recent advances in computerized aerospace structural analysis and design
RECENT ADVANCES IN COMPUTERIZED AEROSPACE STRUCTURAL ANALYSIS AND DESIGN? RALFW E. MILLER,JR., B.
F. BACKMAN,H. B. HANSTEEN,$ C. M. LEWIS, R. A. SAMUELand S. R. VNUNASI (Receiued15 Nouember 1975)
At&a&-This paperpresentsaccomplishmentsat the BoeingCommercialAirplaneCompanyin the areasof (1) large problemsolutions, (2) practicalstructuraldesign automationand (3) a hypothesis for stable crack growth. Aturbine blade (3200nodes 350elements,9500freedoms, 15millionwords of storage)and a sportsstadium(3400 nodes, 9600 elements, 20,ooOfreedoms, 70 millionwords of storage)are presentedin terms of problemdefinition, strategyof approach,data managementand computerresources. The Boeing automatedstrengthdesign capabilityin the ATLAS System is discussed in terms of user specified constraintson the automatedstrengthresixingand local optimixation,strengthcriteria,user controland convergence criteria.Cost and weight data are presentedfrom large aircraftdesign studies (20,000design variables). A new concept of crack stabilitybased on non-linearstress/strainanalysis is presented.Stable crack growthis modeled by a suitable failure criterionin the stress analysis procedure.A crack is extended by a simultaneous unloadingof the newly created crack surface and the loading of the untracked region. Test comparisonsand numericalexperimentssupporta new hypothesisbased on materialstrengthcharacteristics,plasticzone size/history and the residualplastic strains. LARGE PRORLEM WLITl’lONS
Fig. 2. In addition to the applied loads shown, support boundary conditions were applied along the fir tree attachment lands and at the blade tip shroud. Geometry. Turbine blade geometry presents a modeling problem much more extensive than the size of the part indicates. Detailed geometric representation and stresses are required in the region of the airfoil root and the lir tree attachment to the turbine disk These areas were modeled with higher order isoparametric elements having 32 nodes per element. Substructuring and data management. The large bandwidth produced by these finite elements caused great amounts of data to be stored on the computer disk tiles. To execute the analysis on the Boeing Computer Services CDC 6600 computers as a production run, it was necessary to substructure theblade as shown in Fig. 3. The alternative to substructuring, a full model of the blade, would have resulted in prohibitive block (weekend) computer time costs, and excessive analysis flowtime. Two criteria governed the choice of the substructure boundaries (size). One criteria ensured efficient reanalysis for different boundary conditions. The second constrained the volume of data associated with various steps of the analysis to practical sizes.
Turbine blade analysis The turbine blade shown in Fig. 1, even though only a small part of the turbine stage presents one of the most complex design problems in the gas turbine engine. Even recently, turbine designers were forced to rely on expensive trial and error testing to optimize blade design. Recent developments on the isoparametric solid finite element, have made numerical 3dimensional stress analysis of turbine blades a feasible design tool. The following discusses an application of the ATLAS system[l, 21 to a three dimensional stress analysis of a turbine blade. The analysis was performed as part of a contract with the U.S. Army[3]. The approach to the problem, data management and costs are emphasized. Loads. The operating environment of the blade includes temperatures, inertia loads due to rotation and pressure loads. These loadiis are shown schematically in
AT~lp
TURBINE -A’%
TYPuALBLmE
Fig. 1. 3-D Stressanalysisof a turbineblade. tPresented at the Second National Symposiumon Computerixed StructuralAnalysis and Design at the School of Engineering and AppliedScience, GeorgeWashingtonUniversity,Washington D.C. 29-31 March 1976. $BoeingComputerServices, Inc. 315
31 PRESSURE
THERMAL i
6
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-
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2. Substructurescheme.
INERTIA
316
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words was required because tiles may not overlap disk boundaries. These criteria required that the portion of the blade between the tip and fir tree substructures be broken into multiple substructures, as shown in Table 1. Table1. Turbinebladesubstructures
9RlClS “WOE9
92
NoDrs/srlcr
32
TOTAL 0.o.r. II,EIACTI"O D.O.F.
81
23
0
II
66
90
32
32
-
3220
657
171
2685
2859
433
138
48
007
462
-
I4&
8b
76
1281
732
531
26
13
44
48
26
3527
946
244
945
AYfRAGE HlLF 9110 10. or D.0.F.
SLBSTRIJCTURES
TOTAL MOCEL
I or TOT&L IO. OF ,lPUT CARD,
Fig.
2913
56 -
3. Bladeloading.
The stresses in the lir treeand tip shroud areas are sensitive to details of the boundary conditions in those areas. Thus multiple analyses, each with different boundary conditions, would be required. Re-analysis cost was minimized by chasing one substructure for the tip and one for the tir tree. Bach substructure was as small as possible while still completely containing all boundary freedoms. This scheme required recalculating for the second and subsequent analyses, only the reduced stiffness and loads matrices for these two substructures. The second set of criteria minimized computer costs and flowtime by ensuring regular production runs on the CDC 6600 computers. Regular production runs were advantageous because they; (a) could be made at least once each day and thereby minim& the total flowtime for the analysis, and (b) reduce the single run residency time upon the computer thereby minGzing the potential of machine unreliability. Because the amount of disk space available to store the data during execution was limited, the management of disk storage became the key to the selection of substructure size and analysis solution steps. Disk storage. At the time of the analysis the theoretical disk storage capacity of the four available disks was approximately 40 million words. Practicality reduced this capacity because; defects made portions of disks unavailable; one disk was shared among the three central processors (CP’s) and multiple jobs were executing on each CP. Thus a practical limit of 16 million words of disk storage was calculated as available for any single execution. In addition, a disk single tile limit of 5 million
The choice of analysis steps also aiIected disk storage requirements. Only the following four steps involved signiticant amount of stored data: generation of element stiffness and stress matrices assembly (merging) of gross stiffness matrices for lowest level substructures calculation of reduced stiffness and loads matrices for lowest level substructures displacement backsubstitution and element stress calculation. The total disk storage requirements for the critical substructure during each of these steps are shown in Pig. 4. As shown in that figure, a single substructure model was impractical since its disk requirements would have exceeded the practical capacity. Several important parameters describing such a single substructure model are presented in Table 2. Table 2. Turbine blade single substructuremodel No. of nodes No. of bricks
2885
352 8655 870
TotalDOF Averagehalf band
The maximum size of any single file for the critical substructure during each step of the analysis is shown in Fig. 5. While the maximum file size for multiple substructures is only approximately half the practical
PEAK TOTAL DISK
3254
STORAGE
SUBSTRUCTURED
REQUIREMENTS.
PRACTICAL
LIMIT
(IO6 WORDS)
'
STIFFNESS A LOADS
'
LOWEST LEVEL MERGE
'
REDUCTION A SOLUTION
BACK-
'
SUBSTITUTION A STRESS
Fig.4. Totaldisk requirements,turbinebladeanalysis.
'
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Recent advancesin computerizedaerospacestructuralanalysisand design
THEORETICAL
LIMIT
PEAK SINGLE
DISK
FILE SIZE, (IOS WOROS)
‘r-
STIFFNESS a
I
LOWEST LEVEL
LOADS
I
REDUCTION b SOLUTION
MERGE
I
BACK SUBSTITUTION
I
6 STRESS
Fig. 5. Maximumfile size, turbine blade analysis.
limit, a single substructure analysis would have significantly exceeded the available capacity. Cost. The cost of the analysis execution is given in terms of CP seconds in Table 3. The percentage of the total problem cost associated with each major ATLAS module execution is also given. Note that more than 60% of the total cost of execution is spent in the REDUCE phase. This emphasizes the need for the most efficient possible code for the matrix decomposition and matrix multiplication routines.
I1
CP
TIME.
1696 56 I 31 I2 973
13 6 14
a9 3a 47 220
5.9 2.7 15.0 4.5
x WATER
SEE
BEARINGS
FlGuRE7a~~
Fig. 6. Plan view and support configuration, Denver Mile High Stadium.
+----
145 FEET
.a .2 .3 I.1
1149 391 6276 761
4.2 2.1 34.0 3.8
I6
1551 SOS 2929 654
1 I .5 3.5
460 322
1.7 I.1
21554 (0 6 NRS)
i
UNIT
SEC
I9
21
FfMER
i-
Table3. Costs,turbine blade analysis SVRSTRVCTVRE
DIRECTION
t Mo"&T
5.3 2.3
Fig. 7. Cross-sectional frame, Denver Mile High Stadium.
