concrete shell structures

Anton TedeskoChairman

Frank BaronDavid P. BillingtonRichard R. BradshawFelix CandelaJohn V. Christiansen

ACI 334.1R-92(Reapproved 2002)

Concrete Shell StructuresPractice and Commentary

Reported by ACI Committee 334

Committee mebers voting on the 1992 revisions:

Phillip L. GouldChairman

Jack ChristiansenJohn F. AbelDavid P. BillingtonArthur J. Boyt, Jr.Eli W. CohenMorris N. FialkowAjaya K. GuptaRobert B. Haber

Alfred L. ParmeSecretary

Wilhelm FluggeRichard M. GensertOtto GruenwaldMilo S. Ketchum, Jr.James A. McCarthy

Kye J. HanHarry G. HarrisMark A. KetchumMilo S. KetchumStefan J. MedwadowskiLuis F. MeyerEldon F. MockryJohn K. Parsons

A report on the practical aspects of shell design including recommen-dations and a commentary for designers of thin concrete shells. Generalguidance based on current practice is given on analysis, proportioning,reinforcing, and construction. A selected bibliography on analyticalmethods featuring design tables and aids is included to assist the engineer.

Keywords: ACI committee report; aggregate size; buckling; construction; design;double-curvature shell; edge beam; folded plate; formwork; model; prestressing;reinforcement; shell; single-curvature shell; splice; stiffening member; supportingmember, thickness; thin shell.

Eric C. MolkeVice Chairman

Stefan J. MedwadowskiMario G. SalvadoriJohn B. SkillingBruno ThurlimannRobert Zaborowski

Stuart E. SwartzSecretary

William C. SchnobrichAlexander ScordelisDavid B. SouthAnton TedeskoBing-Yuan TingDaniel F. TullyArnold Wilson

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CONTENTS

Preface, pg. 334.1R-2

CRITERIAChapter l-General, pg. 334.1R-2

1.l-Definitions1.2-Scope

ACI Committee Reports, Guides, Standard Practices andCommentaries are intended for guidance in designing,planning, executing, or inspecting construction and inpreparing specifications. References to these documentsshall not be made in the Project Documents. If items foundin these documents are desired to be a part of the ProjectDocuments, they should be phrased in mandatory languageand incorporated into the project documents.

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Chapter 2-Analysis, pg. 334.1R-22.1-Assumptions2.2-Analysis of thin shells2.3-Analysis of supporting members2.4-Model analysis

Chapter 3-Stability analysis, pg. 334.1R-33.1-Buckling

Chapter 4-Proportioning, pg. 334.1R-34.1-Allowable stresses and load factors4.2-Shell thickness4.3-Shell reinforcement4.4-Prestressing4.5-Concrete cover over reinforcement

Chapter 5-Construction, pg. 334.1R-4

ACI 334.1R-92 replaces ACI 334.1R-64 (Revised 1982) (Reapproved 1956)effective Mar. 1.1992. The 1992 revision included minor editorial corrections andthe deletion of the year designation for ACI 318 to make the current code theapplicable reference.

Copyright 0 2002 American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by

any means, including the making of copies by any photo process, or by any elec-tronic or mechanical device, printed or written or oraI, or recording for sound orvisual reproduction or for use in any knowledge or retrieval system or device,unless permission in writing is obtained from the copyright proprietors.

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334.1R-2 ACI COMMlTTEE REPORT

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5.1-Aggregate size5.2-Forms

PART II-COMMENTARYChapter I-General, pg. 334.1R-4

Chapter 2-Analysis of shell, pg. 334.1R-42.1-Thin shells of single curvature2.2-Folded plates2.3-Thin shells of double curvature

Chapter 3-Analysis of supporting members,pg. 334.1R-6

Chapter 4-Prestressing, pg. 334.1R-6

Chapter 5-Stability, pg. 334.1R-6

Chapter 6-Proportioning, pg. 334.1R-7

Chapter 7-Construction, pg. 334.1R-87.1-Forms and decentering7.2-Concrete placing7.3-Curing of concrete

Chapter 8-Models, pg. 334.1R-8

Chapter 9-Standards and ACI documents cited in thisreport, pg. 334.1R-9

PART III-SELECTED BIBLIOGRAPHY

PREFACE

With the increased use of thin shells has come anincreased understanding of their behavior through fieldobservations, laboratory tests, and mathematical refine-ment of analytic procedures. However, because of thewide range of geometry possible with thin shells, the ac-cumulated understanding is still limited. For some thinshell systems, such as cylindrical barrel shells, the designcan be made with the same degree of accuracy as forconventional reinforced concrete construction. For otherthin shell systems, such as those of double curvature, thedesign must be at times based on less refined analyses inthe same sense as the empirical design of flat platefloors. Therefore, it was felt desirable to divide thisreport into two parts.

The first part, entitled “Criteria,” covers generaldesign recommendations. The second part, entitled“Commentary,” contains data of general interest to thedesigners of thin shells and reflects current practice. A“Selected Bibliography” is included as reference materialon current methods of analysis.

The analysis, design, and construction of thin shellstructures requires a thorough knowledge in this field.Therefore, the recommendations contained herein are

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not sufficient in themselves for the satisfactory executionof thin shell structures. Designers are referred to themany texts and technical papers that are readily available.

