Consistent Design Of Structural Concrete - PCI
Special ReportToward aConsistent Design ofStructural ConcreteJorg Schlaich,Dr.-Ing.Kurt Schafer,Dr.-Irtg.Mattias Jennewein,Dipl.-Ing.Professor at the Instituteof Reinforced ConcreteUniversity of StuttgartWest GermanyProfessor at the Instituteof Reinforced ConcreteUniversity of StuttgartWest GermanyResearch AssociateUniversity of StuttgartWest GermanyThis report (which is being considered by Comite Euro-International du Bt tonin connection with the revision of the Model Code) represents the latest andmost authoritative information in formulating a consistent design approach forreinforced and prestressed concrete structures.74
CONTENTSSynopsis. 771. Introduction — The Strut-and-Tie-Model . 762. The Structure's B- and D-Regions . 773. General Design Procedure and Modelling . 843.1 Scope3.2 Comments on the Overall Analysis3.3 Modelling of Individual B- and D-Regions4. Dimensioning the Struts, Ties and Nodes . 974.1 Definitions and General Rule4.2 Singular Nodes4.3 Smeared Nodes4.4 Concrete Compression Struts — Stress Fields C.4.5 Concrete Tensile Ties --- Stress Fields T,4.6 Reinforced Ties T,4.7 Serviceability: Cracks and Deformations4.8 Concluding Remarks5. Examples of Applications . 1105.1 The B-Regions5.2 Some D-Regions5.3 Prestressed � Notation .150PCI JOURNAL May-June 198775
1. INTRODUCTION -THE STRUT-AND-TIE-MODELThe truss model is today consideredby researchers and practitioners to bethe rational and appropriate basis for thedesign of cracked reinforced concretebeams loaded in bending, shear and torsion. However, a design based on thestandard truss model can cover onlycertain parts of a structure.At statical or geometrical discontinuities such as point loads or frame corners, corbels, recesses, holes and otheropenings, the theory is not applicable.Therefore, in practice, procedureswhich are based on test results, rules ofthumb and past experience are usuallyapplied to cover such cases.Since all parts of a structure includingthose mentioned above are of similarimportance, an acceptable design concept must be valid and consistent forevery part of any structure. Furthermore, since the function of the experiment in design should be restricted toverify or dispute a theory but not to derive it, such a concept must be based onphysical models which can be easilyunderstood and therefore are unlikely tobe misinterpreted.For the design of structural concrete*it is, therefore, proposed to generalizethe truss analogy in order to apply it inthe form of strut-and-tie-models to everypart of any structure.This proposal is justified by the factthat reinforced concrete structures carryloads through a set of compressive stressfields which are distributed and interconnected by tensile ties. The ties maybe reinforcing bars, prestressing tendons, or concrete tensile stress fields.For analytical purposes, the strut-and-'Following a proposal by Dr. J. E. Breen andDr. A. S. C. Bruggeling, the term "structuralconcrete" covers all loadbearing concrete, includingreinforced, prestressed and also plain (unreinforced concrete, if the latter is part of a reinforcedconcrete structure.76tie-models condense all stresses in compression and tension members and jointhem by nodes.This paper describes how strut-andtie-models can be developed by following the path of the forces throughouta structure. A consistent design approach for a structure is attained whenits tension and compression members(including their nodes) are designedwith regard to safety and serviceabilityusing uniform design criteria.The concept also incorporates themajor elements of what is today called"detailing," and replaces empirical procedures, rules of thumb and guess workby a rational design method. Strut-andtie-models could lead to a clearer understanding of the behavior of structuralconcrete, and codes based on such anapproach would lead to improvedstructures,The authors are aware of the encouraging fact that, although they published papers on this topic earlier,1.2.3they are neither the first nor the onlyones thinking and working along theselines. It was actually at the turn of the lastcentury, when Ritter*' and Mcirsch s introduced the truss analogy. This methodwas later refined and expanded byLeonhardt, Rusch, 7 Kupfer, 8 and othersuntil Thurlimann's Zurich school, a withMarti lu and Mueller," created its scientific basis for a rational application intracing the concept back to the theory ofplasticity.Collins and Mitchell further considered the deformations of the truss modeland derived a rational design method forshear and torsion."In various applications, Bay, Franz,Leonhardt and Thurlimann had shownthat strut-and-tie-models could be usefully applied to deep beams and corbels.From that point, the present authorsbegan their efforts to systematically expand such models to entire structures