100.0
Analysis of the Denver Mile High Stadium extension In a project for the City and County of Denver, Colorado, a new East Stand is added to the Denver Mile High Stadium increasing the current seating capacity by approximately 35,000 seats (consulting engineers are DMJM-Phillips Reister, Inc., Denver, Colorado). The superstructure of the new stand can move as a single unit to enlarge the playing field in the case of baseball, or to reduce the playing field for football or soccer. The moving system is composed of a water bearing support subsystem which will reduce friction and facilitate movement, a guiding system and a moving and control subsystem. The water bearing supports glide on a system of concrete runways which are placed below the surface of the playing field. Problem description. The structural analysis discussed in this paper applies to the moveable part of the new stands shown in Fiis. 6 and 7. This structure is mostly made of steel and consists of 19 straight bays of 17-ft
width each, and of 5 radial bays of 5” width each at both ends. Stadium dimensions include a length of approximately 534 ft, a width of 192ft and a height of 127ft. The distance between the two extreme positions of the stadium is 145ft. The structure is supported by 67 water bearings as indicated by the x’es in Fig. 6. When the stadium is being moved the bearings can only transmit vertical reactions. The bearings will in addition act as flexible supports for forces in the horiiontal plane when the stadium is in a fixed position. The two power units located near the two outer ends of the structure and the guide rail near the middle provide horizontal supports for the stadium when it is moved. Since these power/support mechanisms are located in the fixed substructure of the stadium, their position relative to the superstructure will change. The stadium was analyzed for 13 different load cases. This included dead load, live load, four cases of wind loads, a sway load in the longitudinal direction of the stands, four differential deflection load cases, inertia load
MILLRR, JR. et ai.
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in direction of stadium movement and a friction load acting at the water bearing points. The analysis also included a study of 13 different sets of boundary conditions. Twelve of these conditions applied to different locations of the power units and the guide mechanism due to movement of the stands. The last condition applied to the stands in a fixed service position. Particular interest in the analysis was devoted to the water bearing system, and the effects of differential deflections of these points. To study these effects, the reduced stiffness matrix with regard to the vertical displacement component of these points was computed. Solution strategy. The factors determining the computational task of performing a finite element analysis of the stadium is shown in Fig. 8. The solution for one of the 13
A static condensation was performed on this system with respect to the r2 displacements. The mathematical operations carried out are described in Fig. 9, task 5. Phase 2. The remaining equation system &=l? was used as a basis to impose the 13 different sets of
boundary conditions. This includes a re-partitioning of the
Task
1 Value
Description
Symbol used in eqn (2)
2 3
3152 9606 13 13
Number of structural nodes Number of finite elements Number of load cases Number of independent sets of boundary conditions Dimension of global stiffness matrix Average bandwidth of globalst8ness matrix
4 4
20712
R
600
b
5
Fig. 8. Basic parameters, Denver Mile High Stadium. sets of boundary
conditions
involves
solving a set of
6
linear equations Kr=R
(1)
where K is the global stiffness matrix of the structure, t the nodal displacements and R the nodal loads. The work required to perform this solution may be measured by the number of multiplications and additions to be carried out. This may be approximated by the formula s = l/2& *+ 2nbq.
(2)
Substituting the actual values from Fii. 8 yields S = 4x 109 operations. Considering that each of the 13 sets of boundary conditions represents a new solution, it would appear that the complete analysis requires more than 50 billion (50 x 10 basic operations. This constitutes a very considerable amount of work even for the most modern electronic computers. The approach chosen for the actual solution reduced the amount of re-analysis required for each new set of boundary conditions. The solution was carried out in two phases as follows: Phase 1. The nodal freedoms of the structure were partitioned intq the two subsets r, and rz, where r-The muon of all freedoms which are being supported iu any of the structurai con@uations. This includes the x, y and z-translational freedoms at the water bearing points, the y-translational freedoms at 12 locations along each power unit track and the xtranslational freedoms at the corresponding 12 locations along the guide rail. r2contains a total of 237 freedoms. r,-The remaining 20,475 freedoms of the structure. The linear system of eqns (1) was partitioned accordi@Y,
I
Description Read input data Generate structural plots and produce various verification printing to establish correctness of input data Generate finite element stiffness matricies and nodal load matricies Assemble the governing linear system of equations for the total structure, represented by the stiffness matricies K,,, K,,, and Kz2,and the load matrices R, and R, Perform static condensation of the system of equations with respect to the displacements r,. The operation is carried out in 4 subtasks: (A) Decompose matrix K,, +L,, (B) Compute C12= L;,‘K,, (C) Compute HI = L;,‘L;iR,
g* = L ;,% (D) Compute K = Kz2- C& R = Rz- KBH,
Perform a solution for each of the 13 support conhgurations. Each solution constitutes a subtask performing the following operations: (A) Apply the boundary conditions to form the matrix partitioning;
r, = Unconstrained freedoms rs = Support freedoms (B) Solve for r, (C) Compute rl = H, - H2r2 (D) Compute member stresses and reaction forces Compute the reduced stiffness matrix with respect to the Z-displacements at the neoprene pad nodes. The following operations are performed: (A) Partitioning the condensed system arrived at in Task 5 into:
rp = Unconstrained freedoms rR = Z-displacements at neoprene pad nodes rs = y;;i-y supports to assure a non-singular 8
(B) gmpute I?- = Km - KmKgKm Produce printouts of subsets of displacements, stresses and reaction forces
Fig. 9. Computer task description, Denver Mile High Stadium.
319
Recent advances in computerized aerospace structural analysis and design
system (4) and, for the futed stadium con8guration, addition of x- and y-spring supports at all water bearing points. The displacement partitions I, and rz were then computed as described in Fig. 9, task 6. The work required to perform the two-phased solution described above is again measured in terms of the number of multiplications and additions to obtain r, and rz. With reference to Fig. 9 the requirements are as follows: Task 5 Performed once Task 6 Performed 13times
10.2X 109operations 0.9 X 109operations 11.3X 109operations
This shows (50x 109 to 11x 109 that the selected approach reduced the amount of computational work required for the solution by approximately 80% compared to a conventional analysis. Task management. The computer analysis was carried out using the ATLAS program[l] which operates on CDC 6008 and CYBER series machines. Because of the size of the project, the necessity of checking intermediate results and varying requirements with regard to computer resources at diierent points of the analysis, it was necessary to perform the computer running as a series of subtasks. The data generated in the course of the analysis was maintained in an off-line data base. At the beginning of one subtask the data required by that task was loaded by the ATLAS program. Similarly, at the end of a task the data base was updated by any new information generated. The task breakdown is defined in Fii. 9. Tasks 1 and 2 were concerned with preparation and checking of data input. Data generators were used extensively in the data preparation to take advantage of repetitive information or patterns in the structure. Thus the data volume required for the analysis was “limit& to about 5000 cards. Without data generators it would have been in the range of 10 times more. Verification of correctness of data volumes of this size constitutes a major problem. For the stadium a sign&ant part of the data checking was achieved by inspecting plots of the structure. A total of about 65 plots showing different regions of the structure was required to perform a complete data verification. The bulk of the computer running was performed in tasks 3-7. The main problems encountered in the planning and execution of these tasks were related to the data volume that had to be maintained. The size of this volume at the end of each computational task is shown in Fig. 10.
A peak value of about 70 million 60 bit “words” was required during execution of task 5. Special provisions had to be made to support a data volume of this size. During execution of task 5 the computer had to be specifically con&red due to the extreme size of the matrices being processed. In the reconliguration seven 844 disk units were combined as one logical equipment. While performing task 6 special care had tc be taken to balance the data volume on the available disk units. Whereas no particular problems were encountered in maintaining the off-line data base, it was found that the data copying involved in starting and checkpointing of a task was considerably more expensive than anticipated. Report generating of results was performed in task 8. At this point all “raw” data were available in the data base, and only the final data selection and formatting remained to be done. Typically, task 8 will be performed several times as the analysis of the results proceed and requirements for more information become apparent. The computing resources required to complete the analysis are illustrated in Fig. 11 by the accumulated central processor time at the end of each computer task. The total amount at the end of the project was about 32OOOsecs,or roughly 9hr of CP time. The actual residency time in the computer was much higher than this, probably in the range of 40-50hr. The flowtime for the whole project, including data preparation and checkout, was about 3 months. The relative costs of each computer task is shown in Fig. 12. Practical-automated structuml design The basic automated structural design capability
developed at Boeing for the ATLAS System is a modified
L&II!__ ‘Onrn '2345676
TA5K
Fig. 11. AccumulatedCP time, Denver MileHigh Stadium. 50 -
40
40 -
DATA mw, wRDs
30
30-
Xlos
%
20
IO
OF COST 20
1
IO
6
10. Accumulateddata volume, Denver MileHigh Stadium.
7
6
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TASK Fig.
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Relativecost, Denver MileHighStadium.
