is' ESL-TR-82-50

EVALUATION OF NASA STRUCTURALANALYSIS (NASTRANI TO PREDICT THEDYNAMIC RESPONSE OF REINFORCED CONCRETE

WILLIAM W. PAYFE. JR.

AIR FORCE ENGINEERING AN9 SERVICES CENTERENGINEERING AND SERVICES LABORATORYTYNDALL AIR FORCE BASE, FLORIDA 324 13

DECEMB3ER 1983

FINAL REPORT27 MAY 1981 - 5 AUGUST 1981

- JAN 26 1984

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C. REPORT NUMBER 2. GCVT ACCESSION NO. a RECIPIENT'S CATALOG NUMBERESL-TR-82-50 A D/r, 2 "I

4. TITLE (and SubtJII.j 5. TYPE OF REPORT & PLRIOD COVERED

Evaluation of NASA Structural Analysis Final Report(NASTRAN) to Predict the Dynamic Response May-iOai . R AUMB98

6. PERFORMING 01G. REPORT NUMBERof Reinforced Concrete

7. AUTHOR(s) 89 CONTRACT OR GRANT NUMBER(*)

Payne, William W., Jr. F49620-79-C-0038

9. PERFO;1MING ORGANIZATION NAME AND ADDRESS IC. PROGRAM ELEMENT. PROJECT, TASK

AREA & WORK UNIT NUMBERSHeadquarters, Air Force Engineering and ServicesCanter (AFESC), Tyndall AFB FL 32403 PE 62601F

JON : 26730003

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19. KEY WORDS (Contlnue, on reverse side If nece.emry and Identify by block no, b•r)

Dynamic AnalysisFinite Element AnalysisNASTRANReinforced ConcreteStructural Analysis

20. ABSTRACT (Continue on rererme sIde If necessary and Identify by blocir number)

This report evaluates the ability of the finite element program, NASTRAN, toanalyze reinforced concrete structures under dynamic loads. Experimental datafran a quarter ecale modc.t. test of an underground shelter was used to validatethe computer projections.

NASTRAN is a general-purpose structural analysis program containing severaltypes of finite elements and several displacement analysis approaches.

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For this study five different computer models of reinforced concrete wereuse 1. The models were composed of the following elements;

li• Plate membrane elements,

2. Plate membrane and rod elements.3. Plate-bending elementc.

4. Plate-bending-membrane elements.5. Plate-bending-membrane and beam elements.

Static Analysis and Transient Analysis approaches were used to evaluate thecomputer model.

Favorable results were obtained for the plate membrane and rod element modelusing the Transient Analysis Approach. Strain in the reinforcing rods,time to maximum strain, and time to return to zero strain were used tocompare the experimental data to the computer prediction.

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PREFACE

This report documents the work accomplished dur.i,,g the USAF SummerFaculty Research Program (1981) by the Southeastern Ce:stei for Electrical

Engineering Education (SCEEE), 1101 Massachusetts A .. ? St. Cloud, FL 32769

under USAF Contract Number F49620-79-C-0038. This .'. t is sponsored by

the Air Force Office of Scientific Research, Air For<cc Systems Command,

Bolling AFB, Washington, DC and was accomplished at the Ili- Force Engineering

and Services Center, Engineering and Services Laboratory, (Av"ESC/RD) Tyndall

AFB, FL.

This is a reprint of a report previously published by SCEEE in the 1980USAF/SCEEE Summer Faculty Research Prograo.. Research Reports Volume I, and

covered the period between 27 May 1981 and 5 August 1981. Captain Paul L.Rosengren, Jr. was the AFESC/RDCS project officer.

The author would like to express his appreciation to the Air Force Engin-eering and Services Center, the Engineering Research Division, Airbase Surviv-

ability Branch, and the Southeastern Center for Electrical Engineering Educa-tion for the opportunity to participate in the 1981 Summer Faculty ResearchProgram and the personnel of the Airbase Survivability Branch for their hospi-tality to him and his family, and the excellent working conditions the branchmembers provided. Special credit is given to Mr. David R. Colthorp, Struct-ural Mechanics Division, Waterways Experiment Station, for perfo)rming thescale model testing and supplying the experimental results.

This report have been reviewed by the Public hffairs Office (PA) and isreleasable to the National Technical Information service (NTIS). At NTIS itwill be available to the general public, including foreign nationals.

This technical report has been reviewed and is app ved for publication.

PU L. ROSENGRE JR., Capt, tISAF HN A. CENTRONE, Maj, USAFP ectofiAirbase Survivability Branch

•OHN E. GOIN, Lt Col, USAF R EBADON utyhief, Engineering Research Division Director, Engineering & Services

Laboratory

t.. ; , ; , . . ,

(The Reverse of Th~s Page is Blank)

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TABLE OF CONTENTS

Section Title Page

I INTRODUCTION .......... ...... ......... ....... 1

II OBJECTIVES ............................... .............. 2

III FINITE ELEMENT MODELS AND RESULTS ........................... 3

PLATE MEMBRANE MODEL ..................................PIATE MEMBRANE AND ROD MODEL ..................... ,...... 5

PLATE-BENDING MODEL...................................... 7

PLATE-BENDING - MEMBRANE MODEL .......................... 7

PLATE-BENDING - MEMBRANE AND BEAM MODEL .................... 8

IV CONCLUSIONS. ....................................................................... 9

V RECOMMENDATIONS ......................................... 10

REFERENCES .............................................. 12

Appendix

A FIGURES ................................................. 15

B TABLES ................... ............................. 22

C DISPLACEMENT APPROACH RIGID FORMATS .......................... 36

D NASTRAN FINITE FLEMENTS............................... 40

E QUARTER-SCALE UNDERGROUND SHELTER TEST ........................ 43

F RESISTANCE - DEFLECTION FUNCTIONS ............................. 47

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-- - I-- - j - '•i .. .. • i i ..... ... ............. i • i ' .. .. .i .. .i ... ... ..