SynopsisCertain parts of structures are designed with almost exaggerated accuracy while other parts are designedusing rules of thumb or judgmentbased on past experience. However, all parts of a structure are ofsimilar importance.A unified design concept, which isconsistent for all types of structuresand all their parts, is required. To besatisfactory, this concept must bebased on realistic physical models.Strut-and-tie-mode Is, a generalizationof the well known truss analogymethod for beams, are proposed asthe appropriate approach for design-and all structures.The approaches of the various authorscited above differ in the treatment of theprediction of ultimate load and thesatisfaction of serviceability requirements. From a practical viewpoint, truesimplicity can only be achieved if solutions are accepted with sufficient(hat not perfect) accuracy. Therefore, itis proposed here to treat in general theultimate limit state and serviceability inthe cracked state by using one and thesame model for both. As will be shownlater, this is done by orienting theing structural concrete, which includesboth reinforced and prestressed concrete structures.This report shows how suitablemodels are developed and proposescriteria according to which the model'selements can be dimensioned uniformly for all possible cases. The concept is explained using numerous design examples, many of which treatthe effect of prestress.This report was initially prepared fordiscussion within CEB (ComitdEuro-International du Beton) in connection with the revision of the ModelCode,geometry of the strut-and-tie-model atthe elastic stress fields and designingthe model structure following the theoryof plasticity.The proposed procedure also permitsthe demonstration that reinforced andprestressed concrete follow the sameprinciples although their behaviorunder working loads is quite distinct.It should be mentioned that only theessential steps of the proposed methodare given here. Further support of thetheory and other information may befound in Ref. 3.2. THE STRUCTURE'S B- AND D-REGIONSThose regions of a structure, in whichthe Bernoulli hypothesis of plane straindistribution is assumed valid, are usually designed with almost exaggeratedcare and accuracy. These regions arereferred to as B-regions (where B standsfor beam or Bernoulli). Their internalstate of stress is easily derived from thesectional forces (bending and torsionalmoments, shear and axial forces).PCI JOURNAL'May-June 1987As long as the section is uncracked,these stresses are calculated with thehelp of section properties like crosssectional areas and moments of inertia.If the tensile stresses exceed the tensilestrength of the concrete, the truss modelor its variations apply.The B-regions are designed on thebasis of truss models as discussed lateron in Section 5.1.77
as Shz'h1-1 —h2hh-4.—h-4tnt,- b)IIFH2. h—LhIFig. 1 D-regions (shaded areas) with nonlinear strain distribution due to (a) geometricaldiscontinuities; (b) statical and/or geometrical discontinuities.The above standard methods are notapplicable to all the other regions anddetails of a structure where the straindistribution is significantly nonlinear,e.g., near concentrated loads, corners,bends, openings and other discontinuities (see Fig. 1). Such regions arecalled D-regions (where D stands for78discontinuity, disturbance or detaiI).As long as these regions are uncracked, they can be readily analyzed bythe linear elastic stress method, i.e., applying Hooke's Law. However, if thesections are cracked, accepted designapproaches exist for only a few casessuch as beam supports, frame corners,