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COST EFFICIENCY FLOW TIME REDUCTION
I
CONVERGENCEIYPRUVEMENT PSYCOLOQICALADVANTAGE
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CONCEPTS I
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Fii. 13. Designautomation;objectivestructure.
fully stressed design (F.S.D.)[l]. The selection and development of this baseline function is a logical consequence of the objectives illustrated in Fig. 13. These objectives demand the implementation of a method that satisfies the following requirements: The typical aerospace structure is characterized by a huge number of variables and an enormous amount of data. The method must therefore be particularly feasible in a “large problem environment.” The method must be especially suitable for applications in the field of preliminary design. The flexibility in usage of the method should be based on user control in execution and data handling. Cost efficiency must be the dominating concern in selection of method and architecture of data management. Flowtime reduction is obviously a natural consequence of automation. However, since data preparation and changes constitute a very time consuming phase, it is essential to emphasize this aspect. Functions for checking of input, modification of data and display of output therefore becomes especiahy important. Input efficiencv is YI importance from a user feasibility standpoint. Thus regional input structure and update capabilities are indispensable functions. A method that is based on traditional concepts ensures preservation of existing experience and promotes fast user acceptance. It therefore is highly desirable that this facet be considered. ‘Ibis set of objectives is displayed in Fig. 13 and evaluated in terms of pay-off potential and user appeal and acceptance. The block-diagrams also illustrate the relative emphasis on the different goals. The dominating characteristic of each aspect in the evaluation is represented by key-words in each block. User Control. As the most feasible type of design automation in a “large problem environment” involves optimality criteria and in particular, the concept of a fully stressed design, it becomes imperative to design a process that provides maximum control of convergence related phenomena. The traditional F.S.D.-method has been modified and
complemented with a structure of functions for user control which requires: Data control Execution control Convergence control. Figure 14 shows the functions and their relative importance in supporting these three requirements. INPUT COilSTRAINTS: "Pm UD L6sERwJ"OS,FlXEDDATA.,"W, l!AllXS OF wm
Fig. 14. Relative emphasis on the design of user-control functions.
The data control is achieved by an input structure that is characterized by: A variety of con&ant data subsets A data hierarchy An override capability at time of execution A regional input definition (in addition to element specifications)
Recentadvancesin computerizedaerospace structuralanalysisanddesign
An input update capability A regional resize definition The execution control is supported by: Selective execution of technical algorithms System procedures for iteration control System restart capability Regionalresizing The convergencecontrol is exercisedand supportedby: A history function recordingand displayingmarginsof safety and maximumchanges The input structure and update capability A regional technical or user specitied resize function User specified convergence criteria. Usage experience.Usage experience with design automation in the “large problem environment” has until recently been very liited. Results from Boeing design studies involving problems of a size of 15800420,000 designvariablesare shownin Fii. 15.The relative cost per design variable are related to pre-automated levels, as baseline.Fire 16shows the solutiontimes for ~erent segments of the design process. Three different approaches to the use of automated designare illustrated.Project I used, as a startingpoint, a detailed manual sizing and a highly differentiatedinput. The execution took place in steps in the sense that one cycle was broken up into: prepocessing,analysis,partial
RELATIVE
321
postprocessing, design and total postprocessing. Each step was followed by checkingof results and evaluation. This procedure accounts for the relatively high cost. Despite this, a sign&ant cost reduction was realized. The second project had a startingpoiut that was defined by a regionalmanualsizingand less di#Terentiatedinput. The execution was performed for one cycle at a time and the checking was considerably less extensive. Project III finally had a completely arbitrary starting point and the input was of regional character. The execution was performed for more than one cycle at a time. It is barest to note that, ~dependent of starting point, sign&ant cost-savings were realized. It is also promising to note how similar the convergence characteristics for these three designs were. Figure 17 illustrates this point in terms of weight change relative to starting point. This experience seems to indicate that very little benefit is derived from the effort of establishing a “good” starting point. The behaviour of aerospace structures from a convergence standpoint is very complex. Present experience, however, seems to support the idea that even though a whole family of designs with different element properties represents potential candidates only small weight differences exist between them.
COST
PER OESIQW CYCLE (WE LINE : PRE-ATLAS COST LEVEL)
FIRST
CVCLE
..
,.”
WECTIVE
Fii. 1%Relativecost for diierent approachesto design.
2500
cp, SECONDS
Fig. 16. Computertimerequirement.
CYCLES
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RELATIVE "5b"#' DESIGNS .50
0
I
2
3
DESIGN CYCLE
Fig. 17. Relative weight change for different approaches to design.
Ben& characteristics. The automated design process that has been discussed this far is of the optimality criterion type with stress constraints defined by strength and buckling algorithms. The cost-reductions realized in applications have been significant but are, of course, size-dependent. Figure 18 shows an intensity description of pay-off. The dark regions represent high potential for cost-savings. The !igure illustrates two cases, single structure idealization and single level substructuring. The black regions represent maximum pay-off from design automation. Unfortunately these high payoff regions coincide with limitations of the computing system. Capacity of bulk storage devices generally limits the size of the maximum payoff regions. Substructuring capability, however, allows a dramatic extension of the feasible payoff region. An introduction of multi-level substructuring would extend the payoff regions indefinitely, limited then by cost. Figure 18 could also be interpreted as an illustration of the success of the automation of the structural design process. The black and to some extent the dark regions represent costs that are one order of magnitude lower than pre-automation levels. It furthermore illustrates the capacity for solving problems of a size that only a few years ago was considered to be out of reach.
NUMBER OF DESIGN VARIABLE
0
0
I0000 20&O ANALYSIS NUMBER OF DEGREES OF FREEDOM (DARK = HIGH INTENSITY)
Fig. 18. Design automation: intensity description of feasibility and payoff potential.
Current status and future. The design process contains considerably more aspects than discussed this far. This section will illustrate the overall picture with regard to status and development of the automation of design as seen from the standpoint of need for an immediate future. The experiencegained (illustrated in Figs. 15and 17),the potential realized, (shown in Fig. 18)and the requirements imposed by advances in the state-of-the-art constitute the basis for the direction, emphasis and character of the development plans illustrated in Fig. 19. Two basis subjects are considered: 1. Optimality criteria 2. Direct optimization. The development for item 1 involves: More technical algorithms for stress constraints Algorithms for introduction of stiffness constraints into the design process Practical supporting functions. In the field of direct optimization two items are considered: Component optimization Regional optimization. Component optimization is intended to complement to the optimality criteria. It is directed toward a lower level involving detailed structural definitions. It attacks the problem from a concept standpoint and lets finite elements be associated with concepts like skin-stringer panels, sandwich panels, composite laminates, etc. The regional optimization is intended as an alternative to optimality criteria and involves: Selection of key-elements representing structural regions Substructuring with non-element related design variables Introduction of practical constraints (e.g. tapered properties). A new stable crack growth hypothesis Boeing research on a ductile fracture criterion has produced a new concept of crack stability based on nonlinear stress-strain analysis using the finite element method. The finite element method has been applied previously in the study of a steadily growing crack under monotonically increasing load [4,5], and in the study of a fatigue crack growth under cyclic loading[6].
Recent advances in computerized aerospace structural analysis and design
FULLY
323
STRESSED DESIGN
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COMPOSITES 0 STRENGTH
SMOOTHING *PRACTICAL
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REGIONAL
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l f4~4~MENT
MILES
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DIRECT OPTIMIZATIONS EXISTING, m
SHORT-RANGE PLANS(<
I YEAR), j--I
LONG-RANGE PLANS
Fii. 19. Design functions: status and plans.
The problem of stable and catastrophic crack growth in a center-cracked plate of a ductile material, subjected to a monotonically increasing applied stress is discussed here. An incremental elastic-plastic plane stress analysis of this cracked plate is performed using a Boeing finite element program PEPSAD] which uses a tangent stiffness formulation and a two-step Runge-Kutta integration scheme for each load-step. Constant strain triangles are used for the idealization of the cracked plate. The ductile material of the cracked plate is assumed to be an isotropically work hardening material which can be characterized by Von Mises yield condition and PrandtlReuss incremental stress-strain equations. Failure criterion-crack extension. The phenomenon of stable crack growth prior to instability is modeled by incorporating a failure criterion in the stress analysis procedure. This failure criterion permits crack growth when the maximum principal stress, u,, at the crack tip node, reaches the material ultimate tensile strength, F,,,. At the end of the stress analysis for any load-step, this failure criterion is applied to the cracked plate to ascertain whether crack growth is indicated at that load level. If crack growth is not indicated, the stress
analysis proceeds in the usual way with the application of the next load-increment. If the crack is to be extended, then the following procedure is used for unloading the newly cre@ed crack surface and calculating the redistribution of stresses as a result of crack extension (Fig. 20). 1. The crack tip node, which was previously constrained in the direction perpendicular to the crackline, is released and the crack tip is advanced to the next node on the crackline. 2. The newly released crack tip node is subjected to a force which is equal in magnitude but opposite in direction to the accumulated reaction force at that node. A stress analysis is performed with this nodal force as the only external loading. The increments of stresses and displacements due to this step are added to the accumulated stresses and displacements of the cracked sheet respectively. 3. After the crack extension and the redistribution of stresses in step 2, the failure criterion is applied again to determine if further crack growth is indicated. If further crack growth occurs, the above procedure (steps 1 and 2) is repeated. If a crack extension causes a redistribution of
RALPA E. MILLER,JR. et al.
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t t t t f
HALF CRACK 1
J’7
CRACK TIP BEFORE CRACK EXTENSION
APPLIED NODAL
FORCES NEW NODAL RESTRANlf’& FORCES
RESTRAININGFORCE
Fig.