PART I-CRITERIACHAPTER l-GENERAL

1.1-Definitions1.1.1 Thin shells-Curved or folded slabs whose thick-

nesses are small compared to their other dimensions.They are characterized by their three dimensional loadcarrying behavior which is determined by their geomet-rical shape, their boundary conditions and the nature ofthe applied load. Thin shells are usually bounded by sup-porting members and edge members.

1.1.2 Auxiliary members-In a broad sense any mem-ber located along the boundary of a shell or shell seg-ment, with a capacity to stiffen the shell and distribute orcarry load in composite action with the shell. They areclassified as follows in accordance with established usage,although for certain shells a member may serve in a com-bination of capacities:

a) Supporting members-Beams, arches, trusses, dia-phragms, etc., along the edges of thin shells whichserve both to support and to stiffen the thin shell

b) Edge members-Beams, trusses, etc., along theedges of thin shells which do not form part of themain supporting structure, but serve to stiffen andact integrally, i.e., in composite action with thethin shell to carry loads to the supporting members

c) Stiffening members-Ribs which serve only to stiff-en the thin shell or to control local deformations

1.1.3 Elastic analysis-Any structural analysis based onelastic behavior and involving assumptions which are suit-able to approximations of three dimensional elastic be-havior. Analyses based on the results of elastic modeltests, when conducted properly, are considered as validelastic analyses.

1.2-Scope1.2.1 These recommendations cover the design of thin

shell concrete structures and only apply to the thin shellportions of such structures unless otherwise stated.

1.2.2 All applicable sections of the ACI Building Code(ACI 318), including the precast and prestressed concretesections of the Code, should be followed in the design ofshell structures except where they conflict with the fol-lowing provisions.

CHAPTER 2-ANALYSIS

2.l-Assumptions2.1.1 For an elastic analysis, concrete may be assumed

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CONCRETE SHELL STRUCTURES 334.1R-3

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uncracked, homogeneous, and isotropic.2.1.2 Poisson’s ratio may be assumed equal to zero.

2.2-Analysis of thin shells2.2.1 Elastic behavior is the commonly accepted basis

for determining stresses, displacements, and stability ofthin shells.

2.2.2 The rigor and the necessary degree of accuracyrequired in the analysis of any specific thin shell structuredepend on certain critical factors. These include: theconfiguration of the surface and the degree of curvature(geometry), the size of the structure, the type of boun-dary conditions and the nature of the loading. Because ofthe complex interrelationship between these factors, spe-cific recommendations regarding rigor and accuracy ofanalysis are not given.

2.2.3 Approximate methods of analysis which do notsatisfy compatibility of strains or stresses in the shell maybe used in cases where authoritative sources and experi-ence have shown them to be applicable within the rangeemployed.

2.2.4 Equilibrium checks of internal stresses andexternal loads are to be made to insure consistency ofresults.

2.2.5 An ultimate strength analysis may be used onlyas a check on the adequacy of the design. It is not to beused as a sole criterion for design, except where it can beproven to be applicable.

2.3-Analysis of supporting members2.3.1 Supporting members shall be designed in accor-

dance with a recognized elastic analysis.2.3.2 A portion of the thin shell shall be assumed to

act with the supporting member.

2.4-Model analysis2.4.1 Models may be used as the basis for a design

and/or to check the validity of assumptions involved in amathematical analysis. When models are used, only thoseportions which significantly affect the items under studyneed be simulated. Every attempt must be made to en-sure that these tests reveal the quantitative behavior ofthe prototype structure.

CHAPTER 3-STABILITY ANALYSIS

3.1-Buckling3.1.1 When investigating shells for stability, consid-

eration shall be given to the possible reduction in thevalue of the buckling load caused by large deflections,creep effects, and the deviation between the actual andtheoretical shell surface.

CHAPTER 4-PROPORTIONING

4.1-Allowable stresses and load factors
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4.1.1 Unless otherwise stated, concrete and steelstresses and load factors will be as specified in theBuilding Code (ACI 318).

4.1.2 Minimum standard cylinder strength,f,‘, shall be3000 psi.

4.2-Shell thickness4.2.1 Shell thickness is not always dictated by strength

requirements but often by deformation of edge members,stability, and cover over reinforcing steel.

4.2.2 Stress concentrations due to abrupt changes insection shall be considered and, where necessary, the thinshell shall be gradually thickened.

4.3-Shell reinforcement4.3.1 The stress in the reinforcement may be assumed

at the allowable value independently of the strain in theconcrete.

4.3.2 Where the tensile stresses vary greatly in mag-nitude over the shell, as in the case of cylindrical shells,the reinforcement capable of resisting the total tensionmay be concentrated in the region of maximum tensilestress. Where this is done, the percentage of crack con-trol reinforcing in any 12 in. width of shell shall be notless than 0.35 percent throughout the tensile zone.

4.3.3 The principal tensile stresses shall be resistedentirely by reinforcement.

4.3.4 Reinforcement to resist the principal tensilestresses, assumed to act at the middle surface of theshell, may be placed either in the general direction of thelines of principal tensile stress (also referred to as par-allel to the lines of principal tensile stress), or in two orthree directions. In the regions of high tension it is advis-able, based on experience, to place the reinforcing in thegeneral direction of the principal stress.