Fig. 2. Stress trajectories in a B-region and near discontinuities(D-regions).corbels and splitting tension at prestressed concrete anchorages. And eventhese approaches usually only lead tothe design of the required amount ofreinforcement; they do not involvea clear check of the concretestresses.The inadequate (and inconsistent)treatment of D-regions using so-called"detailing," "past experience" or "goodpractice" has been one of the main reasons for the poor performance and evenfailures of structures. It is apparent,then, that a consistent designphilosophy must comprise both B- andD-regions without contradiction.Considering the fact that several decades after MOrsch, the B-region design is still being disputed, it is only reasonable to expect that the more complexD-region design will need to be simplified with some loss of accuracy.However, even a simplified methodicalconcept of D-region design will be preferable to today's practice. The preferredconcept is to use the strut-and-tie-modelapproach. This method includes theB-regions with the truss model as a special case ofa strut-and-tie model.In using the strut-and-tie-model approach, it is helpful and informative tofirst subdivide the structure into its Band D-regions. The truss model and thePCI JOURNALMay-June 1987design procedure for the B-regions arethen readily available and only thestrut-and-tie-models for the D-regionsremain to be developed and added.Stresses and stress trajectories arequite smooth in B-regions as comparedto their turbulent pattern near discontinuities (see Fig. 2). Stress intensitiesdecrease rapidly with the distance fromthe origin of the stress concentration.This behavior allows the identificationof B- and D-regions in a structure.In order to find roughly the divisionlines between B- and D-regions, thefollowing procedure is proposed, whichis graphically explained by four examples as shown in Fig. 3:1. Replace the real structure (a) by thefictitious structure (b) which is loaded insuch a way that it complies with theBernoulli hypothesis and satisfiesequilibrium with the sectional forces.Thus, (b) consists entirely of one or several B-regions. It usually violates theactual boundary conditions.2. Select a self-equilibrating state ofstress (c) which, if superimposed on (b),satisfies the real boundary conditions of(a).3. Apply the principle of Saint-Wnant(Fig. 4) to (c) and find that the stressesare negligible at a distance a from theequilibrating forces, which is approxi79
{ a1(d)(c)lb)hFr B/hF h1 dht Bd hFig. 3.1. Column with point loads.{c1hlMI l t iM3 h2V VMM1M3Vcb1 i U.(c)-dr r –d 2 h2{ d1iBFig. 3.3. Beam with a recess.80/B
a]IL l I 4 3 TTTTTr I-mhtt (C) d h ,rth Itf1:!II!1IIIIle!!!:Fig. 3.2. Beam with direct supports.(a] .D I d bfFig. 3.4. T-beam.(a) real structure(b) loads and reactions applied inaccordance with Bernoulli hypothesis(c) self-equilibrating state of stress(d) real structure with B- andD-regionsFig. 3. Subdivision of four structures into their B- and D-regions, using SaintVenant's principle (Fig. 4).PCI JOURNAL/May-June 198781
Q)F a6 0\i06b) d hFy42y! 2? iTIhHIdd h' Y"1.0xh10xh1,0hdydy1,0hFig. 4. The principle of Saint-Venant: (a) zone of a bodyaffected by self-equilibrating forces at the surface; (b)application to a prismatic bar (beam) loaded at one face.mately equal to the maximum distancebetween the equilibrating forces themselves. This distance defines the rangeof the D-regions (d).It should be mentioned that crackedconcrete members have different stiffnesses in different directions. This situ82ation may influence the extent of theD-regions but needs no further discussion since the principle of Saint-V nantitself is not precise and the dividinglines between the B- and D-regionsproposed here only serve as a qualitative aid in developing the strut-and-
phB B Bt/a2h}c) aB ' BBB0 hf 4hB01"Okol UIF hI h BBI Zh 4h1wrf frFig. 5. The identification of their B- and D-regions (according to Fig. 3) isa rational method to classify structures or parts thereof with respect totheir Ioadbearing behavior: (a) deep beam; (b) through (d) rectangularbeams; (e) T-beam,tie-models.The subdivision of a structure into Band D-regions is, however, already ofconsiderable value for the understanding of the internal forces in thestructure. It also demonstrates, that simple 1fh rules used today to classifyPCI JOURNALMay-June 1987beams, deep beams, short/long/highcorbels and other special cases are misleading. For proper classification, bothgeometry and loads must be considered(see Figs. 3, 5 and 6).If a structure is not plane or of constant width, it is for simplicity sub83