20.
Release of nodal reshining force as crack grows in a
center-crackedsheet. stresses which causes successive crack extensions, then the process is considered as having reached the point of instability. If there is no further crack growth after a redistribution of stresses, then it is a stable crack and the external load must be increased for further crack growth. Element loading-unloading modeling. In step 2, the relaxation of the crack tip nodal force produces unloading in the newly created crack surface behind the new crack tip while loading the region ahead of the crack tip. This simultaneous loading and unloading behavior necessitates the inclusion in the stress analysis procedure of routines for changing the element stiffhess according to the incremental theory of plasticity. Therefore for an unloading element, elastic stress-strain equations are used in the calculation of the tangent stiffness, while for a loading element, elastic-plastic stress-strain relations are used. During crack growth, the determination of whether an element loads or unloads is done iteratively by an extra (trial step) application of step 2. Initially, all elements are assumed to continue loading, and the tangent stiffness equations are solved with crack tip relief force as the only external load and the stresses due to the trial load-step are accumulated in the usual manner. Then, the element equivalent stresses for before and after the load step are compared. If & 2 b,, then the element is assumed to continue loading. If ~7~c 6, then the element is assumed to unload (Fig. 21). This iteration trial step process is implemented twice in the present investigation, so as to provide an accurate (stable field) state for element loading and unloading calculations. Test comparison. The results of comparison of the
1
(a) Looding
EQUIVALENT STRAIN Z
ECUMLENT
STRAb4 Z
in an element
the relaxation of crack tip restmining force.
642-
INITIAL CRAWi 1 LENGTH = 5 IN
HALF CRACK LENGTH, a, (INCHES)
Fig. 22. Correlation of analysis results with test panel results.
REACTION FOffCE
PRINUPAL STRESS
2
// (b) -
Fig. 21. Determination of loading or unlow
analysis method to a 2024-T3 ahuuinum test panel failure are shown in Fig. 22. The panel failure stress predicted by the analysis method when the crack tip force is relaxed in a single step is 28 ksi and when the crack tip force is relaxed in ten equal steps is 30 ksi. This single, step of crack tip unloading force causes the material to behave more elastically. The discrepancy between the test panel failure stress and the analysis panel failure stress may be due to thickness effect of the test panel. Numerical experiments-load step size. The effect of the load-unload step size on the results of the analysis method is shown in Fig. 23. In this numerical experiment, the crack tip force is relaxed in one single step and a number of equal steps. The accumulated reaction force and the maximum principal stress at the new crack tip are plotted as a function of the number of steps used for the relaxation of the crack tip force. It may be seen that the solution converges rapidly and there is no appreciable change in the solution after 4 steps for this case. Numerical experiments-elastic us elastic-plastic. It is found that the determination of the elasto-plastic, load-unload characteristics of the finite element assemblage during crack extension procedure is one of the vital elements in the fomulation of a crack stability concept for a ductile material. Fiies 24 and 25 compare the results of a numerical experiment in which the crack is extended by (a) an elasto-plastic analysis in which the crack tip force is relaxed following the determination of the load-unload character of each element as described earlier and (b) an elastic analysis in which the same crack tip force is relaxed elastically for all the elements. Figure 24 shows (a) the distribution of reaction forces at the nodes along the crackline, accumulated from incremental
for
3
4
5
6
7
6
9
10
NUdBER OF STEPS USED FOR RELAXING THE CRACK TR NODAL REACTION FORCE
Fig. 23. The effect of load-unload step size on analysis results.
325
Recent advances in computerizedaerospacestructnralanalysisand design
Fig. 24. Distribution of reaction forces at the nodes along the crack line.
)LASTlC
01 25
DISTANCE Fig.
teristics as retlected by the state of stress in the vicinity of the crack tip. The tangent stiffness of the material immediately ahead of the crack tip is small due to the local plastic zone. Thus the material there absorbs a comparatively smaller amount of the released crack tip force, depending on such factors as material ductility, strainhardening characteristics, stress field history and crack length. Plastic zones-forward and behind crack tip. As the crack extends at constant applied stress, the unloading of the newly created crack surface results in the translation of the plastic zone ahead of the crack, while leaving a wake of permanently strained material behind the moving crack tip (Fii. 26). When a crack reaches a stable length and external load-increments are applied, the plastic zone grows larger at the crack tip of the stable crack and the failure criterion is finally reached at the immediate crack tip. At this time the crack extends again under constant load leaving behind a permanently strained zone. This process is repeated until catastrophic crack growth @ii. 27). The wake of residual plasticity left behind a moving crack is directly related to the stress history, material ductility and strain hardening characteristics. For less ductile materials, this wake will be smaller and the applied stress at which crack instability occurs will be smaller. Thus, the stable crack growth behavior of a material is directly attributable to its ability to form a permanently strained zone behind a moving crack.
Y - DISTANCE FROM CENTER OFPLATE (IN.)
I
010:
I (IJ mm ,
.&-7Y
c
35
45
55
65
FROM CRACK CENTER,
75
X, (IN)
25. Distribution of max. principal stress at the net section due to the release of the crack-tip nodal force. I
Ok’
elastic-plastic analysis, (b) the incremental reactions at the nodes due to the release of the crack tip nodal force by the elastoplastic analysis and (c) the incremental reactions at the nodes due to the release of the crack tip nodal force elastically. Figure 25 shows the distribution of the maximum principal stress, CT,,at the net section due to the release of the crack tip force elasto-plastically as well as elastically. It may be seen from these figures that there is a marked ditference between elastic and elastic-plastic crack extension procedures. When a crack is extended elastically, the unloading of the newly created crack surface loads very locally the new crack tip region. Thus, for an elastic material, the new crack tip is severely loaded, crack initiation is catastrophic and crack stability does not occur. In contrast, when a crack is extended elasto-plastically, the unloading of the newly created crack surface causes loading not only of the region immediately ahead of the new crack tip but loads also an appreciable portion of the remaining net section. This loading is not confined to the immediate vicinity of the crack tip because the new load distribution is dependent on the element stiffness charac-
HALF
L 2.6
I
25
2.6
CRACK
LENGTH,
27
a. (IN)
Fig. 26. Shift of plastic zone as the crack moves at constant applied stress and the development of permanent strain in the wake of moving crack tip.
Ax MI)
AL* 1 25
1% 30
3.5
HALF cRAQ(
40
LEWTH,
4.5
50
a. (IM
Fig. 27. Plastic zone growth for the rising load test.
RALPH E.
326
MILLBR,JR. et al.
Stmin energy zones. Strain energy zones both elastic and plastic correspond to (a) the plastic zones ahead of the crack tip, (b) the wake of residual strain behind the crack tip and the rest of the plate. Quantitative changes in the amount of elastic and plastic strain energy occur at the various states of element load-unloading, and crack growth or stability. Present research is directed toward establishing these quantitative strain energy relationships to the new hypothesis presented. CONCIXJSIONS Large problems.
structures as well as large structures can result in large analysis problems size. Problem size results in lots of: input data computation cost output data. Lots-of requires especial attention: to use efficient data generators and processing to manage data storage to fit computer constraints -utilize substructures of proper size -phase the analysis steps -identify the computer system constraints upon the analysis plan. Our experience with large problems has shown that even when the computer tool is technically adequate, and its computer usage cost is acceptable, the management of data storage for large problems in a particular computer installation can easily become the critical task to ensure successful analyses. Automated structural design. The basic ATLAS capability developed at Boeing has become a powerful tool in preliminary design because realistic analysis models can be made for the entire aircraft, and these analyses can be executed for a cost that is one order of magnitude less than preautomated levels. The flowtime for a design cycle has also been significantly reduced. The future trend for
development stress/stiffness
Snd
is expected to emphasize additional constraints on the F.S.D. approach and
utilization of direct optimization for detail structural components or regions within the overall F.S.D. process. This attention to detail structural components will set the
stage for automatic communication between the preliminary design stage and the CAD efforts-in detail design. Crack growth hypothesis. The numerical evidence presented from elastic-plastic analyses has established a feasible relationship between material properties, loading history and crack growth. The hypothesis is advanced that crack growth is a function of critical stress and that stable crack growth depends not only upon the plastic zone ahead of the crack tip but most importantly upon the wake of residual strain behind the crack tip. These phenomena are also naturally related to quantitative changes in the elastic and plastic strain energy fields, as the crack grows, becomes stable or becomes unstable. Detailed, micro, experiments seem desirable to verify the presence and changes in the plastic zones and permanenent strain wakes as well as the quantitative correlation to crack growth and stability.
I R. M. Thomas, B. F. Backman, F. D. Flood, F. P. Gray, H. B.