4.3.5 The reinforcement may be considered parallel tothe line of principal stress when its direction does notdeviate from the direction of the principal stress morethan 15 deg. Variations in the direction of the principalstress over the cross section of a shell due to momentsneed not be considered for the determination of the max-imum deviation. In areas where the stress in the rein-forcing is less than the allowable stress a deviationgreater than 15 deg can still be considered parallelplacing; a stress decrease of 5 percent shall be consideredto compensate for each additional degree of deviationabove 15 deg. Wherever possible, such reinforcing mayrun along lines considered most practical for construc-tion, such as straight lines.

4.3.6 Where placed in more than one direction, thereinforcement shall resist the components of the principaltension force in each direction.

4.3.7 In those areas where the computed principaltensile stress in the concrete exceeds 300 psi, placementof at least one layer of the reinforcing shall be parallel tothe principal tensile stress, unless it can be proven that adeviation of the reinforcing from the direction parallel tothe lines of principal tensile stress is permissible because

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of the geometrical characteristics of a particular shell andbecause for reasons of geometry only insignificant andlocal cracking could develop.

4.3.8 Where the computed principal tensile stress (psi)in the concrete exceeds the value 2fl (where& is alsoin psi), the spacing of reinforcement shall not be greaterthan three times the thickness of the thin shell. Other-wise the reinforcement shall be spaced at not more thanfive times the thickness of the thin shell, nor more than18 in.

4.3.9 Minimum reinforcement shall be provided asrequired in the Building Code (ACI 318) even where notrequired by analysis.

4.3.10 The percentage of reinforcement in any 12 in.width of shell shall not exceed 30 f,‘/f& However, themaximum percentage shall not exceed 6 percent if fs =20,000 psi, 5 percent if fs = 25,000 psi, or 4 percent if fs= 30,000 psi when the latter values are acceptable. If thedeviation of the reinforcing from the lines of principalstress is greater than 10 deg, the maximum percentageshall be one-half of the above values.

4.3.11 Splices in principal tensile reinforcement shallbe kept to a practical minimum. Where necessary theyshall be staggered with not more than one-third of thebars spliced at any one cross section. Bars shall be lappedonly within the same layer. The minimum lap for shellreinforcing bars, where draped, shall be 30 diameterswith a 1 ft 6 in. minimum unless more is required by theBuilding Code (ACI 318), except that the minimum maybe 12 in. for reinforcement not required by analysis. Theminimum lap for welded wire fabric shall be 8 in. or onemesh, whichever is greater, except that Building Code re-quirements shall govern where the wire fabric at thesplice must carry the full allowable stress.

4.3.12 The computed stress of the shell reinforcing atthe junction of shell and supporting member or edgemember shall be developed by anchorage within or be-yond the width of the member.

4.3.13 Reinforcement to resist bending moments shallbe proportioned and provided in the conventional man-ner with proper allowance for the direct forces.

4.4-Prestressing4.4.1 Where prestressing tendons are draped within a

thin shell, the resulting tendon profile not lying in oneplane will exert a force on the shell which can be re-solved into components. The design shall account forthese force components.

4.4.2 Where prestressing tendons are anchored, spe-cial reinforcing shall be added to assure that no localoverstressing occurs due to the application of these con-centrated reactions.

4.5-Concrete cover over reinforcement4.5.1 The concrete cover over reinforcement at sur-

faces protected from weather and not in contact with theground shall be at least ½ in. for bars (K in. when pre-

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cast), Y’S in. for welded wire fabric, and 1 in. forprestressed tendons. In no case shall the cover be lessthan the diameter of bar, prestressed tendon, or duct.

4.5.2 If greater concrete cover is required for fireprotection, such cover requirements shall apply only tothe principal tensile and moment reinforcement whoseyielding would cause failure.

CHAPTER 5-CONSTRUCTION

5.1-Aggregate size5.1.1 The maximum size aggregate shall not exceed

one-half the shell thickness, nor the clear distancebetween reinforcement bars, nor ½ times the cover.Where formwork is required for two faces, the maximumsize of aggregate shall not exceed one-quarter the min-imum clear distance between the forms nor the coverover the reinforcement.

5.2-Forms5.2.1 Removal of thin shell concrete forms shall be

considered a matter of design and the form removal se-quence shall be specified or approved by the engineer.

5.2.2 The minimum strength of concrete f,', based onfield-cured cylinders, at the time of decentering and ofreshoring, when required, shall be designated by theengineer.

5.2.3 Where, in the opinion of the designing engineer,stability of short or long-time deflections are importantfactors, the modulus of elasticity at the time of decen-tering, based on field-cured beams, shall be specified bythe engineer. The proportions and loading of these speci-mens shall insure action which is primarily flexural.

5.2.4 The batter on vertical elements, or other ele-ments, if desired or required for stripping, and the con-struction tolerances shall be designated by the engineer.

PART II-COMMENTARYCHAPTER l--GENERAL

This commentary contains data of general interest tothe designers of thin shells and reflects current practice.The following sections are not to be considered designcriteria for use in all cases, but discuss general methodsof analysis and limitations which have been found help-ful.