divided into its individual planes, whichare treated separately. Similarly, threedimensional stress patterns in plane orrectangular elements may be looked atin different orthogonal planes. Therefore, in general, only two-dimensionalmodels need to he considered. However, the interaction of models indifferent planes must be taken into account by appropriate boundary condi-tions.Slabs may also be divided into B-regions, where the internal forces are easily derived from the sectional forces,and D-regions which need further explanation, If the state of stress is notpredominantly plane, as for example inthe case with punching or concentratedloads, three-dimensional strut-and-tiemodels should be developed.3. GENERAL DESIGN PROCEDURE ANDMODELLING3.1 ScopeFor the majority of structures it wouldbe unreasonable and too cumbersome tobegin immediately to model the entirestructure with struts and ties. Rather, itis more convenient (and common practice) to first carry out a general structuralanalysis. However, prior to starting thisanalysis, it is advantageous to subdividethe given structure into its B- and D-regions. The overall analysis will, then,include not only the B-regions but alsothe D-regions.If a structure contains to a substantialpart B-regions, it is represented by itsstatical system (see Fig. 6). The generalanalysis of linear structures (e.g., beams,frames and arches) results in the supportreactions and sectional effects, thebending moments (M), normal forces(N), shear forces (V) and torsional moments(M r ) (see Table 1).The B-regions of these structures canthen be easily dimensioned by applyingstandard B-region models (e.g., the trussmodel, Fig. 8) or standard methodsusing handbooks or advanced codes ofpractice. Note that the overall structuralFig. 6. A frame structure containing a substantial part of B-regions, its statical system andits bending moments.84
Fig. 7, Prismatic stress fields according to the theory ofplasticity (neglecting the transverse tensile stressesdue to the spreading of forces in the concrete) areunsafe for plain concrete.analysis and B-region design providealso the boundary forces for the D-regions of the same structure.Slabs and shells consist predominantly of B-regions (plane strain distribution) - Starting from the sectionaleffects of the structural analysis, imaginary strips of the structure can be modelled like linear members.sectional effects by a statical system maybe omitted and the inner forces or stresses can be determined directly from theapplied loads following the principlesoutlined for D-regions in Section 3.3.However, for structures with redundantsupports, the support reactions have tobe determined by an overall analysisbefore strut-and-tie-models can beproperly developed.In exceptional cases, a nonlinear fi-If a structure consists of one D-regiononly (e.g., a deep beam), the analysis ofTable 1 Analysis leading to stresses or strut-and-tie-forces.Structure consisting of:StructureAnalysisor stressesin individualregionState I(uncracked)M, N, V. MrA,Js,JrD-regionsBoundary forces:Sectional effectsVia sectional valuesD-regions onlye.g., deep beamsf3-regionsB-regionsOverall structural analysis(Table 2) gives:Analysis ofinner forcesB- and D-regionse.g., linear structures, slabs and shellsSectional effectsISupport reactionsLinear elastic analysis*(with redistributed stress peaks)Strut-and-tie-modelsand/or nonlinear stress analysis *State II(cracked)Usuall y trussMay Le cuuwbiuct[ with overall .uiak his.PCI JOURNAL/May-June 198785
p)simple span with cantileverhrte,b)O-regionreg an9,in the bolt on chordforce/ftruss—multiple trussMlzt t4.-single trussmultiple truss (steps)C)beamVIx alI) xoocT-1 11Cclx)vix a} TwwxTs(x-a) o8',h—a-z cot 8 —S8 ——ZmodelCclx-a1Tvx IxbCc Ix }-- EaI i Mxtt—-ll - V2x Co t oTslx)—MCw I x } sTwlx- l V (x}––V(x)cwlxl b z snV(xJt w o) z rat 9BIsmeo ed dogcrd stress)(pEra nil length of beam)TS (xl M x) V cot 8V Ix) may nclude shear forces from torqueaccording to fig 28Fig- 8. Truss model of a beam with cantilever: (a) model; (b) distribution of inner forces;(c) magnitude of inner forces derived from equilibrium of a beam element.86