Hansteen. C. R. Pratt-Barlow and S. 0. Wahlstrom. Aircraft Strength ‘and Stiffness Design Automation. U.S.A.-Japan Design Automation Symposium, Tokyo, Japan (Aug. 1975). 2. R. E. Miller, Jr., Structures Technology and the impact of Computer Systems. Winter Annual Meeting ASME, Houston, Texas (Nov. 1975). 3. 3-D Stress Analysis of Turbine Blades, Contract DAAG46-75C-0072.U.S. Armv Materials and Mechanics Research Center. Watertown, Ma. (Mar. 1975). 4. A. S. Kobayashi, S. T. Chiu and R. Beeuwkes, A numerical and experimental investigation on the use of J-integral. &gag Fracf. Mech.5, 293-305 (1973). 5. H. Anderson, A finite element representation of stable crack growth. 1. Mech.Phys. Solids 21, 331-356 (1973). 6. J. C. Newman, Jr. and H. Armen, Jr., Elastic-plastic analysis of a propagating crack under cyclic loading. A.I.A.A.Paper No. 74-366, AJAA/ASME/SAE 15th Structures, Structural Dynamics and Materials Co& Las Vegas (17-19Apr. 1974). 1. S. R. Varanasi and G. F. Carey, Elasto-plastic analysis of plane structures by the finite element displacement method (Program PEPSA). Boeing Document W24524 (Jk. 1970). 8. R. L. Dreisbach, ATLAS-an &grated analysis and design system, user manual. Boeing Document D6-254UO-ooO3TN (lY74).
F. BACKMAN,H. B. HANSTEEN,$ C. M. LEWIS, R. A. SAMUELand S. R. VNUNASI (Receiued15 Nouember 1975)
At&a&-This paperpresentsaccomplishmentsat the BoeingCommercialAirplaneCompanyin the areasof (1) large problemsolutions, (2) practicalstructuraldesign automationand (3) a hypothesis for stable crack growth. Aturbine blade (3200nodes 350elements,9500freedoms, 15millionwords of storage)and a sportsstadium(3400 nodes, 9600 elements, 20,ooOfreedoms, 70 millionwords of storage)are presentedin terms of problemdefinition, strategyof approach,data managementand computerresources. The Boeing automatedstrengthdesign capabilityin the ATLAS System is discussed in terms of user specified constraintson the automatedstrengthresixingand local optimixation,strengthcriteria,user controland convergence criteria.Cost and weight data are presentedfrom large aircraftdesign studies (20,000design variables). A new concept of crack stabilitybased on non-linearstress/strainanalysis is presented.Stable crack growthis modeled by a suitable failure criterionin the stress analysis procedure.A crack is extended by a simultaneous unloadingof the newly created crack surface and the loading of the untracked region. Test comparisonsand numericalexperimentssupporta new hypothesisbased on materialstrengthcharacteristics,plasticzone size/history and the residualplastic strains. LARGE PRORLEM WLITl’lONS
Fig. 2. In addition to the applied loads shown, support boundary conditions were applied along the fir tree attachment lands and at the blade tip shroud. Geometry. Turbine blade geometry presents a modeling problem much more extensive than the size of the part indicates. Detailed geometric representation and stresses are required in the region of the airfoil root and the lir tree attachment to the turbine disk These areas were modeled with higher order isoparametric elements having 32 nodes per element. Substructuring and data management. The large bandwidth produced by these finite elements caused great amounts of data to be stored on the computer disk tiles. To execute the analysis on the Boeing Computer Services CDC 6600 computers as a production run, it was necessary to substructure theblade as shown in Fig. 3. The alternative to substructuring, a full model of the blade, would have resulted in prohibitive block (weekend) computer time costs, and excessive analysis flowtime. Two criteria governed the choice of the substructure boundaries (size). One criteria ensured efficient reanalysis for different boundary conditions. The second constrained the volume of data associated with various steps of the analysis to practical sizes.
Turbine blade analysis The turbine blade shown in Fig. 1, even though only a small part of the turbine stage presents one of the most complex design problems in the gas turbine engine. Even recently, turbine designers were forced to rely on expensive trial and error testing to optimize blade design. Recent developments on the isoparametric solid finite element, have made numerical 3dimensional stress analysis of turbine blades a feasible design tool. The following discusses an application of the ATLAS system[l, 21 to a three dimensional stress analysis of a turbine blade. The analysis was performed as part of a contract with the U.S. Army[3]. The approach to the problem, data management and costs are emphasized. Loads. The operating environment of the blade includes temperatures, inertia loads due to rotation and pressure loads. These loadiis are shown schematically in
AT~lp
TURBINE -A’%
TYPuALBLmE
Fig. 1. 3-D Stressanalysisof a turbineblade. tPresented at the Second National Symposiumon Computerixed StructuralAnalysis and Design at the School of Engineering and AppliedScience, GeorgeWashingtonUniversity,Washington D.C. 29-31 March 1976. $BoeingComputerServices, Inc. 315
31 PRESSURE
THERMAL i
6
E=
ATEM Fii.
-
e
2. Substructurescheme.
INERTIA
316
RALPH
MILLER, JR. ef al.
E.
words was required because tiles may not overlap disk boundaries. These criteria required that the portion of the blade between the tip and fir tree substructures be broken into multiple substructures, as shown in Table 1. Table1. Turbinebladesubstructures
9RlClS “WOE9
92
NoDrs/srlcr
32
TOTAL 0.o.r. II,EIACTI"O D.O.F.
81
23
0
II
66
90
32
32
-
3220
657
171
2685
2859
433
138
48
007
462
-
I4&
8b
76
1281
732
531
26
13
44
48
26
3527
946
244
945
AYfRAGE HlLF 9110 10. or D.0.F.
SLBSTRIJCTURES
TOTAL MOCEL
I or TOT&L IO. OF ,lPUT CARD,
Fig.
2913
56 -
3. Bladeloading.
The stresses in the lir treeand tip shroud areas are sensitive to details of the boundary conditions in those areas. Thus multiple analyses, each with different boundary conditions, would be required. Re-analysis cost was minimized by chasing one substructure for the tip and one for the tir tree. Bach substructure was as small as possible while still completely containing all boundary freedoms. This scheme required recalculating for the second and subsequent analyses, only the reduced stiffness and loads matrices for these two substructures. The second set of criteria minimized computer costs and flowtime by ensuring regular production runs on the CDC 6600 computers. Regular production runs were advantageous because they; (a) could be made at least once each day and thereby minim& the total flowtime for the analysis, and (b) reduce the single run residency time upon the computer thereby minGzing the potential of machine unreliability. Because the amount of disk space available to store the data during execution was limited, the management of disk storage became the key to the selection of substructure size and analysis solution steps. Disk storage. At the time of the analysis the theoretical disk storage capacity of the four available disks was approximately 40 million words. Practicality reduced this capacity because; defects made portions of disks unavailable; one disk was shared among the three central processors (CP’s) and multiple jobs were executing on each CP. Thus a practical limit of 16 million words of disk storage was calculated as available for any single execution. In addition, a disk single tile limit of 5 million
The choice of analysis steps also aiIected disk storage requirements. Only the following four steps involved signiticant amount of stored data: generation of element stiffness and stress matrices assembly (merging) of gross stiffness matrices for lowest level substructures calculation of reduced stiffness and loads matrices for lowest level substructures displacement backsubstitution and element stress calculation. The total disk storage requirements for the critical substructure during each of these steps are shown in Pig. 4. As shown in that figure, a single substructure model was impractical since its disk requirements would have exceeded the practical capacity. Several important parameters describing such a single substructure model are presented in Table 2. Table 2. Turbine blade single substructuremodel No. of nodes No. of bricks
2885
352 8655 870
TotalDOF Averagehalf band
The maximum size of any single file for the critical substructure during each step of the analysis is shown in Fig. 5. While the maximum file size for multiple substructures is only approximately half the practical
PEAK TOTAL DISK
3254
STORAGE
SUBSTRUCTURED
REQUIREMENTS.
PRACTICAL
LIMIT
(IO6 WORDS)
'
STIFFNESS A LOADS
'
LOWEST LEVEL MERGE
'
REDUCTION A SOLUTION
BACK-
'
SUBSTITUTION A STRESS
Fig.4. Totaldisk requirements,turbinebladeanalysis.
'
317
Recent advancesin computerizedaerospacestructuralanalysisand design
THEORETICAL
LIMIT
PEAK SINGLE
DISK
FILE SIZE, (IOS WOROS)
‘r-
STIFFNESS a
I
LOWEST LEVEL
LOADS
I
REDUCTION b SOLUTION
MERGE
I
BACK SUBSTITUTION
I
6 STRESS
Fig. 5. Maximumfile size, turbine blade analysis.
limit, a single substructure analysis would have significantly exceeded the available capacity. Cost. The cost of the analysis execution is given in terms of CP seconds in Table 3. The percentage of the total problem cost associated with each major ATLAS module execution is also given. Note that more than 60% of the total cost of execution is spent in the REDUCE phase. This emphasizes the need for the most efficient possible code for the matrix decomposition and matrix multiplication routines.
I1
CP
TIME.
1696 56 I 31 I2 973
13 6 14
a9 3a 47 220
5.9 2.7 15.0 4.5
x WATER
SEE
BEARINGS
FlGuRE7a~~
Fig. 6. Plan view and support configuration, Denver Mile High Stadium.