LIST OF FIGURES

Figure Titie Itlq-

A-i Undetormed Shape for Membrane and Membrane Rod Finite

Element Models ................................................. 16

A-2 Undeformed Shape for Plate-Bending and Plate-Bending-BeamFinite Element Models ........................................ 11

A-3 NASTRAN Plot of Deformed Membrane Model ...................... 18

A-4 NASTRAN Plot of Deformed Plate-Bending Model ................. )A-5 Typical. NASTRAN Stress vs Time Plot. . ........A-6 Typical NASTRAN Deflection vs Time Plot ...................... 21

LIST OF TABLES

Table Ti tle l'*

B-I MEMBRANE FINITE ELEMENT (QDMEM2) MODEL VS TEST 1 RES4LT,

FOR GAGE ET4 ....................................... .........B-2 MEMBRANE FINITE ELEMENT (QDMEM2) MODEL VS TEST 1 RESULTS

FOR GAGE FT5 ............... .................................. 24B-3 MEMBRANE-ROD FINITE ELEMENT (QDMEM2, ROD) MODEL VS TESTA

POR GAGE ET4 ..................................................B-4 MEMBRANE-ROD FINITE ELF24ENT (QDMEM2, ROD) MODEL VS TEST I

FOR GAGE ET5 .................................................B-5 MEMBRANE-ROD FINITE ELEMENT (QDMEM2, ROD) MODEL VS TEST 1

FOR GAGE ET4 .................................................. 2/

B-6 MEMBRANE-ROD FINITE ELEMENT (QDMEM2, ROD) MODEL VS TEIST 1FOR GAGE hT55 ................................................. 2B.

8-7 MEMBRANE-ROD FINITE ELEMENT (QDMFM2, ROD) MODEt. Vs TEST IFOR GAGE ET3 ................................................. All

B--8 MEMBRANE-ROD FINITE ELEMENT (QDMEM2, ROD) MODEL VS TEST IFOR GAGE T2 .................................................

B-9 MEMBRANE-BENDING FINITE EILFEENT (QUADI) MODEL VS TEST I

RESULTS FOR GAGE ET4 .........................................B-10 MEMBRANE-BENDING FINITE ELEMENT (QUADI) MODEL VS TES',' 1

RESULTS POP? GAGE 'F.5 ......................................... 32B-11 MEMBRANE-BENDING AND ROD FINITE ELEMENT kQ[UAD1 , BAR) MODEl

VS TEST 1 RESULTS FOR GAGE ET4 ............................... 31

B-12 MEMBRANE-BENDING AND Rod FINITE ELEMENT (QUADI, BAR) MODIE.VS TEST 1 RESULTS FOR GAGE ET5 .............................

B-13 MATERIAL AND SECTION PROPERTIES FOR QUARTFR--,' •E MODEl ....... 35

JV

SECTION I

i NTRODUCTI ON

The United States Air Force is investigating ways of predicting the

response of reinforced concrete in hardened structures due to blast (dynamic)

loads. The dynamic loads are caused by the nearby explosion of conventional(non-nuclear) weapons. Current-generation hardened structures are reinforcedconcrete construction, covered with layer(s) of soil, rock rubble, and/or aburster slab. The steuctures have boxy shapes with heavily rei nforcedconcrete walls, floors, and roofs.

The Air Force hopes to reduce the costs of construction by improving the.analysis and design techniques. Substantial savings can be made by reducingthe amount of steel required in the shelters.

Present design and analysis procedures use "limit-state" theory todetermine the ultimate st -ength of the structure and idealize the structure asa single degree-of-freedom system to predict the required resi,'tance. Theseassumptions neglect the interaction between the structural cntiponents such asthe roof and the supporting walls. Finite element techniques offer the bestmeans of modeling the structural interaction between components and predicting

the capacity of e~ch component.

Commonly available structural analysis programs are not strictly appli-cable to this problem. Most programs do not contain nonlinear materialbehavior and provisions for soil elements, concrete-soil interaction, orsteel-concrete interaction. The few programs that address these areas aredesigned for nuclear weapons where the loading has a long-duration shockfront. Conventional weapons have a short-duration pressure wave that de-creases rapidly with distance from the detonation (Reference 1).

Many current investigations attempt to accurately model reiiforced con-crete for use in finite element programs. Investigators are developing modelsof the concrete that will exhibit the nonlinear stress-strain behavior and thelow tensile strength that result in cracking and loss of stiffness in thereinforced concrete. More sophisticated models are including strain-rateeffects and load histories so that repeated detonations can be modeled(Reference 2). Present design criteria for hardened structures do not callfor repeated explosions from conventional weapons.

Besides the modeling problems of concrete, the interaction between thereinforcing teel and concrete is not completely understood. An accuratereinforced concrete element must model the bond stress and friction frominterlocking between the deformed reinforcing bars and the concrete.

Until a complete reinforced concrete model is developed, analysis techni-ques will be able to predict failure mechanisms related to the response ofthe complete structure only. For those explosions where the structure canrespond as ýt un't, developing bending and membrane forces, finite elementUVŽt (.;an ulnumodul the rosponse with sufficient accucacy. Structural responsewhere the pressures ain high enough to cause failure of a localized area canktot be modeled (Reference 1).

•J1

SECTION II

OBJECTIVES

The main objective of this project was to determine if the finite elementprogram, NASTRAN, is a useful tool in modeling the response of hardenedreinforced concrete structures. The accuracy of the finite element model wasof primary importance and results from tests on a quarter-scale model of anunderground structure were used to evaluate the computer predictions. Theutility of the program was also judged on the usefulness of the output and theease of input.

A secondary objective was to understand the full capabilities of NASTRANand to relate them to the problems associated with modeling reinforcedconcrete. This would identify areas of NASTRAN that cou].d be changed toimprove its ability to model reinforced concrete.

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SECTION III

FINITE ELEMENT MODELS AND RESULTS

NASTRAN (NASA Structural Analysis) was developed to aid in the linearanalysis of structures for the loading conditions associated with flight.Different static and dynamic analysis approaches are available and limitednonlinear capability is included. Nonlinear behavior nay arise from changesin geometric shape or material properties.

To evaluate NASTRAN, different finite element models of a quarter-scaleunderground shelter were constructed and used in different static and dynamicanalysis approaches. The finite element models were representative of theelement types available in NASTRAN and were not intended to be sophisticatedmodels of reinforced unncrete, since an accurate reinforced concrete elementdoes not exist. The wodels used were a combination of rod and plate elementsin which the rods represented the reinforcing steel and the plates representedthe concrete and a plate element alone that modeled both the steel andconcrete.