CHAPTER 2-ANALYSIS OF SHELL

Analytical procedures should be consistent with thecomplexity and importance of the structure. Judgmentand experience may be guides as reliable as analyticalprocedures based on simplifying assumptions. Factorssuch as size, curvature, boundary conditions, config-

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uration and distribution of load must be consideredcollectively by the designer in the choice of analyticalprocedures. For the purpose of discussion, a few com-ments will be made on each individually.

The most important factor to be considered is size.Shells generally possess a large reserve strength. Thecompressive stresses in a shell are usually a fraction ofthe allowable stresses. This is due to the fact that shellthickness is usually dictated by construction and code re-quirements. Moreover, for the major portion of the shellthe reinforcement is determined by requirements fortemperature and shrinkage. Therefore, the load carryingcapacity of a shell is generally greater than required. Asspans increase or curvatures decrease the compressionstresses increase, whereas the construction or coderequirements generally remain the same. Thus reservecapacity tends to decrease with larger sizes and flattershells. Therefore, approximations which are valid forrelatively small structures may not be at all valid whenthe same type of structure is increased in size.

Of almost equal importance to size is the manner inwhich the shell is supported. If proper support is pro-vided, capable of taking the shell reaction withoutappreciable deformations, the shell may respond to loadsin direct compressive or tensile stresses which for theusual spans will be less than the allowable stress. If thesupports are flexible, the stresses in the shell can behigher than those which would prevail in a properly sup-ported shell, and thus a more precise determination ofstresses is needed.

While size and support conditions have an importantbearing on the degree of accuracy needed in the analysis,the distribution of load has a less important effect onstresses. This is due to the fact that bending moments inthe shell are more closely related to the boundary con-ditions than to the load. Hence, it is usually unnecessaryto analyze a thin shell for partial live loads even thoughthe supporting members must be analyzed for such par-tial loads. For this reason, snow load on thin shells maybe assumed either uniformly distributed on the horizontalprojection or uniformly distributed over the surface ofthe shell. On the other hand, local bending moments dueto large concentrated loads on the shell must be con-sidered.

Finally, the degree of analytical accuracy required isalso dependent on the geometry of the shell, especiallythe type and amount of curvature. With the variety ofpossible shapes, generalized recommendations of the ef-fect of shape cannot be made. A few pertinent remarkson some frequently used shapes follow.

2.1-Thin shells of single curvatureExamples of shells of single curvature are segments of

cylinders and cones. These shell systems tend to act likebeams in the longitudinal direction where the span islong in proportion to the radius. Their behavior departsfrom that of beams in proportions as the valley deflectsrelative to the crown. The smaller such relative deflection

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is, the more nearly the direct longitudinal stresses will bedistributed linearly as in a beam section. The range overwhich the beam analysis is applicable has been dis-cussed.5,9

For simply supported cylindrical shells with para-meters in a range where beam analysis does not giveresults of sufficient accuracy, a number of design tableshave been presented.7,8,10,11,13 The use of these tables isrecommended.

No analysis comparable to that for simply supportedcylindrical shells is available for the case of shellscontinuous over supports. For the latter cases, the effectsof continuity may be estimated by first calculating thelongitudinal stresses and transverse shearing stresses asfor simply supported spans and then multiplying these bythe ratio of the moment and shear coefficients for con-tinuous to simple spans. It is pointed out however thatthe shear deformation in certain shells may have a stronginfluence on the distribution of longitudinal stresses.*

For singly curved shells moderately tapered in plan,i.e., those in which the chords at each end are different,experience indicates that an analysis based on a cylin-drical shell with a chord width equal to the average issatisfactory. However, shear conditions at the smaller endshould be investigated on the basis of the actual crosssection. For shells tapered markedly, a more precisesolution is available.34

2.2-Folded platesAnalysis of prismatic folded plates is based on the

conventional flexural theory for longitudinal action andon a one way continuous slab action for transverse be-havior. A number of satisfactory analytic procedures, allbased on the same fundamental approach, are avail-able.16 Attention is called to the fact that the iterationprocedures that are available may not always converge.

Folded plates are more sensitive to longitudinal sup-port conditions than curved thin shells. The transversebending moments in the outer bays of multiple foldedplate structures are apt to be much larger than those inbarrel shells of the same size. For this reason, foldedplates are often subjected to critical stresses duringconstruction when certain plates may temporarily beshored or act as exterior plates.

Although a folded plate analysis is recommended todetermine the effects produced in the two or three outerbays of a multiple folded plate structure, the behavior ofthe composite interior plates is similar to that of a beam.Hence a rigorous folded plate analysis need be appliedto only at most the exterior and first interior bays of amultiple folded plate system except when the depth isless than l/15 of the span.15

2.3-Thin shells of double curvatureShells of double curvature both the synclastic (sur-

faces in which the curvatures in the two principal direc-

• Reference 1 pp. 138-140


334.1R-6 ACI COMMITTEE REPORT

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tions have the same sign) and anticlastic (surfaces inwhich curvatures in the two principal directions have op-posite signs) are inherently better suited to resist loads bydirect forces than are shells of single curvature. Thereason for this is obvious from the fact that this type ofshell possesses arch action along both curvatures. But inorder that surfaces curved in two directions behave as ashell, it is important that proper support or edge mem-bers be provided.