+----
145 FEET
.a .2 .3 I.1
1149 391 6276 761
4.2 2.1 34.0 3.8
I6
1551 SOS 2929 654
1 I .5 3.5
460 322
1.7 I.1
21554 (0 6 NRS)
i
UNIT
SEC
I9
21
FfMER
i-
Table3. Costs,turbine blade analysis SVRSTRVCTVRE
DIRECTION
t Mo"&T
5.3 2.3
Fig. 7. Cross-sectional frame, Denver Mile High Stadium.
100.0
Analysis of the Denver Mile High Stadium extension In a project for the City and County of Denver, Colorado, a new East Stand is added to the Denver Mile High Stadium increasing the current seating capacity by approximately 35,000 seats (consulting engineers are DMJM-Phillips Reister, Inc., Denver, Colorado). The superstructure of the new stand can move as a single unit to enlarge the playing field in the case of baseball, or to reduce the playing field for football or soccer. The moving system is composed of a water bearing support subsystem which will reduce friction and facilitate movement, a guiding system and a moving and control subsystem. The water bearing supports glide on a system of concrete runways which are placed below the surface of the playing field. Problem description. The structural analysis discussed in this paper applies to the moveable part of the new stands shown in Fiis. 6 and 7. This structure is mostly made of steel and consists of 19 straight bays of 17-ft
width each, and of 5 radial bays of 5” width each at both ends. Stadium dimensions include a length of approximately 534 ft, a width of 192ft and a height of 127ft. The distance between the two extreme positions of the stadium is 145ft. The structure is supported by 67 water bearings as indicated by the x’es in Fig. 6. When the stadium is being moved the bearings can only transmit vertical reactions. The bearings will in addition act as flexible supports for forces in the horiiontal plane when the stadium is in a fixed position. The two power units located near the two outer ends of the structure and the guide rail near the middle provide horizontal supports for the stadium when it is moved. Since these power/support mechanisms are located in the fixed substructure of the stadium, their position relative to the superstructure will change. The stadium was analyzed for 13 different load cases. This included dead load, live load, four cases of wind loads, a sway load in the longitudinal direction of the stands, four differential deflection load cases, inertia load
MILLRR, JR. et ai.
RALPHE.
318
in direction of stadium movement and a friction load acting at the water bearing points. The analysis also included a study of 13 different sets of boundary conditions. Twelve of these conditions applied to different locations of the power units and the guide mechanism due to movement of the stands. The last condition applied to the stands in a fixed service position. Particular interest in the analysis was devoted to the water bearing system, and the effects of differential deflections of these points. To study these effects, the reduced stiffness matrix with regard to the vertical displacement component of these points was computed. Solution strategy. The factors determining the computational task of performing a finite element analysis of the stadium is shown in Fig. 8. The solution for one of the 13
A static condensation was performed on this system with respect to the r2 displacements. The mathematical operations carried out are described in Fig. 9, task 5. Phase 2. The remaining equation system &=l? was used as a basis to impose the 13 different sets of
boundary conditions. This includes a re-partitioning of the
Task
1 Value
Description
Symbol used in eqn (2)
2 3
3152 9606 13 13
Number of structural nodes Number of finite elements Number of load cases Number of independent sets of boundary conditions Dimension of global stiffness matrix Average bandwidth of globalst8ness matrix
4 4
20712
R
600
b
5
Fig. 8. Basic parameters, Denver Mile High Stadium. sets of boundary
conditions
involves
solving a set of
6
linear equations Kr=R
(1)
where K is the global stiffness matrix of the structure, t the nodal displacements and R the nodal loads. The work required to perform this solution may be measured by the number of multiplications and additions to be carried out. This may be approximated by the formula s = l/2& *+ 2nbq.
(2)
Substituting the actual values from Fii. 8 yields S = 4x 109 operations. Considering that each of the 13 sets of boundary conditions represents a new solution, it would appear that the complete analysis requires more than 50 billion (50 x 10 basic operations. This constitutes a very considerable amount of work even for the most modern electronic computers. The approach chosen for the actual solution reduced the amount of re-analysis required for each new set of boundary conditions. The solution was carried out in two phases as follows: Phase 1. The nodal freedoms of the structure were partitioned intq the two subsets r, and rz, where r-The muon of all freedoms which are being supported iu any of the structurai con@uations. This includes the x, y and z-translational freedoms at the water bearing points, the y-translational freedoms at 12 locations along each power unit track and the xtranslational freedoms at the corresponding 12 locations along the guide rail. r2contains a total of 237 freedoms. r,-The remaining 20,475 freedoms of the structure. The linear system of eqns (1) was partitioned accordi@Y,
I
Description Read input data Generate structural plots and produce various verification printing to establish correctness of input data Generate finite element stiffness matricies and nodal load matricies Assemble the governing linear system of equations for the total structure, represented by the stiffness matricies K,,, K,,, and Kz2,and the load matrices R, and R, Perform static condensation of the system of equations with respect to the displacements r,. The operation is carried out in 4 subtasks: (A) Decompose matrix K,, +L,, (B) Compute C12= L;,‘K,, (C) Compute HI = L;,‘L;iR,
g* = L ;,% (D) Compute K = Kz2- C& R = Rz- KBH,
Perform a solution for each of the 13 support conhgurations. Each solution constitutes a subtask performing the following operations: (A) Apply the boundary conditions to form the matrix partitioning;
r, = Unconstrained freedoms rs = Support freedoms (B) Solve for r, (C) Compute rl = H, - H2r2 (D) Compute member stresses and reaction forces Compute the reduced stiffness matrix with respect to the Z-displacements at the neoprene pad nodes. The following operations are performed: (A) Partitioning the condensed system arrived at in Task 5 into:
rp = Unconstrained freedoms rR = Z-displacements at neoprene pad nodes rs = y;;i-y supports to assure a non-singular 8
(B) gmpute I?- = Km - KmKgKm Produce printouts of subsets of displacements, stresses and reaction forces
Fig. 9. Computer task description, Denver Mile High Stadium.
319
Recent advances in computerized aerospace structural analysis and design
system (4) and, for the futed stadium con8guration, addition of x- and y-spring supports at all water bearing points. The displacement partitions I, and rz were then computed as described in Fig. 9, task 6. The work required to perform the two-phased solution described above is again measured in terms of the number of multiplications and additions to obtain r, and rz. With reference to Fig. 9 the requirements are as follows: Task 5 Performed once Task 6 Performed 13times
10.2X 109operations 0.9 X 109operations 11.3X 109operations
This shows (50x 109 to 11x 109 that the selected approach reduced the amount of computational work required for the solution by approximately 80% compared to a conventional analysis. Task management. The computer analysis was carried out using the ATLAS program[l] which operates on CDC 6008 and CYBER series machines. Because of the size of the project, the necessity of checking intermediate results and varying requirements with regard to computer resources at diierent points of the analysis, it was necessary to perform the computer running as a series of subtasks. The data generated in the course of the analysis was maintained in an off-line data base. At the beginning of one subtask the data required by that task was loaded by the ATLAS program. Similarly, at the end of a task the data base was updated by any new information generated. The task breakdown is defined in Fii. 9. Tasks 1 and 2 were concerned with preparation and checking of data input. Data generators were used extensively in the data preparation to take advantage of repetitive information or patterns in the structure. Thus the data volume required for the analysis was “limit& to about 5000 cards. Without data generators it would have been in the range of 10 times more. Verification of correctness of data volumes of this size constitutes a major problem. For the stadium a sign&ant part of the data checking was achieved by inspecting plots of the structure. A total of about 65 plots showing different regions of the structure was required to perform a complete data verification. The bulk of the computer running was performed in tasks 3-7. The main problems encountered in the planning and execution of these tasks were related to the data volume that had to be maintained. The size of this volume at the end of each computational task is shown in Fig. 10.
A peak value of about 70 million 60 bit “words” was required during execution of task 5. Special provisions had to be made to support a data volume of this size. During execution of task 5 the computer had to be specifically con&red due to the extreme size of the matrices being processed. In the reconliguration seven 844 disk units were combined as one logical equipment. While performing task 6 special care had tc be taken to balance the data volume on the available disk units. Whereas no particular problems were encountered in maintaining the off-line data base, it was found that the data copying involved in starting and checkpointing of a task was considerably more expensive than anticipated. Report generating of results was performed in task 8. At this point all “raw” data were available in the data base, and only the final data selection and formatting remained to be done. Typically, task 8 will be performed several times as the analysis of the results proceed and requirements for more information become apparent. The computing resources required to complete the analysis are illustrated in Fig. 11 by the accumulated central processor time at the end of each computer task. The total amount at the end of the project was about 32OOOsecs,or roughly 9hr of CP time. The actual residency time in the computer was much higher than this, probably in the range of 40-50hr. The flowtime for the whole project, including data preparation and checkout, was about 3 months. The relative costs of each computer task is shown in Fig. 12. Practical-automated structuml design The basic automated structural design capability
developed at Boeing for the ATLAS System is a modified
L&II!__ ‘Onrn '2345676
TA5K
Fig. 11. AccumulatedCP time, Denver MileHigh Stadium. 50 -
40
40 -
DATA mw, wRDs
30
30-
Xlos
%
20
IO
OF COST 20
1
IO
6
10. Accumulateddata volume, Denver MileHigh Stadium.
7
6
TASK
TASK Fig.