Table 2. Overall structural behavior and method of overall structural analysis of staticallyindeterminate structures.LimitstateCorresponding method of analysisof sectional effects and support reactionsOverallstructural behaviorMost adequateAcceptableEssentially uncrackedLinear elasticServiceabilityConsiderably cracked,with steel stresses belowyieldNonlinearLinear elastic (or plasticif design is oriented atelastic behavior)UltimatecapacityWidely cracked,tbrming plastic hingesPlastic with limitedrotation capacityor elastic withredistributionLinear elastic ornonlinear or perfectlyplastic withstructural restrictionsnite element method analysis may beapplied. A follow-up check with a strutand-tie-model is recommended, especially if the major reinforcement is notmodelled realistically in the FEManalysis.3.2 Comments on the OverallAnalysisIn order to be consistent, the overallanalysis of statically indeterminatestructures should reflect the realisticoverall behavior of the structure. Theintent of the following paragraph (summarized in Table 2) is to give someguidance for the design of statically indeterminate structures. Some of thisdiscussion can also be applied to statically determinate structures especiallywith regard to determining deformations.Plastic methods of analysis (usuallythe static method) are suitable primarilyfor a realistic determination of ultimateload capacity, while elastic methods aremore appropriate under serviceabilityconditions. According to the theory ofplasticity, a safe solution for the ultimateload is also obtained, if a plastic analysisis replaced by a linear or nonlinearanalysis. Experience further shows thatthe design of cracked concrete strucPCI JOURNALIMay -June 1987tures for the sectional effects using alinear elastic analysis is conservative.Vice versa, the distribution of sectionaleffects derived from plastic methodsmay for simplification purposes also beused for serviceability checks, if thestructural design (layout of reinforcement) is oriented at the theory of elasticity.3.3 Modelling of Individual B- andD-Regions3.3.1 Principles and General DesignProcedureAfter the sectional effects of the B-regions and the boundary forces of the Dregions have been determined by theoverall structural analysis, dimensioningfollows, for which the internal flow offorces has to be searched and quantified:For uncracked B- and D-regions,standard methods are available for theanalysis of the concrete and steel stresses (see Table 1). In the case of highcompressive stresses, the linear stressdistribution may have to be modified byreplacing Hooke's Law with a nonlinearmaterials law (e.g., parabolic stressstrain relation or stress block).If the tensile stresses in individual Bor D-regions exceed the tensile strengthof the concrete, the inner forces of those87
regions are determined and are designed according to the following procedure:1. Develop the strut-and-tie-model asexplained in Section 3.3. The struts andties condense the real stress fields byresultant straight lines and concentratetheir curvature in nodes.2. Calculate the strut and tie forces,which satisfy equilibrium. These are theinner forces.3. Dimension the struts, ties andnodes for the inner forces with due consideration of crack width limitations (seeSection 5).This method implies that the structureis designed according to the lowerbound theorem of plasticity. Since concrete permits only limited plastic deformations, the internal structural system(the strut-and-tie-model) has to be chosen in a way that the deformation limit(capacity of rotation) is not exceeded atany point before the assumed state ofstress is reached in the rest of the structure.In highly stressed regions this ductility requirement is fulfilled by adaptingthe struts and ties of the model to thedirection and size of the internal forcesas they would appear from the theory ofelasticity.In normally or lightly stressed regionsthe direction of the struts and ties in themodel may deviate considerably fromthe elastic pattern without exceedingthe structure's ductility. The ties andhence the reinforcement may be arranged according to practical considerations. The structure adapts itself tothe assumed internal structural system.Of course, in every case an analysis andsafety check must be made using the finally chosen model.This method of orienting the strutand-tie-model along the force paths indicated by the theory of elasticity obviously neglects some ultimate loadcapacity which could be utilized by apure application of the theory of plasticity. On the other hand, it has the major88advantage that the same model can beused for both the ultimate load and theserviceability check. If for some reasonthe purpose of the analysis is to find theactual ultimate load, the model can easily be adapted to this stage of loading byshifting its struts and ties in order to increase the resistance of the structure. Inthis case, however, the inelastic rotationcapacity of the model has to be considered. (Note that the optimization ofmodels is discussed in Section 3.3.3.)Orienting the geometry of the modelto the