For most shells of double curvature distributed loadson the shell can be transmitted to the supports by tan-gential shearing and normal forces acting along theperiphery of the shell irrespective of the load distri-bution. When this can be done, the internal forces actingin the shell are all direct forces, generally of smallmagnitude except in the region of the column supportswhere often critical tensile forces and moments aredeveloped.

The direct stresses throughout the major portion ofthe shell are usually of little significance except as theyrelate to buckling. A careful evaluation should be madeof the bending moments produced in the vicinity of theedge members by the interaction of the edge memberand the shell. For moderate size shells, this effect usuallyis confined within a few feet of the vicinity of the edgemember. For such determination the formulas for cyl-indrical shells4 have been used in certain cases.

An exception to this are some anticlastic shells, likethe hyperbolic paraboloid, wherein bending can prevailthroughout a greater portion of the shell. To a limitedextent, this also occurs in domes, when the supports donot provide a reaction tangent to the shell surface. Inthese cases, the bending moments may extend a signi-ficant distance into the shell.

Because of the popularity of the hyperbolic para-boloid stiffened only in the conventional way by unsup-ported edge beams, it becomes important to recognizethat its ability to carry loads by direct force is restrictedto uniform load. For other loadings, as for example thedead load produced by sharply tilted saddle shape shells,the shell is subject to bending moments. The momentsare not significant for conventional size shells but theymay become very significant for large spans.

CHAPTER 3-ANALYSIS OFSUPPORTING MEMBERS

The analysis of the supporting members may be basedon the consideration of either the total shell load or onthe shearing forces acting between shell and supportingmember. When the total load is used the axial thrust andmoment in the supporting member shall be adjusted bysubtracting (algebraically) the direct thrust and momentin the thin shell. If total shell loads are used, themoments due to the eccentricity of the shell with respectto the centroidal axis of the supporting member shall betaken into account.13

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Supporting members shall be designed for partial liveloads when such loadings control the design.

CHAPTER 4-PRESTRESSING

Prestressing can often be employed advantageously onshells covering large spans to reduce the bending mo-ments in the shell under service loads and to controldeflection and cracking. As with conventional prestressedstructures, the analysis must be made at design and atultimate loads to ensure both proper behavior at designloads and an adequate overload capacity. With respect tooverload capacity, it is important to determine the shellbending moments at ultimate loads since they may beseveral times the bending moments produced by the com-bination of design load and prestressing. In all cases, theshell must be designed for the shell bending momentproduced by the design loads times the appropriate over-load factor minus (algebraically) the bending momentproduced by the prestressing force.

CHAPTER 5-STABILITY

In common with other structures, shells should be in-vestigated for buckling. In some simple two-dimensionalstructures it is sufficient to determine only the lowestvalue of the load at which buckling commences. In thecase of shells, however, it may be also necessary toinvestigate the postbuckling behavior since it has animportant bearing on the magnitude of the failure load.

As a thin shell deforms under load, principal mem-brane forces develop. If one of these forces is tensile, ittends to return the shell back to its original position, thusenabling it to carry loads greater than the initial bucklingload. If, on the other hand, both the principal membraneforces are compressive, they tend to increase with defor-mation of the shell. After the initial buckling, the shellcan only transmit loads smaller than the initial bucklingload. This is particularly true for concrete shells becauseof creep and deviation of the actual shape of the shellfrom the assumed theoretical surface.

The importance of the postbuckling behavior, whichsmall deflection theory of buckling is not capable of pre-dicting, was discovered as a consequence of attempts tocorrelate experimental results with analytical predictions.

Some surfaces subjected to various load combinationswhich have been studied analytically and experimen-tally22,25,27 with respect to their buckling characteristicsare as follows:

1. Circular cylinders (ribbed and unribbed)2. Segments of circular cylinders3. Elliptical cylinders4. Cones5. Spherical domes6. Translational surfaces



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Good correlation exists when one of the principalmembrane forces is tensile; such cases include:

1. Cylindrical shells under torsional loading2. Cylindrical shells under radial inward pressure

As stated, poor correlation between the results oftheory and experiment exists when both principal mem-brane forces are compressive, as in the case of:

1. Cylindrical shells under axial compressive load2. Cylindrical shells under distributed load normal to

the surface which causes bending3. Domes under inward radial pressure

In extreme cases the buckling load obtained experi-mentally has been found to be as little as 10 percent ofthat predicted by the small deflection theory.

The effect of creep of concrete is similar to loweringE, the modulus of elasticity. The effect may be estimatedas follows:

Assume a reduced value of E or, knowing the princi-pal membrane forces at any point of a given shell, de-termine the tangent modulus of elasticity. Divide thetangent modulus of elasticity by a multiplier for long-term deflections. The multiplier shall not be less than 2.The use of a reduced E accomplishes the same purposeas a buckling load reduction factor.

A deviation of the shell dimensions from the geo-metry used in stress analysis is associated with a changein the principal radii of curvature of the shell surface.Elastic and creep deformations may also lead to a changeof curvature. If the actual radii are larger, (i.e., the shellis flatter), the membrane forces are generally greater.Hence the value of the buckling load will be lower, pos-sibly substantially lower.