_L L_lA 12345
12345676
Fii. 12.
Relativecost, Denver MileHighStadium.
RALPH
320
E. MILLER, JR. er al.
LARQE PROBLEM
COST EFFICIENCY FLOW TIME REDUCTION
I
CONVERGENCEIYPRUVEMENT PSYCOLOQICALADVANTAGE
MElHoD F&xlslLllT
USER CONTROL
CCWUlING STANDARDS
SETTER PERFORUNCE
ImMVED
PRDUJCTIVITY
CAPAclrV
L
INPUT EFFICIENCY TRADITIONAL
LESS RISK FOR ERRORS
ISTIllS EXPERIENCE
FAST ACCEPTANCE
CONCEPTS I
I
I
I
Fii. 13. Designautomation;objectivestructure.
fully stressed design (F.S.D.)[l]. The selection and development of this baseline function is a logical consequence of the objectives illustrated in Fig. 13. These objectives demand the implementation of a method that satisfies the following requirements: The typical aerospace structure is characterized by a huge number of variables and an enormous amount of data. The method must therefore be particularly feasible in a “large problem environment.” The method must be especially suitable for applications in the field of preliminary design. The flexibility in usage of the method should be based on user control in execution and data handling. Cost efficiency must be the dominating concern in selection of method and architecture of data management. Flowtime reduction is obviously a natural consequence of automation. However, since data preparation and changes constitute a very time consuming phase, it is essential to emphasize this aspect. Functions for checking of input, modification of data and display of output therefore becomes especiahy important. Input efficiencv is YI importance from a user feasibility standpoint. Thus regional input structure and update capabilities are indispensable functions. A method that is based on traditional concepts ensures preservation of existing experience and promotes fast user acceptance. It therefore is highly desirable that this facet be considered. ‘Ibis set of objectives is displayed in Fig. 13 and evaluated in terms of pay-off potential and user appeal and acceptance. The block-diagrams also illustrate the relative emphasis on the different goals. The dominating characteristic of each aspect in the evaluation is represented by key-words in each block. User Control. As the most feasible type of design automation in a “large problem environment” involves optimality criteria and in particular, the concept of a fully stressed design, it becomes imperative to design a process that provides maximum control of convergence related phenomena. The traditional F.S.D.-method has been modified and
complemented with a structure of functions for user control which requires: Data control Execution control Convergence control. Figure 14 shows the functions and their relative importance in supporting these three requirements. INPUT COilSTRAINTS: "Pm UD L6sERwJ"OS,FlXEDDATA.,"W, l!AllXS OF wm
Fig. 14. Relative emphasis on the design of user-control functions.
The data control is achieved by an input structure that is characterized by: A variety of con&ant data subsets A data hierarchy An override capability at time of execution A regional input definition (in addition to element specifications)
Recentadvancesin computerizedaerospace structuralanalysisanddesign
An input update capability A regional resize definition The execution control is supported by: Selective execution of technical algorithms System procedures for iteration control System restart capability Regionalresizing The convergencecontrol is exercisedand supportedby: A history function recordingand displayingmarginsof safety and maximumchanges The input structure and update capability A regional technical or user specitied resize function User specified convergence criteria. Usage experience.Usage experience with design automation in the “large problem environment” has until recently been very liited. Results from Boeing design studies involving problems of a size of 15800420,000 designvariablesare shownin Fii. 15.The relative cost per design variable are related to pre-automated levels, as baseline.Fire 16shows the solutiontimes for ~erent segments of the design process. Three different approaches to the use of automated designare illustrated.Project I used, as a startingpoint, a detailed manual sizing and a highly differentiatedinput. The execution took place in steps in the sense that one cycle was broken up into: prepocessing,analysis,partial
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postprocessing, design and total postprocessing. Each step was followed by checkingof results and evaluation. This procedure accounts for the relatively high cost. Despite this, a sign&ant cost reduction was realized. The second project had a startingpoiut that was defined by a regionalmanualsizingand less di#Terentiatedinput. The execution was performed for one cycle at a time and the checking was considerably less extensive. Project III finally had a completely arbitrary starting point and the input was of regional character. The execution was performed for more than one cycle at a time. It is barest to note that, ~dependent of starting point, sign&ant cost-savings were realized. It is also promising to note how similar the convergence characteristics for these three designs were. Figure 17 illustrates this point in terms of weight change relative to starting point. This experience seems to indicate that very little benefit is derived from the effort of establishing a “good” starting point. The behaviour of aerospace structures from a convergence standpoint is very complex. Present experience, however, seems to support the idea that even though a whole family of designs with different element properties represents potential candidates only small weight differences exist between them.
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DESIGN CYCLE
Fig. 17. Relative weight change for different approaches to design.
Ben& characteristics. The automated design process that has been discussed this far is of the optimality criterion type with stress constraints defined by strength and buckling algorithms. The cost-reductions realized in applications have been significant but are, of course, size-dependent. Figure 18 shows an intensity description of pay-off. The dark regions represent high potential for cost-savings. The !igure illustrates two cases, single structure idealization and single level substructuring. The black regions represent maximum pay-off from design automation. Unfortunately these high payoff regions coincide with limitations of the computing system. Capacity of bulk storage devices generally limits the size of the maximum payoff regions. Substructuring capability, however, allows a dramatic extension of the feasible payoff region. An introduction of multi-level substructuring would extend the payoff regions indefinitely, limited then by cost. Figure 18 could also be interpreted as an illustration of the success of the automation of the structural design process. The black and to some extent the dark regions represent costs that are one order of magnitude lower than pre-automation levels. It furthermore illustrates the capacity for solving problems of a size that only a few years ago was considered to be out of reach.
NUMBER OF DESIGN VARIABLE
0
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I0000 20&O ANALYSIS NUMBER OF DEGREES OF FREEDOM (DARK = HIGH INTENSITY)
Fig. 18. Design automation: intensity description of feasibility and payoff potential.
Current status and future. The design process contains considerably more aspects than discussed this far. This section will illustrate the overall picture with regard to status and development of the automation of design as seen from the standpoint of need for an immediate future. The experiencegained (illustrated in Figs. 15and 17),the potential realized, (shown in Fig. 18)and the requirements imposed by advances in the state-of-the-art constitute the basis for the direction, emphasis and character of the development plans illustrated in Fig. 19. Two basis subjects are considered: 1. Optimality criteria 2. Direct optimization. The development for item 1 involves: More technical algorithms for stress constraints Algorithms for introduction of stiffness constraints into the design process Practical supporting functions. In the field of direct optimization two items are considered: Component optimization Regional optimization. Component optimization is intended to complement to the optimality criteria. It is directed toward a lower level involving detailed structural definitions. It attacks the problem from a concept standpoint and lets finite elements be associated with concepts like skin-stringer panels, sandwich panels, composite laminates, etc. The regional optimization is intended as an alternative to optimality criteria and involves: Selection of key-elements representing structural regions Substructuring with non-element related design variables Introduction of practical constraints (e.g. tapered properties). A new stable crack growth hypothesis Boeing research on a ductile fracture criterion has produced a new concept of crack stability based on nonlinear stress-strain analysis using the finite element method. The finite element method has been applied previously in the study of a steadily growing crack under monotonically increasing load [4,5], and in the study of a fatigue crack growth under cyclic loading[6].