elastic stress distribution is also asafety requirement because the tensilestrength of concrete is only a small fraction of the compressive strength. Caseslike those given in Fig. 7 would be unsafe even if both requirements of thelower bound theorem of the theory ofplasticity are fulfilled, namely, equilibrium and F'IA --f,. Compatibility evokestensile forces, usually transverse to thedirection of the loads which may causepremature cracking and failure. The"bottle-shaped compressive stressfield," which is introduced in Section4.1, further eliminates such "hidden"dangers when occasionally the modelchosen is too simple.For cracked B-regions, the proposedprocedure obviously leads to a trussmodel as shown in Fig. 8, with the inclination of the diagonal struts orientedat the inclination of the diagonal cracksfrom elastic tensile stresses at the neutral axis. A reduction of the strut angle by10 to 15 degrees and the choice of vertical stirrups, i.e., a deviation from theprincipal tensile stresses by 45 degrees,usually (i.e., for normal strength concrete and normal percentage of stirrupreinforcement) causes no distress. Sinceprestress decreases the inclination ofthe cracks and hence of the diagonalstruts, prestress permits savings of stirrup reinforcement, whereas additionaltensile forces increase the inclination.The distance z between the chordsshould usually be determined from theplane strain distribution at the points of
- 10 l-a La lha)I IHL//Cjt-/--.-k i T Jl I IC1 lilZ---Ta -struttieab)0,51 t5QG Z'A dMfra/1-0170 0,3Q100.5 0,5 0.7 9.8 09r2F/P I art/pt1.0 1.1 1,? 1,3 1,6 1,5 lbd/tFig. 9. A typical D-region: (a) elastic stress trajectories, elasticstresses and strut-and-tie-model; (b) diagram of internal forces,internal lever arm z and strut angle 0.maximum moments and zero shear andfor simplicity be kept constant betweentwo adjacent points of zero moments.Refinements of B-region design will bePCI JOURNALIMay-June 1987discussed later in Section 5.1.For the D-regions it is necessary todevelop a strut-and-tie-model for eachcase individually. After some training,89
AABBpiloodpothcjrT1IAti'–— rlBABFig. 10. Load paths and strut-and-tie-model.IF15'S/ 'A1IBBFIIPF BBFig. 11. Load paths (including a "U-turn") andstrut-and-tie-model.this can be done quite simply. Developing a strut-and-tie-model is comparable to choosing an overall staticalsystem. Both procedures require somedesign experience and are of similar relevance for the structure.Developing the model ofa D-region ismuch simplified if the elastic stressesand principal stress directions are available as in the case of the example shownin Fig. 9. Such an elastic analysis isreadily facilitated by the wide variety ofcomputer programs available today. Thedirection of struts can then be taken inaccordance with the mean direction of90principal compressive stresses or themore important struts and ties can be located at the center of gravity of the corresponding stress diagrams, C and T inFig. 9a, using the y diagram giventhere.However, even if no elastic analysis isavailable and there is no time to prepareone, it is easy to learn to develop strutand-tie-models using so-called "loadpaths." This is demonstrated in moredetail by some examples in the nextsection.3.3.2 The Load Path MethodFirst, it must be ensured that the outer
CIr I 1 7 I II I I iiIILI-iJIII I k IIII I I IIIIIIIIILL Y-iT-l-rClhialedrl7Q fx LVt 1C7Vic, :c.Y -1 I} 1II- -1DDIIs pPL1IIc ,11III Fig. 12.1. A typical D-region: (a) elastic stress trajectories; (b) elastic stresses;(c) strut-and-tie-models.aaIF"tr,,dI4d'r8AICT jCr1tt!IIFig. 12.2. Special case of the D-region in Fig. 12.1 with the load at thecorner; (b) elastic stresses; (c) strut-and-tie-models.equilibrium of the D-region is satisfiedby determining all the loads and reactions (support forces) acting on it. In aboundary adjacent to a B-region theloads on the D-region are taken from theB-region design, assuming for examplethat a linear distribution of stresses (p)PCI JOURNAL/May-June 1987exists as in Figs, 10 and 11.The stress diagram is subdivided insuch a way, that the loads on one side ofthe structure find their counterpart onthe other, considering that the loadpaths connecting the opposite sides willnot cross each other. The load paths91