In general, the value of the buckling load depends onshell geometry, type of restraint at boundary, materialproperties of shell, the location of reinforcing steel, andthe type of load. With respect to shell geometry, thedominant factor is whether the surface is synclastic oranticlastic. For

a) Synclastic surfaces, such as most shells of revo-lution, cylinders and cones, the buckling resistanceusually lies somewhere between a sphere and a cyl-inder, the two shell forms which have been exten-sively studied. Hence an estimate of the criticalstress may be sometimes obtained by comparingthe given shell to a cylinder or a sphere

b) Anticlastic surfaces, such as hyperbolic para-boloids, the buckling resistance is much greaterthan that offered by synclastic surfaces becausetheir shape results in principal tensile forces in onedirection. Hence it is often possible to use thelinear buckling theory

Resistance to buckling is increased when reinforcing
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steel is provided on both faces of the shell.Some methods which may be used to estimate failure

loads are as follows:

1. Energy methods, based on minimization of theenergy of the system. This approach permits com-putation of the lower as well as the upper boun-dary of the buckling load, and is applicable to alltypes of shells

2. Equilibrium and compatibility methods. Thisapproach is based on the classical equations ofequilibrium and compatibility

3. Experimental methods yield very good estimates ofbuckling load for any type of shell. However, itshould be noted that most experiments have beenmade with materials other than concrete.

4. Recommended procedures such as those advocatedfor cylindrical shells and domes with use ofbuckling graphs.1,4,26 Where the influences ofcreep, deformations, and decentering conditionsare not expected to be unusual and are not sepa-rately evaluated, it is recommended to calculatebuckling stability in two directions separately (withthe use of graphs) by computing the buckling loador critical stress based on the unreduced value ofthe modulus of elasticity expected at the time ofdecentering. It is further recommended to calcu-late the apparent safety factors (F1, F2) for the twodirections by dividing buckling load (or criticalstress) by working load (or working stress) and tocompute the combined apparent safety factor forbuckling as:

Fl F2F=-Fl + F2

This apparent safety factor must be greater than 5 forthe above described conditions.

CHAPTER 6-PROPORTIONING

It is often desirable to increase gradually the thicknessof a thin shell where it is connected to supporting mem-bers or edge members. In barrel shells and domes, thisgradual thickening usually commences at a distance fromthe edge of about ten times the minimum thickness un-less otherwise dictated by stress analysis. The increasedthickness is usually twice the minimum thickness requiredat the center of the span unless otherwise determined byanalysis.

The use of high-strength steel reinforcement withworking stresses above 20,000 psi is acceptable in thinshell structural systems. However, in the shell portion itmay be used only when the main reinforcement followsthe direction of principal stress. The working stressshould not exceed 50 percent of the yield point or thepermissible working stress. Where such high strengthsteel is used, careful attention must be paid to crack

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control and to deformations.At the edges of thin shells, it is usually desirable to

provide top and bottom reinforcement. The torsional re-sistance of the supporting members or edge membersoften governs the top and bottom reinforcement whenthe shell terminates with these members.

As the principal tensile stress in the shell increases, itis desirable to provide an increasing amount of steel inthe direction of the principal stress or in two or threedirections within the shell.

CHAPTER 7-CONSTRUCTION

7.1-Forms and decenteringFor thin shell structures of large spans or unusual

proportions, the engineer should prepare a decenteringdrawing before the method of shoring is determined bythe contractor. In the preparation of the decenteringdrawing, the decentering procedure should be studied toavoid any temporary support including a concentratedreaction on the shell. Generally, decentering should beginat points of maximum deflections and should progresstowards points of minimum deflection, with the decen-tering of edge members proceeding simultaneously withthe adjoining shell. Where edge members are to be re-shored, the spacing of the shoring should be specified bythe engineer.

The supporting members should be decentered afterthe forms under the thin shell have been decentered.When this is done, the stresses created by the decen-tering procedure should be considered.

Where the time of decentering is based on the attain-ment of a prescribed modulus of elasticity of field-curedbeams, it has been found satisfactory to use lightlyreinforced beams, 4 x 6 in. x 6 ft long, for the fieldcontrol beams.28

It is desirable to have the engineer specify the maxi-mum deviations permissible from the true surface of ashell, such as, for example ½ in. deviation for an arc ofx ft in a shell of a radius of y ft.

When movable forms are used, a batter of % in. perft minimum is recommended for vertical surfaces topermit easy removal.

7.2-Concrete placingConcrete should be placed in a symmetrical pattern,

otherwise the shoring must be braced for the effect ofunbalanced loads on the forms. It is recommended thatconcreting commence at the low point or points of theshell and proceed upward to the high point. The concreteshould be deposited as nearly as possible in its finalposition.

When deep beams or diaphragms are employed assupporting members, and when these members are to becast integrally with the thin shell, care should be taken touse a rate of concreting which will permit consolidationof concrete in the deep beams or diaphragms and pre-

ight American Concrete Institute
ed by IHS under license with ACI L
Nroduction or networking permitted without license from IHS

vent cold joints in the shell.Construction joints should be shown in detail on

design drawings and should be located preferably in areasof compressive stresses.