The models were used in different analysis approaches and the predictionscompared to the recorded data from the actual test. The computer projectionswere output in numerical and graphical form and the usefulness of the outputwas evaluated. Because of the limited experimental data, the model predic-tions for maximum strain at specific locations of reinforcing steel, time tothat maximum strain, and time to return to zero strain were used to judge theaccuracy of the model.

The points used for evaluation were located on the top layer ofreinforcing steel at the middle of the roof span and at the face of theinterior wall (see Figure A-i). The gage points are denoted by ET4 and ET5for Test 1 and by ET2 and ET3 for Test 2. These points had the bestexperimental records for both Test 1 and Test 2.

A better measure of the accuracy of the model would have been deflectionsrather than strains. However, deflection data were unavailable. For themodel composed of plates only, strain output from NASTRAN was not possible andsome bias was introduced when converting deflections to strain.

The finite element models were used in two displacement analysisapproaches, static and transient. The static approach used the static.'equivalent" uniform loads provided in Reference 4. No mass effects areconsidered in static analysis. This analysis was usedl tu develop aresistance-deflection function for each finite element model.

The transient analysis used a linear representation of the load historytaken from the pressure-time traces in Reference 4. A grid point was placedat each pressure gage location (seL Fiqure A-7) for each finite element model.The loads could be applied directly W each grid point by assuming the

k3

pressure was uniform', distributed over one-.half the element lenqth in both

directions. The loads were stopped at the face of each wall.

Because the transient approach is a dyn~amic analysis, structural dampingand nonstructural mass effects of the covex.'ing soil and burster slab car. eincluded, using NASTRAN. For this investigation, they were omitted,excelt to,,rtK,ý model composed of plate membrane and rod finite elements, Varying amounts3of damping and nonstructural mass were added to the model to determine theireffects. Damping and nonstructural mass were applie3 independently ot eachother.

No rational way of predicting the amount of structural daulping is avdil-able, so the amounts used were chosen in an attempt to fine-tune the model.The denominator in the damping coefficient fraction is the second natural fre-quency from an eigenvalue analysis and the numerator was chosen to improvemodel response. A convenient way of converting this damping to percentage otcritical damping was not found.

Some of the mass above the shelter root acts with the roof and therefoikcaffects the mass terms in the dynamic equilibrium equation. The nonstructuralmass of the concrete buirster slab, plus the covering soil, and the nonstruc-tural mass of the soil only, was placed on the roof of the model. Since thereis no way of determining how much nonstructural mass acts with the shelter,these two amounts seem to be the loqical. choices.

The effects of damping and nonstructural mass were added to the TransientAnalysis Approach on a model that included nonlinear loads. All NASTRANdynamic approaches assume linear material properties. As the stress in theconcrete increases, the concrete cracks in the tension zone and responds non-linearly in the compression zone. Presently, the only way of inducing non-linear behavior in the NASTRAN dynamic appioaches is by including loads trig-gered by the displacements and/or velocities of grid points.

To represent the loss of stiffness in a reinforced concrete member due tocracking; nonlinrar loads based on the resistance deflection function wereapplied in the dLrection of the blast loading. The load was equal to the dif-ference between the NASTRAN elastic prediction (static analysis) for theresistance-deflection function and the resistance-deflection function as des-cribed in Reference 10. A detailed description of the resistance-deflectionfunction is found in Aplnendix P.

The additional load used was assumed to start at the deflection where thefinite element model strength and the ultimate moment strength from Peference10 coincided. The load increased linearly to the maximum deflection of theshelter. This neglected the portion of the resistance-deflection function,whereas the finite element model underestimnted the flexural strength of thereinforced concrete structure.

4

'fTe finite element models require cert;tin rimecetric ei'd material proper-ties. Table B-13 containe; the section properties of the quarter-icale model.

Appendices C and D contain brief discussions uf the NASTRAIN analysis3pproaches and finite elements. 'ite discussions deal only with those parts ofthe approaches nnd element;,; applicable to modelinq reinforced concrete underblast loaxdings. More detailed information is available in ReferenceF 12 and13. Appendix E descrihes the quarter-scale model.

PLATE MEMBRANE MODEL

The initial model was composed of plate membrane elements (QDMEM2)representing both tho steel and the concrete. The model was formed by passingtwo parallel planes a tinit distance apart through the cross section of theshelter (See Figure A-i). The nodes of each element were on the outermostconcrete fiber, halfway between the planes. No attempt was made to model thereinforcing steel, and the concrete material properties were used for allelements.

For this element, output of strain at the location of the reinforcingsteel was not availahle. The difference in the horizontal displacements wasconverted to strain by dividing by the length of the elements. This was thestrain on the outermost top and bottom concrete fibers. By assuming planesections remain plane, these strains were converted to strain at the locationof the reinforcing steel by using similar triangles. Since the grid mesh isvery coarse, this strain was a crude average. In the Transient Analysis Ap-proach, the time to maximum strain and the time to return to zero strain wereassumed to occur at the time to maximum horizontal displacement, and at thetime to return to zero horizontal displacement, respectively.

The results from this model are in Tables B-i and B-2. The maximum strainin the static and transient approaches is too small, particularly near theinterior wall, and the time to zero strain is much too short for the transientapproach. Nonlinear loads increase the strains and the time to return to zerostrain; however, the loads did not improve the overall accuracy of the model.

The results from this model do not indicate that membrane elements alonecould successfully model a reinforced concrete shelter under blast loads. Thestructure has a thick roof with very short spans, so that inplane forces arepredominant. If membrane elements alone cc uld model a hardened shelter, theyshould work for this particular structure. As the vertical deflectionsincrease or as the span of the roof lengthens, bending would become moreimportant and the membrane model would become worse.

PLATE MEMBRANE AND ROD MODEL

To improve the representation of reinforced concrete, rod elements (ROD)with linear steel nroperties were added to the membrane finite elements. The

I

nodes for the membrane and roxJ elements were moved to the location of the

reinforcing bars. An advantage of this model over the plate membrane model

alone, is that although many new elements were added, the solution times

should not ba substantially increasei because no additional nodes were added.

The nembrane elements represent the concrete between the top and bottom rein-

forcement. This neglects the concrete in compression outside the steel bars;however, it includes the portion of the concrete in tension between the rein-forcin,5 steel layers.

This model worked very well up to the maximum striin using a transientanalysis approach (see Tables B-3 and B-4). The percentage difference for themaximum strain and the time to maximum strain were small enough to heconsidered negligible. Even when the most sophisticated methods are usea. todetermine material properties, the coefficielLt of variation for deflectionpredictions is in the ordpr of 15 to 20 percent (Reference 3). For this case,the worse percentage difference (31 percent) should be considered excellent.