Recent advances in computerized aerospace structural analysis and design
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The problem of stable and catastrophic crack growth in a center-cracked plate of a ductile material, subjected to a monotonically increasing applied stress is discussed here. An incremental elastic-plastic plane stress analysis of this cracked plate is performed using a Boeing finite element program PEPSAD] which uses a tangent stiffness formulation and a two-step Runge-Kutta integration scheme for each load-step. Constant strain triangles are used for the idealization of the cracked plate. The ductile material of the cracked plate is assumed to be an isotropically work hardening material which can be characterized by Von Mises yield condition and PrandtlReuss incremental stress-strain equations. Failure criterion-crack extension. The phenomenon of stable crack growth prior to instability is modeled by incorporating a failure criterion in the stress analysis procedure. This failure criterion permits crack growth when the maximum principal stress, u,, at the crack tip node, reaches the material ultimate tensile strength, F,,,. At the end of the stress analysis for any load-step, this failure criterion is applied to the cracked plate to ascertain whether crack growth is indicated at that load level. If crack growth is not indicated, the stress
analysis proceeds in the usual way with the application of the next load-increment. If the crack is to be extended, then the following procedure is used for unloading the newly cre@ed crack surface and calculating the redistribution of stresses as a result of crack extension (Fig. 20). 1. The crack tip node, which was previously constrained in the direction perpendicular to the crackline, is released and the crack tip is advanced to the next node on the crackline. 2. The newly released crack tip node is subjected to a force which is equal in magnitude but opposite in direction to the accumulated reaction force at that node. A stress analysis is performed with this nodal force as the only external loading. The increments of stresses and displacements due to this step are added to the accumulated stresses and displacements of the cracked sheet respectively. 3. After the crack extension and the redistribution of stresses in step 2, the failure criterion is applied again to determine if further crack growth is indicated. If further crack growth occurs, the above procedure (steps 1 and 2) is repeated. If a crack extension causes a redistribution of
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center-crackedsheet. stresses which causes successive crack extensions, then the process is considered as having reached the point of instability. If there is no further crack growth after a redistribution of stresses, then it is a stable crack and the external load must be increased for further crack growth. Element loading-unloading modeling. In step 2, the relaxation of the crack tip nodal force produces unloading in the newly created crack surface behind the new crack tip while loading the region ahead of the crack tip. This simultaneous loading and unloading behavior necessitates the inclusion in the stress analysis procedure of routines for changing the element stiffhess according to the incremental theory of plasticity. Therefore for an unloading element, elastic stress-strain equations are used in the calculation of the tangent stiffness, while for a loading element, elastic-plastic stress-strain relations are used. During crack growth, the determination of whether an element loads or unloads is done iteratively by an extra (trial step) application of step 2. Initially, all elements are assumed to continue loading, and the tangent stiffness equations are solved with crack tip relief force as the only external load and the stresses due to the trial load-step are accumulated in the usual manner. Then, the element equivalent stresses for before and after the load step are compared. If & 2 b,, then the element is assumed to continue loading. If ~7~c 6, then the element is assumed to unload (Fig. 21). This iteration trial step process is implemented twice in the present investigation, so as to provide an accurate (stable field) state for element loading and unloading calculations. Test comparison. The results of comparison of the
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analysis method to a 2024-T3 ahuuinum test panel failure are shown in Fig. 22. The panel failure stress predicted by the analysis method when the crack tip force is relaxed in a single step is 28 ksi and when the crack tip force is relaxed in ten equal steps is 30 ksi. This single, step of crack tip unloading force causes the material to behave more elastically. The discrepancy between the test panel failure stress and the analysis panel failure stress may be due to thickness effect of the test panel. Numerical experiments-load step size. The effect of the load-unload step size on the results of the analysis method is shown in Fig. 23. In this numerical experiment, the crack tip force is relaxed in one single step and a number of equal steps. The accumulated reaction force and the maximum principal stress at the new crack tip are plotted as a function of the number of steps used for the relaxation of the crack tip force. It may be seen that the solution converges rapidly and there is no appreciable change in the solution after 4 steps for this case. Numerical experiments-elastic us elastic-plastic. It is found that the determination of the elasto-plastic, load-unload characteristics of the finite element assemblage during crack extension procedure is one of the vital elements in the fomulation of a crack stability concept for a ductile material. Fiies 24 and 25 compare the results of a numerical experiment in which the crack is extended by (a) an elasto-plastic analysis in which the crack tip force is relaxed following the determination of the load-unload character of each element as described earlier and (b) an elastic analysis in which the same crack tip force is relaxed elastically for all the elements. Figure 24 shows (a) the distribution of reaction forces at the nodes along the crackline, accumulated from incremental
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Fig. 24. Distribution of reaction forces at the nodes along the crack line.
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teristics as retlected by the state of stress in the vicinity of the crack tip. The tangent stiffness of the material immediately ahead of the crack tip is small due to the local plastic zone. Thus the material there absorbs a comparatively smaller amount of the released crack tip force, depending on such factors as material ductility, strainhardening characteristics, stress field history and crack length. Plastic zones-forward and behind crack tip. As the crack extends at constant applied stress, the unloading of the newly created crack surface results in the translation of the plastic zone ahead of the crack, while leaving a wake of permanently strained material behind the moving crack tip (Fii. 26). When a crack reaches a stable length and external load-increments are applied, the plastic zone grows larger at the crack tip of the stable crack and the failure criterion is finally reached at the immediate crack tip. At this time the crack extends again under constant load leaving behind a permanently strained zone. This process is repeated until catastrophic crack growth @ii. 27). The wake of residual plasticity left behind a moving crack is directly related to the stress history, material ductility and strain hardening characteristics. For less ductile materials, this wake will be smaller and the applied stress at which crack instability occurs will be smaller. Thus, the stable crack growth behavior of a material is directly attributable to its ability to form a permanently strained zone behind a moving crack.
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elastic-plastic analysis, (b) the incremental reactions at the nodes due to the release of the crack tip nodal force by the elastoplastic analysis and (c) the incremental reactions at the nodes due to the release of the crack tip nodal force elastically. Figure 25 shows the distribution of the maximum principal stress, CT,,at the net section due to the release of the crack tip force elasto-plastically as well as elastically. It may be seen from these figures that there is a marked ditference between elastic and elastic-plastic crack extension procedures. When a crack is extended elastically, the unloading of the newly created crack surface loads very locally the new crack tip region. Thus, for an elastic material, the new crack tip is severely loaded, crack initiation is catastrophic and crack stability does not occur. In contrast, when a crack is extended elasto-plastically, the unloading of the newly created crack surface causes loading not only of the region immediately ahead of the new crack tip but loads also an appreciable portion of the remaining net section. This loading is not confined to the immediate vicinity of the crack tip because the new load distribution is dependent on the element stiffness charac-
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Stmin energy zones. Strain energy zones both elastic and plastic correspond to (a) the plastic zones ahead of the crack tip, (b) the wake of residual strain behind the crack tip and the rest of the plate. Quantitative changes in the amount of elastic and plastic strain energy occur at the various states of element load-unloading, and crack growth or stability. Present research is directed toward establishing these quantitative strain energy relationships to the new hypothesis presented. CONCIXJSIONS Large problems.
structures as well as large structures can result in large analysis problems size. Problem size results in lots of: input data computation cost output data. Lots-of requires especial attention: to use efficient data generators and processing to manage data storage to fit computer constraints -utilize substructures of proper size -phase the analysis steps -identify the computer system constraints upon the analysis plan. Our experience with large problems has shown that even when the computer tool is technically adequate, and its computer usage cost is acceptable, the management of data storage for large problems in a particular computer installation can easily become the critical task to ensure successful analyses. Automated structural design. The basic ATLAS capability developed at Boeing has become a powerful tool in preliminary design because realistic analysis models can be made for the entire aircraft, and these analyses can be executed for a cost that is one order of magnitude less than preautomated levels. The flowtime for a design cycle has also been significantly reduced. The future trend for
development stress/stiffness
Snd
is expected to emphasize additional constraints on the F.S.D. approach and
utilization of direct optimization for detail structural components or regions within the overall F.S.D. process. This attention to detail structural components will set the
stage for automatic communication between the preliminary design stage and the CAD efforts-in detail design. Crack growth hypothesis. The numerical evidence presented from elastic-plastic analyses has established a feasible relationship between material properties, loading history and crack growth. The hypothesis is advanced that crack growth is a function of critical stress and that stable crack growth depends not only upon the plastic zone ahead of the crack tip but most importantly upon the wake of residual strain behind the crack tip. These phenomena are also naturally related to quantitative changes in the elastic and plastic strain energy fields, as the crack grows, becomes stable or becomes unstable. Detailed, micro, experiments seem desirable to verify the presence and changes in the plastic zones and permanenent strain wakes as well as the quantitative correlation to crack growth and stability.
I R. M. Thomas, B. F. Backman, F. D. Flood, F. P. Gray, H. B.
Hansteen. C. R. Pratt-Barlow and S. 0. Wahlstrom. Aircraft Strength ‘and Stiffness Design Automation. U.S.A.-Japan Design Automation Symposium, Tokyo, Japan (Aug. 1975). 2. R. E. Miller, Jr., Structures Technology and the impact of Computer Systems. Winter Annual Meeting ASME, Houston, Texas (Nov. 1975). 3. 3-D Stress Analysis of Turbine Blades, Contract DAAG46-75C-0072.U.S. Armv Materials and Mechanics Research Center. Watertown, Ma. (Mar. 1975). 4. A. S. Kobayashi, S. T. Chiu and R. Beeuwkes, A numerical and experimental investigation on the use of J-integral. &gag Fracf. Mech.5, 293-305 (1973). 5. H. Anderson, A finite element representation of stable crack growth. 1. Mech.Phys. Solids 21, 331-356 (1973). 6. J. C. Newman, Jr. and H. Armen, Jr., Elastic-plastic analysis of a propagating crack under cyclic loading. A.I.A.A.Paper No. 74-366, AJAA/ASME/SAE 15th Structures, Structural Dynamics and Materials Co& Las Vegas (17-19Apr. 1974). 1. S. R. Varanasi and G. F. Carey, Elasto-plastic analysis of plane structures by the finite element displacement method (Program PEPSA). Boeing Document W24524 (Jk. 1970). 8. R. L. Dreisbach, ATLAS-an &grated analysis and design system, user manual. Boeing Document D6-254UO-ooO3TN (lY74).