cBa}AB }PII IvTiAlshear force'ATccBmomentb)ABHmP')III!f/CATtcstruttie. load pathanchorage length of the bar-mteaFig. 13. Two models for the same case: (a) requiring oblique reinforcement;(b) for orthogonal reinforcement.begin and end at the center of gravity ofthe corresponding stress diagrams andhave there the direction of the appliedIoads or reactions. They tend to take theshortest possible streamlined way inbetween. Curvatures concentrate near92stress concentrations (support reactionsor singular loads).Obviously, there will be some caseswhere the stress diagram is not completely used up with the load paths described; there remain resultants (equal
a) goodb) badUIII i LnT TTY irPI(L;1 lrlul .LU.11111 PI11d IiItd IFig. 14. The good model (a) has shorter ties than the bad model (b).in magnitude but with opposite sign)which enter the structure and leave itagain on U-turn or form a whirl as illustrated by forces B or Figs. 11 and 13a.Until now, equilibrium only in the direction of the applied loads has beenconsidered. After plotting all load pathswith smooth curves and replacing themby polygons, further struts and ties mustbe added for transverse equilibriumacting between the nodes, includingthose of the U-turn.While doing so, the ties must be arranged with proper consideration ofpracticality of the reinforcement layout(generally parallel to the concrete surface) and of crack distribution requireme ntsThe resulting models are quite oftenkinematic which means that equilibrium in a given model is possible onlyfor the specific Ioad case. Therefore, thegeometry of the appropriate model hasto be adapted to the load case and is inmost cases determined by equilibriumconditions after only a few struts or tieshave been chosen.A very powerful means of developingnew strut-and-tie-models for complicated cases is the combination of anelastic finite element method analysiswith the load path method. This comPCI JOURNAL/May-June 1987bined approach is applied in Fig. 12 andthe numerical example in Section 5.2.1.In Fig. 12.1 the vertical struts and tiesare found by the load path method asexplained in the previous examples:The structure is divided into a B-regionand a D-region. The bottom of the D-region is acted on by the stresses (p) asderived for the adjacent B-region.These stresses are then resolved intofour components: The two compressiveforces Cs C. F, which leaves twoequal forces T Z and C Y. The forces C,and C, are the components, respectively, on the left hand and right handside of the vertical plane which is deterrnined by the load F. By laterallyshifting the load components into thegiven positions, transverse stresses aregenerated.The corresponding horizontal strutsand ties are located at the center ofgravity of stress diagrams in typical sections which are derived from an elasticanalysis (Fig. 12.1b). Their nodes withthe vertical struts also determine theposition of the diagonal struts (see Fig.12.lc).The example i
dons, or concrete tensile stress fields. For analytical purposes, the strut-and-'Following a proposal by Dr. J. E. Breen and Dr. A. S. C. Bruggeling, the term "structural concrete" covers all loadbearing concrete, including reinforced, prestressed and also plain (unrein-forced concrete, if the latter is part of a reinforced concrete structure.
Concrete Beams 9 Lecture 21 Elements of Architectural Structures ARCH 614 S2007abn Reinforced Concrete - stress/strain Concrete Beams 10 Lecture 21 Elements of Architectural Structures ARCH 614 S2007abn Reinforced Concrete Analysis for stress calculations steel is transformed to concrete concrete is in compression above n.a. and
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concrete with only secondary reinforcement is not considered reinforced concrete. steel fibers—carbon or stainless steel fibers used in fiber-reinforced concrete meeting th e requirements of ASTM A 820. structural concrete—all concrete used for structural purposes
reinforced concrete, Ultra-high performance concrete, Reactive powder concrete. The most common and well researched material is fibre reinforced concrete using different fibers. The concept of using fibers is to enhance the tensile behaviour of the concrete by bridging the cracks and improving the load carrying capacity of the structural members.
polymer impregnated porous concrete. Accordingly, resin was used to fill the voids partially in porous concrete to increase capability of sustained load, which will help to reliability use of this concrete in structural applications. 3.1.2 Details of Concrete Specimens The specimens of porous concrete were mixed with
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