Since the results for Test 1 loads were so good, this model was used withthe loads from Test 2. Results were excellent except for the maximum trainat the middle of the roof span (Table B-8, Gage ET2). The experiment;lresults for this ýsage. are questionable. Normally, the tirst plastic hir j,:would form at this point and this section would have the largest strain in theroof. Therefore, the computer model is probably more accurate than shown inTable B-8.

Nonlinear loads decreased the accuracy of the model by causing too muchincrease in maximtun strain and the time to maximum strain. Sti in wasimproved.

The effects of nonstructural. mass were investigated using this model andthe results are presented in Tables B-3 and B-4. The amount of strain wasgreatly reduced and the response time of the structure increased. Even withonly the weight of the soil applied to the roof, the model was much too stiffand slow to respond. A small amount of nonstructural mass could improve theresponse of the model; however, a rational way of determining the nonstruc-tural mass is not available.

This model was also used to test the effects of structural damping. Theresults are shown in Tables B-5, B-6, B-7, and B-8. As the structural, dampingincreased, the maximum strain and the time to maximum strain decreased, andthe time to return to zero strain increased. Although the amount of dampingwas chosen in an attempt to fine-tune the model, the results were no betterthan in the transient approach withoal nonlinaer lnaids'

This model works particuarly well up to the point of maximum strain.Beyond that time the i.odel responds too quickly. by including the properamount of structural damping and nonstructural mass the model response couldbe improved for a longer time period.

6 I-

PLATE-BENDING MODEL

A model was made of the shelter using plate-bending elements (QDPLT) only.The model is formed by paEsing two parallel planes 12 inches apart through theshelter cross section. The nodes of the elements are on the middle surface ofthe walls, roof, and floor (see Figure A-2). The material properties usedwere for concrete, Bigg's moment of inertia for a cracked reinforced concretecection was used for the bending stiffness. The distance from the compr-ssionfact, to the centroid ro the tension steel was used as the transverse shearthickness (Reference 2).

The model was used in a static analysis appi:oach only, because the ele-ments require 'mrealistic boundarj conditions. A plate-bending element cannotresist inplace forces, so to be used as an element in a wall the vertical de-flection of the wall must be completely constrained. Other finite elements donot require such restrictive boundary conditions and are more useful. Thiselement may be employed as a roof element when used in combination with otherelement types. The roof ahould '-7ve a large span-to-depth ratio.

PLATF BENDING-MEMBRANE MODEL

By uriag an element that can develop inplane forces, realistic boundaryconditions can be used. A model using plate-bending-membrane finite elements(QUADi) vas constructed in the same manner as for thL plate-bending mcxel.Since the reinforcing steel was not being modeled, concrete material proper-ties were used. Bigg's moment of inertia was used for bending stiffness, thedistance from the compression fact to the centroid of tne tension steel wasused for the transverse shear thickness, and the total concrete thickness wasused for the membrane shear thickness (Reference 2).

For the accuracy -:. _,ulred to -nalyze or design a shelter, the plate-bending membrane model requires approximately the same number of nod,ýs as aplate-membrane element model. Because of the gross mesh size used in thisproject, extra nodes had to be added for the plate-bending-membrane model.The stresses in each element are an avez.ge of the stresses within tne ele-ment. This caused the element at the face of the interior wall to have smalltensile strains rather than the large cospr~ssion strains expected. Placing asmall element at the face of the wall (see Figure A-2) corrected thisproblem.

Tables B-9 and B-10 contain the results from the Static and TransientAnalysis Appruoches. The model is too flexible, does nnt reach the maximumstrain soon enough, and returns to zero strain too quickly. Nonlinear loadsworsen the predictions, as would be expected for an overly flexible model.

A model composed of plate-bending-membrane elements alone did not exhibitthe kinds of characteristics necessary to model re:nforced concrete. Theflexibili. of the model depends on the designer's choice for a moment of in-ertia. Also, input values for the transverse shear and membrane shear thick-ness; are debatable. A better finite element mode] would have fewer and lesscritical choices.

PLATE 3ENDING - MEMBRANE AND BUAM MODEL

By adding a beam (BAR) element to 'he bending-membrar',2 model the impor-tance of the value of the moment ot inertia is reduced. The plate element wasused to represent the concrete and was given concrete material properties.The bending stiffness was the moment of inertia of the concrete bc-ween thecenter of gravity of the total section and the top liyer of steel about thecenter of gravity of the total section (neglects concrete outside the Stee]layer). The transverse shear thickness was the distance from the compressionface to the centroid of the tension steel and the membrene shear thickness wasthe distance between the steel layers.

Two beam elements were offset the proper distance in each direction, oneach side of the middle surface of the plate. By releasing the the proyerconstraints only axial extension of each beam was allowed and bending \.:aseliminated. Each beam had a cross-sectional area qual to one-half the rein-forcing steel area in each layer and each element had the standard steelmaterial properties.

The results from this model are in Tables B-I1 and B-12. The model workswell in a Transient AnalysiP Approach, without nonlinear loads. Good results.are obtained at the midspan of the roof; where bending forces arepredominant. Applying nonlinear loads increased the maximum strain aridincreased the time to maximum strain; however, it did not improve the overailaccuracy of the model. Near the interior support the model was too stiff,even with nonlinear loads applied. As with all models, this one returned tozero strain much too quickly.

The plate bending-membrane and beam model waý; the second most accuratemodel in this investigation. Indications are the model would be more accuratefor longer span, thinner roof sections. The depth-to-spun ratio of 3/16 isnot small deflection plate theory and this model may be more applicable tofull-size structures (Reference 11). The input requirements for this modelare large because of the large number of elements; therefore, significant

gains in accuracy would have to be achieved before this would be the model ofchoice.

/B

SECTION IV

V CONCLUSIONS

The finite element program NASTRAN can be extremely useful for analyzingand designing hardened reinforced concrete shelters. When using a finiteelement model composed of plate membrane elements (QDMEM2) and rod elements(ROD), the structure is correctly modeled up to the maximum strain at the roofspan centerline and near interior support. Since the strains are accuratelymodeled, it ii reasonable to assume deflections are also successfully predic-ted.