If the slope of the shell is steeper than about 45 deg,it is desirable to use formwork on both faces except whenthe thin shell is shotcreted or plastered.

Shells which cannot be cast in 1 day should be sec-tioned for two or more placements with 2 to 3 daysbetween each cast in order to reduce shrinkage stresses.

7.3-Curing of concreteThin shells are susceptible to shrinkage cracking if

improperly cured. In hot weather, the use of retarders,preliminary fog spray curing, and wet burlap or watercuring is advisable. In cold weather, use of acceleratorsand special precautions against freezing are usuallyrequired. In moderate weather (40 to 70 F), ordinarymethods such as membrane curing compounds are usu-ally satisfactory although wet curing may produce betterresults.

CHAPTER 8-MODELS

It has been stated in the Criteria that model analysisis acceptable as a basis for design or to check the validityof the assumptions involved in a mathematical analysis.Models should be looked upon as auxiliary tools to aid arational analysis rather than a substitute for a theoreticalanalysis. It must be emphasized that although the modelanalysis technique is valuable for the design or analysisproblems of complex shells, extreme care must be usedin the planning, execution, and interpretation of themodel experiments. Requirements of similitude should befollowed, unless it can be demonstrated that the givendepartures from similitude cause insignificant changes inresults, or changes that can be accurately predicted andaccounted for in the interpretation. The model exper-iment should be so arranged that as many checks onresults can be made as is reasonably possible. Forexample:

1. Can equilibrium of forces in the model bechecked?

2. For symmetrical loads on a symmetrical structureare symmetric readings equal?

3. Are environmental conditions affecting the modelresults?

4. Are the loads of sufficient magnitude to inducepossible secondary effects due to deflections?

Many possible errors may enter into a model analysisgrossly affecting the stress readings. As a matter of goodpractice, after strain measurements are completed it isrecommended that the model be subjected to overloadswhose effects approach the anticipated capacity of themodels or several times the service load. Such high

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loading may reveal weaknesses not apparent from a limited number of strain gages.

CHAPTER 9-STANDARDS AND ACIDOCUMENTS CITED IN THIS REPORT

The document of the standards-producing organi-zation referred to in this document is listed below withits serial designation.

ACI 318 Building Code Requirements for ReinforcedConcrete

The above publication may be obtained from the fol-lowing organization:

American Concrete InstituteP.O. Box 19150Detroit, Michigan 48219

PART III-SELECTED BIBLIOGRAPHY

General considerations for analysisla. Flügge, Wilhelm, Stresses in Shells Springer-

Verlag, Berlin, 1962, 499 pp. (A practical textbookdevoted exclusively to shells, with numerous examples.)

lb. Flügge, Wilhelm, Statik und Dynamik der Schalen,Springer-Verlag, Berlin, 1962. (A practical textbookdevoted exclusively to shells.)

2. Girkmann, K., Flächentragwerke, Springer-Verlag,Vienna, 5th edition, 1959. (A textbook on practicalmethods of analysis of plate and shell structures.)

3. Pflüger, Alf, Elementary Statics of Shells F.W.Dodge Corp., New York, 2nd edition, 1961, 122 pp.

4. Timoshenko, S.P., and Woinowsky-Krieger, S.,Theory of Plates and Shells, McGraw-Hill Book Co., Inc.,New York, 2nd edition, 1959. (One of the classics in theapplication of theory of elasticity.)

Cylindrical shell roofs5. Chinn, James, “Cylindrical Shell Analysis Simplified

by Beam Method,” ACI JOURNAL, Proceedings V. 55, No.10, May 1959, pp. 1183-1192. Discussion by Milo S.Ketchum, A.L. Parme and H.W. Conner, A. Siev, AntonTedesko, and Alfred Zweig, ACI JOURNAL, ProceedingsV. 55, Dec. 1959, Part 2, pp. 1583-1604.

6. Dabrowski, Ryszard, “Analysis of PrestressedCylindrical Shell Roofs,” Proceedings, ASCE, V. 89, No.ST5, Oct. 1963, pp. 91-116.

7. Gibson, J.E., Computer Analyses of CylindricalShells, Span Limited, London, 1961. (A series of tablesgiving stress resultants and moments in circular cylin-drical shells.)

8. Holand, Ivar, “Design of Circular CylindricalShells,” Series 2, No. 3, Norges Tekniske Viten-

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t American Concrete Institute by IHS under license with ACI Licensee

Not for Rduction or networking permitted without license from IHS

skapsakademi, 1957. (A comparison of the equationsused for cylindrical shell analysis.)

9. Lundgren, H., Cylindrical Shells, V. 1-CylindricalRoofs, The Institution of Danish Civil Engineers, TheDanish Technical Press, Copenhagen, 1951. (Applicationof the theory of elasticity and of rupture to the design ofcylindrical roofs, and other thin walled cylindricalstructures.)

10. Parme, A.L., and Conner, H.W., “Design Con-stants for Interior Cylindrical Concrete Shells,” ACIJOURNAL, Proceedings V. 58, No. 1, July 1961, pp. 83-106.(Compact design tables, limited in scope.)

11. Ruediger, D., and Urban, J., Circular CylindricalShells, B.G. Teubner Verlagsgesellschaft, Leipzig, 1955.(Derivation of the shallow shell theory for cylindricalshells with tables for a wide range of shell dimensions.)