Element stresses will not be correct because a linear stress-strain rela-tionship is used by NASTRAN. Element forces are determined directly from theelement displacement matrix and will be as accurate its the displacements.Deflection predictions of reinforced concrete members traditionally have alarge degree of uncertainty and a prediction within 20 percent of the true de-flection should be considered excellent (Reference 3). The large predictionerror is related to material property and construction practices, so reduc-tions in the size of the error cnnot be made by improving the mathematicalmodeling techniques.

Better modeling techniques will help improve the deflection predictionsfor repeated explosions for pressures high enough to cause localized failures,and for time intervals that exceed the time to maximum strain. In particular,bette-f techniques are needed to predict permanent set of a hardened structure.The area,: that need improvement are in the finite element representation ofreinfox. .3 concrete, the amount and type of structural damping in hardenedstructures, and the amount of covering material that acts as nonstructuralmass on the roof.

When dynamic analysis of a hardened structure is conductedhuge amounts ofoutput data are generated. By far the most functional form of output is gra-phical. Numerical output inundates the engineer with data and does not pro-vide a feeling for how the stucture is responding. The graphical capabilitiesof NASTRAN are a tremendous advantage. tndeformed plots (see Figures A-i andA-2) and deformed plots (see Figures A-3 and A-4 may be made separately orsuperimposed. Plots of elements stresses, strains and forces versus time (seeFigure A-5) and plots of grid point deflectiors, velocities, and accelerationsversus time (see Figure A-6) are available. Input loads versus time also canbe plotted (Reference 13).

Input •'or NASTRAN is tedious at best. A large number of cards must bepunched for small problems aad the number increases quickly aa the number ofelements increase. The format of the input cards is rigid, with input datarestricted to certain fields. Where interactive computer service is avail-able, input data could be written to disk as card images and properly formatedby simple user-written programs. This reLieves some of the monotony of input-ing data but would not reduce the number of characters input.

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SECTION V

RECOMMENDATIONS

Results from the analyses performed during this project indicate NASTRANin its present form is useful in analyzing hardened reinforced concrete shel-ters up to the time of maximum strain. The test used in this project had alimited amount of useful data. More validation runs of NASTRAN should be madebefore the program is used for design of hardened structures. The test dataneeded for the validation run are:

(1) blast pressures on the surface of the reinforced concretemember(s).

(2) deflections and velocity of regularly spaced points on theouter surfaces of the reinforced concrete member(s).

(3) strains at selected points on the tension and compresscionsteel.

The boundary conditions on the test :,pecimen should be instrumented sothat any support movement can be determined and used in the analytic model.Any strain gage used on a concrete surface should use at least 3 inches oflength to eliminate bias from cracks and coarse aggregates.

The ability of NASTRAN to model reinforced concrete can be improved byadding nonlinear material behavior to the dynamic analysis approach. Non-linear material behavior is available in a static approach (Piecewise LinearAnalysis) and could he added to the dynamic approaches by changing ti.d rigidfo rma ts.

Accurate prediction of structural response beyond the maximum strain re-quires basic research into the nature of structural damping and nonstructuralmass. The limited analytic study in this project showed thecomputer model to be very sensitive to changes in nonstructural mass andstructural damping. The surrounding and covering soil layers (including bur-ster slab) act with the shelter as nonstructural mass and add structural dam-ping. Since both phenomena are related, a nondimensional model study may beable to develop rational methods of determining stuctural damping and non-structural mass for the specific configuration of a hardened structure (Refer-ence 9).

Intensive work is being done developing a reinforced concrete finite ele-ment and a good element should be available at any time (Reference 5). Mostwork is being done to model nuclear explosions and the finite element willhave to be changed to properly model reinforced concrete subjected to a con-ventional weapon explosion Whenever the model becomes available, it shouldbe added to NASTRAN to improve its ability to predict the response of rein-forced concrete structures.

10

I-

The new element will contain strain/rate (or stress/rate) effects on ma-terial properties. Most of the data available on strain/rate effects are 23years old or older and contain some questionable data. However, the infor-mation indicates gains in ultimate concrete strength are in the range 50 to100 percent (References 6, and 7). An investigation into the strain/rateeffect is needed and should lead to substantial cost savings in reinforcedconcrete shelters.

Investigation into the use of other general purpose finite element pro-grams shoxild continue. Emphasis should be given to those programs that con-tain dynamic analysis capabilities that include structural damping and non-structural mass. The programs should be able to be altered to take newfinite elements. A most important consideration is the ability to generategraphical output. Lack of graphical output would seriously affect the pro-gramss ability for practical use.

[[

11

K...,

REFERENCES

1. Baseheart, T. M., The Response of Reinforced Concrete Structures toNear Field Explosions, USAF Summer Faculty Research Program, Vol 1, AFOSR,Bolling AFE, Washington, DC, 1980.

2. Hegemier, G. A., Evaluation of Material Models for MX Siting Vol II:Reinforced Concrete Models, SSS-R-80-4155, Systems, Science and Software,LaJolla, CA, November 1979.

3. Biggs, J. M., Introduction to Structural Dynamics, McGraw-Hill Book Co,New York, NY, 1964.

4. Branson, D. E., Deformation of Concrete Structures, McGraw-Hill Book Co,New York, NY, 1977.

5. Coltharp, D. A., Analysis of One-Quarter-Scale Model Test Results,Unpublished Report from USA Waterways Experiment Station, StructuresLaboratory, Structure Mechanics Division, Vicksburg, MS.

6. Mindess, S., and Young, J. F., Concrete, Prentice-Hall, Inc., EnglewoodCliffs, NJ, 1981.

7. Neville, A. M., Properties of Concrete, Pitman Publishing Limited, London,Great Britain, 1981.

8. Przemieniecki, J. S., Theory of Matrix Structural Analysis, McGraw-HillBook Co., New York, NY, 1968.

9. Smith, J. H., and Vann, W. P., Theoretical and ExperimentalInvestigation of Buried Concrete Structures VolI.: Analysis and Experiment,AFOSR-TR-76-1070, AFOSR, Bolling AFB, Washington, DC, November 1975.

10. Structures to Resist the Effects of Accidental Explosions, AFM88-22,Departments of the Army, the Navy, and the Air Force, Washington, DC, June1969.

11. Szilard, R., Theory and Analysis of Plates, Prentice-Hall, Inc.,Englewood Cliffs, NJ, 1974.

12. The NASTRAN Theoretical Manual, NASA SP-221(05), National Aeronautics andSpace Administration, Washington, DC, December 1980.

13. The NASTRAN User's Manual, NASA SP-222(05), Nationial Aeronautics andSpace Administration, Washington, DC, December 1980.

14. Timoshenko, S. P., and Goodier, J. N., Theory of Elasticity, McGraw-HillBook Co., New York, NY, 1934,

12

is. venkayya, V. B.# Eastep, F. E., and jobflgon, J. R., NASTR1N

Regi~nner's Course, T1M-FBR-76-12i, Air Force Flight Dynamics Laboratory,

wright-P-t-terson APB, OH, Sept-nber 1976.

16. Wang, C. K., and Salmon, C. J., Re~inforced Concrete ýDes~ign Harper ana

Row Publishers, New York, NY, 1979.

13

(Tue reverse~ of this page is blank)

APPENDIX A

FIGURES

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34

TABLE B-1 3, MATERIAL AND SECTION PROPERTIES FOR QUARTE--SCALE MODEL

Concrete

E - 4.25 X 106 psi

G = 1.77 X 106 psi

Poisson's Ration - 0.2

Unit Weight - 150 pcf

Unit Mass - 0.000216 lb sec 2 /in 4

Steel

E - 29.0 X 106 psi

G = 11.2 X 106 psiI.I

Poisson's Ratio = 0.3

Unit Mass - 0.000735 lb sec 2 /in 4

Soil

Unit Weight = 110 pcf

Structural Clear Thicknkss Flexural SteelElcinent Span Depth Area

(in) (in) (in) (in 2 )

Roof 48 9 8 1.32 each face

Exterior Wall 48 9 8 1.32 each face

Interior Wall 48 6 *5 *0.66 each face

Floor 48 6 *5 *0.82 each face

SAssumed ValUes

35

"PPENDIX C

DISPLACEMENT APPROACH RIGID FORMATS

INTRODUCTION

The NASTRAN (NASA Structural Analysis) Computer Program is a general-pur-pose structural analysis program intended for a wide range of applications.To meet this requirement NASTRAN contains different finite elements to repre-sent common construction members and allows these elements to be comhined tomodel more sophisticated construction materials. NASTRAN currently provides

15 different methods to determine displacements of a ýtructure. Each methodis referred to as a Displacement Approach Rigid Forma.: and the user choosesthe rigid format, depending on the type of analysis required (Reference 15).

To improve the generality of NASTRAN the program is divided into subpro-qrams or modules that can be called independently of each other. A rigid for-mat is a permanently stored sequence of subprogram calls which produce a par-ticular kind of structural analysis. The user may alter the sequence of callsor develop a new sequence for unusual analysis problems (Reference 13).

One of the subprograms available for use with any rigid format is the PlotModule. The plotting routines allow plotting of specific input and outpntdata on a SC-4020 plotter, an Electronic Association Incorporated plotter, andmost Calcomp plotters. Also a user may write his own plotter routines for usewith NASTRAN (Reference 13).

Some rigid formats yield larqe amounts of numerical output. output ingraphical form gives the engineer a better feeling for the response of thestructure. Figures A-I, A-2, A-3, A-4, A-5, and A-6 are samples of the typeof output available through the NASTRAN Plot Module. Undeformed structuralplots aid in the detection of input errors in grid point coordinates and ele-ment connections.

For this investigation two rigid formats were used, Static and Transient.Two other rigid formats, Piecewise Linear and Differential Stiffness may beuseful. The remainder of this appendix is a brief discussion of those rigidformi•ts, emphasizing the portions of those formats that are useful in anal-yzing hardened structures.

STATIC DISPLACEMENT APPROACH RIGID FORMAT

The Statin Analytvih Rigid orpm.at is used to determine deformation, strcs-ses, etc, caused by very slowly applied loads. This rigid format is a goodstarting point in any analysis because the designer can check for input errorsin the finite element model and for conceptual errors in the choice of ele-ments and boundary conditions.

36

K [ .j

The loads may be concentrated at grid points, uniformly distributed overtwo-dimensional finite elements, and generated internally as body forcescaused by gravity. NAS1TRAN will compute the gravity loads from the mass ma-trix if the gravity acceleration vector is input. The mass matrix is composedof the element mass and any nonstructural mass applied to the element, forexample, soil above an underground structure (Reference 12). Different loadfactors may he applied to each load to aid in "limit state" design.

Solving the set of simultaneous equations generated by the finite elementmethod uses large amounts of computer resources. The Static Analysis RigidFormat is set up to analyze the structure for different loading combinationsand/or different boundary conditions without resolving the set of simultaneousequations. Each loading combination and each new set of boundary conditionsis called a subcase.

TRANSIENT APPROACH RIGID FORMAT

To solve structural dynamic problems the Transient Approach Rigid Formatis used. This approach assembles the structural stiffness, mass, and dampingmatrices; generates time history load tables; and solves the differentialequations by a form of the Newmark Beta Method. The stiffness, mass, anddamping matrices may be assembled by NASTRAN and/or directly input by the user(Reference 12).

Because of the large number of load entries required for even a moderatesize problem, NASTRAN provides four different functions, three linear func-tions and one power function, to input dynamic loads. In addition to those

loads NASTRAN can generate nonlinear loads triggered by the deflections and/orvelocities of grid points (Reference 12). Nonlinear loads provide a means ofmodeling nonlinear material behavior.

Mass matrices are generated by either the Lumped Mass or Coupled MassMethods. Some individual elements have restrictions on the method of mass ma-trix generation allowed. Also, masses may be assigned directly to grid points(Reference 12).

The Lumped Mass Method distributes the structural mass and nonstructuralmass evenly between the nodes of the element. This yields a simpler modelbecause there is no inertii coupling between grid points. In most cases theLumped Mass Method will result in natural frequencies below the true value(Reference 12).

The Coupled Mass Method, sometimes called "Consistent" Mass Method, yieldsan inertia coupling between grid points, i.e., the inertia properties of agrid point affect the inertia properies of adjacent grid points. The elementmass matrix is dependent on the elastic properties of the eltmnent. Thismethod normally yields natural frequencies above the exact results (Reference12).