12. Thürlimann, B., and Johnston, B.G., “Analysis andTests of a Cylindrical Shell Roof Model,” Proceedings,ASCE, V. 80, Separate No. 434, 1954.

13. “Design of Cylindrical Concrete Shell Roofs,”Manual No. 31, American Society of Civil Engineers,1952. (Comprehensive series of design tables with math-ematical derivations as background.)

Folded plates14. Yitzhaki, D., The Design of Prismatic and Cylin-

drical Shelf Roofs, Haifa Science Publishers, Haifa, Israel,1958, 253 pp. (Thorough treatment of roofs of simpleand continuous spans.)

15. “Direct Solution of Folded Plate Concrete Roofs,”Advanced Engineering Bulletin No. 3, Portland CementAssociation, Chicago, 1960. (Concise design tables.)

16. “Phase I-Report on Folded Plate Construction,”Proceedings, ASCE, V. 89, ST6, Dec. 1963, pp. 365-406.(A complete bibliography on folded plates is includedwith this report.)

Thin shells of double curvature17. Apeland, K., “Stress Analysis of Translational

Shells,” Proceedings, ASCE, V. 87, EMl, Feb. 1961, pp.111-139.

18. Apeland, K., and Popov, E. P., “Design Tables forTranslational Shells,” Reports 63-1, Structures andMaterials Research, University of California, Berkeley,1963.

19. Bouma, A.L., “Some Applications of the BendingTheory Regarding Doubly Curved Shells,” Proceedings,Symposium on the Theory of Thin Elastic Shells, Delft,1959. (A discussion showing the effects of the type ofshell curvature on the shell behavior, based on shallowshell theory.) (See reference 32.)

20. Candela, Felix, “General Formulas for MembraneStresses in Hyperbolic Paraboloidal Shells,” ACIJOURNAL, Proceedings V. 57, No. 4, Oct. 1950, pp.353-372.

21. Murray, N.W., “A General Design Method forAxisymmetric Shell Roofs,” Proceedings, Institution ofCivil Engineers (London), V. 20, Sept. 1961, pp. 151-162.

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(A general numerical solution for shells of revolutionwith varying thicknesses.)

Stability22. ACI-ASCE Committee 334, Concrete Shell Buck-

ling, E.P. Popov and S.J. Medwadowski, editors, SP-67,American Concrete Institute, Detroit, 1981, 240 pp.

23. Gerard, G., and Becker, H., Handbook of Struc-rural Stability: Part III, NACA N 3783 (1957); and PartVI, NACA N 3786 (1958); National Aeronautics andSpace Administration, Washington, D.C.

24. Nash, W.A., “Recent Advances in Buckling ofThin Shells,” Applied Mechanics Reviews, V. 13, Mar.1960, pp. 161-164.

25. Schmidt, H., “Ergebnisse von Beulversuchen mitdoppelt gekrummten Schalenmodellen aus Aluminum,”Proceedings, Symposium on Shell Research, Interscience,John Wiley & Sons, Inc., New York, and North-HollandPublishing Co., Amsterdam, 1962, pp. 159-181. (A sum-mary of past work on thin shell buckling along withrecommendations for reinforced concrete design.)

26. Timoshenko, S.P., and Gere, J.M., Theory ofElastic Stability, McGraw-Hill Book Co., Inc., New York,2nd edition, 1961.

27. Collected Papers on Instability of Shell Structures,TN-D-1510, National Aeronautics and Space Administra-tion, Washington, D.C., Dec. 1962.

Construction28. Tedesko, Anton, “Construction Aspects of Thin-

Shell Structures,” ACI JOURNAL, Proceedings V. 49, No.

American Concrete Institute y IHS under license with ACI L

Nuction or networking permitted without license from IHS

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6, Feb. 1953, pp. 505-520.

Symposia29. Proceedings of the Symposium on Concrete Shell

Roof Construction, Cement and Concrete Association,London, July 1954.

30. Proceedings of the Conference on Thin ConcreteShells, Massachusetts Institute of Technology, Cambridge,1954, 133 pp.

31. Proceedings of the Second Symposium on ConcreteShelf Roof Construction, Teknisk Ukeblad, Oslo, Norway,1958.

32. Proceedings of the Symposium on the Theory ofThin Elastic Shells (Delft, 1959), Interscience, John Wiley& Sons, Inc., New York, and North-Holland PublishingCo., Amsterdam, 1960, 496 pp.

33. Proceedings of the Symposium on Shell Research,(Delft, 1961), Interscience, John Wiley & Sons, Inc., NewYork, and North-Holland Publishing Co., Amsterdam,1962, 364 pp.

34. Proceedings, Colloquium on Simplified CalculationMethods of Shell Structures (Brussels, 1961), Interscience,John Wiley & Sons, Inc., New York, and North-HollandPublishing Co., Amsterdam, 1962, 529 pp.

35. Proceedings of the World Conference on Shell Struc-tures (San Francisco, 1962), Publication No. 1187,National Academy of Sciences-National Science Foun-dation, Washington, D.C., 1964, 690 pp.

report was submitted to letter ballot of the committee and was approvedaccording to Institute procedures.

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