37

IThe damping matrix is the skim of direct input damping coefficients fron I

viscous damping elements, a percentage of the structure stiffnes;s matrix, anda percentage of any/all element stiffness matrices. The user specifies theappropriate constants for NASTRAN to compute the damping matrix. If t,'h-, inputdata is not specified, structural damping is neglected. Normally strrcturaldamping is expressed as a perc,,ntage of the critical damping (Reference 2).NASTRAN does not contain a convenient way of expressing damping in this man-

For this rigid form~at graphical output is particularly useful. plots of

element stresses and forces versus tim.', and grid point deflections and velo-cities versus time are available. Alco deformed and undeformed structureplots are available in three dimensions.

PIECEWISE LINEAR APPROACH RIGID FORMAT

Piecewise Linear Analysis is used to solve problems involving matezialp].ntl.ci.ty. This rigid format is presently restricted to statically appliedloads. Since the loads cannot be varied with time, it is not applicable tothe analysis of hardened structures in its present form and was not includedin this investigation.

For the Piecewise Linear Approach the user specifies a material stress-strain table and the amount of load to be applied in each increment. The ri-

gid format proceeds to determine the displacements and strain using the firsi.load increment and the user-specified elastic material properties (not theuser provided stress-strain table). NASTRAN then uses extrapolation to deter--mine the strain expected in the next load increment, with this projectedstrain a linear approximation for the modulus of elasticity for each elementis calculated from the stress-strain table. A new stiffness matrix is as--sambled and the deflections 'nd strains for the next load increment are deter- imined. Deflections and strains are accumulated after each load increment. HThe rigid format repeats the sequence begini.ing with the extrapolation to es-timate the next strain and continues until. the total load has been applied(Reference 12). Solving the simultaneous equations for each load incrementrequires huge amounts of computer resources. Choosing too many steps uses anunnecessary amount of computer resources; however, too few steps will make thesolution inaccurate.

Piecewise Linear Analysis also gener.ites large amounts of output data.The plotting capabilities of the plot modules provide output in a more usefulto .0.

DIFFKRENTIAL STIFPNESS APPROACH RIGID FORMAT

In some structural problems deformations occur that adversely affect thestructures' ability to carry the loads. In the design of tall steel buildingsthis is referred to as the P-Delta effect.

iiii

This rigid format introduces nonlinearity into the analysis caused bylarge deflections. The NASTRAN approach is an approximate analysis and maynot be applicable to a particular problem, so that the usi of this rigid for-mat must be carefully reviewed. One important limitation is that appliedloads remain fixed in direction and magnitude during the movement of theAtructur'c (Reforonce 12)o

in some hardened structares this rigid format may prove useful; however,when reinforced concrete is the construction material faijure of the concretewill occur before the deflections affect the structural capacity (Reference16). The Differential Stiffness Approach Rigid Format was not investigated inthis report.

39

APPENDIX D

NASTRAN FINITE EEMENTS

INTRODUCTION

Only a few of the available finite elements were used in this investiga-

tion. This appendix lists those elements and contains a brief description of

the important element properties.

The NASTRAN plate membrane elements were developed by constructing the expres-

sion for strain energy using a linear variation in the inpliane displacements.

The basic element is triangular (TRMEM) and is used to form other quadrilat-

eral elements. The quadrilateral elements are formed by either overlappingfour triangular elements (QDMEM) or by connecting tour triangular elements(QDMEM2) at the center point of the quadrilateral (Reference 12).

The material properties for the elements may he anisotyopic; however, onlyisotropic material was used in this investigation (Referent, 14). For an iso-tropic material two of three material constants, modulus of elasticity, shearmodulus, or Poisctonl's ni tio must be specified.

The Lumped Mass Method for transferring structural and nonstructural massto adjacent grid points is the only method available for membrane elements.The mass matrix and inplane stiffness for the elements are found by assigningone half the thickness of the quadrilateral element to each triangular ele-ment. Although the Coupled Mass Method cannot be used for structural modelswit h membrane L;ifttuuitu only, the method may be specified for elements withboth bending and membrane stiffness. The terms in the mass matrix which cor-respond to inplane motions are computed by the Lumped Mass Method and theother mass terms will be computed by the Coupled Mass Method (Reference 12).

The strains within this element are constant because the deflections wereassumed to be linear, so the stresses are an average wi thin each element. Thestate of stress for a nonoverlapping element is the average sLress in the four

triangular elements. A "shear flow" is calculated for the sides of eachelement. The "shear flow" is the change in the inplane force along the sidedivided by the length of the side (Reference 12).

ROD ElEMENT

The rod element (RoD) developed for NAST'RAq includes extensional' aodtorsional properties only and is based on a linear deformation function. Theelements are straight, loaded at the ends only, and have uniform geometric andmaterial properties. The strains are constant for the element because of thelinear deformation functions and the stresses are on average for the element(Reference 12).

40

For dynamic analysis approaches the structural and nonstructural massassociated with each rod may be transferred to the adjacent grid points by theLumped Mass or Coupled Mass Methods. For this element the coupled mass matrixis the average of the pure lumped mass and the pure coupled mass matrices.This yields a much smaller error in the natural frequency than either theLumped Mass or Coupled Mass Methods alone (Reference 12).

PLATE BENDING ELEMENT

The basic NASTRAN plate-bending element (TRPLT) is a triangle with thi'Qc.,degrees of freedom, one translational and two rotational, at each node. Theout-of-plane deflection is assumed to be the following incomplete third-degrocfunction (Reference 12).

w=ax + by + cx2 + dxy + ey 2 + fx 3 + qxy2 + hy 3

The x 2 y term is omitted because there are only eight indepedent deforma-tions in the triangular element, and only eight constants can be determined inthe out-of-plane deflection field. This displacement does not guarantee slopecontinuity on the edges of adjacent elements; however, elements which haveslope continuity do not necessarily give better results (Reference 8).

The plate-bending element may be an anisotropic material and the user canspecify material properties with respect to any axis orientation. The programwill rotate the properties into the proper coordinate system. As with theplate-membrane element, two of the three material constants must be s,''cified

(Reference 13).

The element may be assumed rigid in transverse shear. This decreases themagnitude of the element stiffness terms by an amount equal to the transverseshear stiffness (Reference 12). In beam elements transverse shear stiffnessis normally omitted because of its small size. In hardened structures trans-verse shear stiffness is important because of the thickness of the construc-tion and should not be neglected.

The internal forces are determined at the center of gravity of the trian-gular element. h linear variation of strain through the thickness of theplate is assumed, so stresses may be determined at any distance from themiddle surface (Reference 12).

The Lumped Mass and Coupled Mass Methods are available to create mass ma-

trices for plate bending elements. The bumped Mass Method places one third ofthe element mass at each node. In the Coupled Mass Method, the mass matrix ofeach element is calculated assuming a uniform mass density within the element.

Thus the mass matrix is depends on the bending properties of the plate element(Reference 12).

41

K.m•

A quadrilateral plate-bending element (QDPLT) is formed by overlappingfour triangular bending elements. The bending stiffness of each triangularelement is one half the bending stiffness of the quaM.rilateral element. Themass matrix is formed by treating the quadrilateral as four triangular ele-ments. The stresses are the average of the stresses in each triangular ele-ment (Reference 12).

PLATE- BENDI.NG-MEMBRANE ELEMENT

Small deflection theory for plates leads to independence between membrane(inplane) forces and bending (out-of-plane) forces (Reference 11). NA:sTRANforms a plate-bending-membrane element by overlapping the quadrilateral mem-brane element, QDMEM (composed of four overlapping triangular elements), andthe bending quadrilateral, QDPLT (compos;ed of four overlapping elements). Twoplate-bending-membrane elements are I )rmed; QUADI in which the differentmaterial properties may be speuified for bending, membrane and transverseshear, and QUAD2 which assumes a solid homogqneous cross section (Reference13).

The mass matrices may be formed by using the Lumped Mans or Coupled MassMethod for the bending stiffness. The Lumped Mass Method will be used for themembrane stiffness (Reference 12).

Element stresses are available at arV location away from the middle Sur- !

face. Also bending stresses of the plat- combined with the membrane stressesare available.

Bean Elements

The beam element (BAR) is derived, assuming a straight beam loaded only atthe ends and having uniform geometric and material properties along itslength. The directions of the principal, axis may be selected by the user andthe ends of the beam element may be offset from the grid point to which thebeam is attziched. The connection between the beam end and the corresponding

grid point may be released for any degree of freedom provided one degree offreedom remains (Reference 12).

The ntl.ffness matrix includes extensional, torsional, and bending in twoplanes. The bending stiffness also includes a contribution due to transverseshea7 (Reference 12).

Internal forces and moments are computed on the ends of the element.Stresses at each end due to bending at user's specified points may be detar-mined along with the average axial stress and the maximum and minimum ext.on-sional stresses (Reference 12).

Lumped Mass and Coupled Mass Methods are available to transfer structuraland nonstructural mass to the adjacent grid points. The center of gravity isassumed to lie on the elastic axis and the inertia effects due to offset ends•i beams are neglected. The Coupled Mass Method does not include the effectoi transverse shear on the mass matrix (Reference 12).

42

1 i I I I I I I I I I

APPENDIX E

QUARTER-SCALE UNDERGROUND SHELTER TEST

The data used to judge the finite element models came from a one-quarterscale model test of an underground shelter (see Reference 4). The test- wezitconducted tc ,erify procedures for predicting shelter response and shelterloading caused by an explosion on a covering burster slab (see Figure E-1).All 14 test were run and a large amount of data was collected. sor thisinvestigation only the interface stresses from pressure gages on the top ,•f;he roof and the strain in the reinforcing steel from Test 1 and Test 2 '.eroused. A more detailed report on the test is being prepared and only prelim-inary data was available for this analysis.

Generally the pressure gages functioned well for both teats and providedthe loads for the finite element model. A typical pressure time curve isshown in Figure E-2. The gage at the face of the exterior wall failed in Test1 and the pressure frrm the gage at the face of the interior wall was substi-tuted (see Figure E-1). I

The strain gage datdi was spotty because many gages failed under the highstrain rates. Good strain data were available fran Tests 1 and 2 at the cen-terline of the roof span and at the fac:e of the interior wall (see Figures E-1and E-3). Since one gage was in a tension zone and the other was in a cow-pression zone, these two points wera rchosen to judge the accuracy of the com-1puter models.

Table B--13 contains the properties u:sed in the finite element models. Thesection propert.y data were incomplete and assumed values were used. Themissing information was for the walls aiid floor so they should not be criticalto the structural analysis.

The quarter scale test contained a weighted average for the interfacedpressure gages for each explosion. No information was given concerning howthe weighted average was computed (Reference 4). This value was used for theStatic Analysis Approaches.

43 1

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45

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46

APPENDIX F

RESISTANCE-DEFLECTION FUNCTIONS

in response to externally applied loads a structural member deflects asufficient amount to develop enough internal force to preserve equilibrium.These internal forces can be considered as an equivalent load. Resistance isin the direction opposite the deflection. Since this equivalent load and thedeflections are based on geometric and material properties, a graphical repre-sentation of resistance versus deflection can be drawn independently of theapplied external loads (Reference 10).

For a reinforced concrete member the initial response is elastic and theelastic response continues until the deflection is sufficient to cause firstcracking in the tension zone. However, the nonlinearity induced by the ini-tial cracks is negligible and a linear resistance deformation function may beused until first yield in the reinforcing steel. After first yield in the re-inforcing steel the response of a reinforced concrete member is termed elasto-plastic and sufficient cracking has occured to require a reduced stiffness inorder to model the response in the reinforced concrete member. The upperlimit of the elasto-plastic region is the formation of the first plastichinge. From this point the member's resistance will remain constant untilincipient failure of the member occurs (Reference 10).

The resistance deflection functions were calculated as described in Refer-

ence 10, except for the ultimate resistance of the reinforced concrete member.The equations given in Reference 10 to calculate the ultimate moment resis-tance of a reinforced concrete member neglect the contribution due to the com-pression concrete area. This would underestimate the strength of the concretemember in the elasto-plastic range. After enough rotation to causecrushing of the compression concrete area the equation given will accuratelypredict the ultimate capacity. The following equation was used to predict theultimate capacity of the shelter (Reference 16).

M11 = Asfy(d-a/2) - A'sf's(d'--a/2)

in which

As = tension steel area

fy = tension steel yield stressI.

d .d.ntancc from comnpression f ace to u.-kuui of tension steel

d' = distance from compression face to centroid of compression steelIIA's = compression steel area

f's = stress in compression steel

I a = depth of Whitney's stress block

Nin = ultimate (nominal) moment

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