Consistent Design of Structural Concrete - PCI€¦ · · 2017-03-13Consistent Design of Structural Concrete Jorg Schlaich, Dr.-Ing. ... verify or dispute a theory but not to de-

Special Report

Toward aConsistent Design ofStructural Concrete

Jorg Schlaich,Dr.-Ing.Professor at the Institute

of Reinforced ConcreteUniversity of StuttgartWest Germany

Kurt Schafer,Dr.-Irtg.Professor at the Institute

of Reinforced ConcreteUniversity of StuttgartWest Germany

Mattias Jennewein,Dipl.-Ing.Research AssociateUniversity of StuttgartWest Germany

This 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

CONTENTS

Synopsis............................................. 77

1. Introduction — The Strut-and-Tie-Model ............... 76

2. The Structure's B- and D-Regions ..................... 77

3. General Design Procedure and Modelling .............. 843.1 Scope3.2 Comments on the Overall Analysis3.3 Modelling of Individual B- and D-Regions

4. 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 Remarks

5. Examples of Applications ............................. 1105.1 The B-Regions5.2 Some D-Regions5.3 Prestressed Concrete

Acknowledgment......................................147

References...........................................146

Appendix— Notation ...................................150

PCI JOURNAL+May-June 1987 75

1. INTRODUCTION -THE STRUT-AND-TIE-MODEL

The 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 tor-sion. However, a design based on thestandard truss model can cover onlycertain parts of a structure.

At statical or geometrical discontinu-ities such as point loads or frame cor-ners, 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 con-cept must be valid and consistent forevery part of any structure. Further-more, since the function of the experi-ment in design should be restricted toverify or dispute a theory but not to de-rive 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 inter-connected by tensile ties. The ties maybe reinforcing bars, prestressing ten-dons, 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 (unrein-forced concrete, if the latter is part of a reinforcedconcrete structure.

tie-models condense all stresses in com-pression and tension members and jointhem by nodes.

This paper describes how strut-and-tie-models can be developed by fol-lowing the path of the forces throughouta structure. A consistent design ap-proach 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 pro-cedures, rules of thumb and guess workby a rational design method. Strut-and-tie-models could lead to a clearer under-standing of the behavior of structuralconcrete, and codes based on such anapproach would lead to improvedstructures,

The authors are aware of the en-couraging fact that, although they pub-lished 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 in-troduced 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 scien-tific basis for a rational application intracing the concept back to the theory ofplasticity.

Collins and Mitchell further consid-ered 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 use-fully applied to deep beams and corbels.From that point, the present authorsbegan their efforts to systematically ex-pand such models to entire structures

76

SynopsisCertain parts of structures are de-

signed with almost exaggerated accu-racy while other parts are designedusing rules of thumb or judgmentbased on past experience. How-ever, 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-

ing structural concrete, which includesboth reinforced and prestressed con-crete structures.

This report shows how suitablemodels are developed and proposescriteria according to which the model'selements can be dimensioned uni-formly for all possible cases. The con-cept is explained using numerous de-sign examples, many of which treatthe effect of prestress.

This report was initially prepared fordiscussion within CEB (ComitdEuro-International du Beton) in con-nection with the revision of the ModelCode,

and all structures.The approaches of the various authors

cited above differ in the treatment of theprediction of ultimate load and thesatisfaction of serviceability require-ments. From a practical viewpoint, truesimplicity can only be achieved if so-lutions 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 the

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-REGIONS

Those regions of a structure, in whichthe Bernoulli hypothesis of plane straindistribution is assumed valid, are usu-ally 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).

As long as the section is uncracked,these stresses are calculated with thehelp of section properties like cross-sectional 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.

PCI JOURNAL'May-June 1987 77

S hz ' has

h1-1^—h2 .—h-4 h-4

t n

-^ t,

b)

IIFH2. h—L I

h

Fig. 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 discon-tinuities (see Fig. 1). Such regions arecalled D-regions (where D stands for

discontinuity, disturbance or detaiI).As long as these regions are un-

cracked, they can be readily analyzed bythe linear elastic stress method, i.e., ap-plying Hooke's Law. However, if thesections are cracked, accepted designapproaches exist for only a few casessuch as beam supports, frame corners,

78

Fig. 2. Stress trajectories in a B-region and near discontinuities(D-regions).

corbels and splitting tension at pre-stressed 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 rea-sons 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 de-cades after MOrsch, the B-region de-sign is still being disputed, it is only rea-sonable to expect that the more complexD-region design will need to be sim-plified with some loss of accuracy.However, even a simplified methodicalconcept of D-region design will be pref-erable 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 spe-cial case ofa strut-and-tie model.

In using the strut-and-tie-model ap-proach, it is helpful and informative tofirst subdivide the structure into its B-and D-regions. The truss model and the

design 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 discon-tinuities (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 exam-ples 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 sev-eral 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 approxi-

PCI JOURNALMay-June 1987 79

{ a1 lb) (c) (d)h F^h1^ r B/h F

d ht^

+ B

d=h

Fig. 3.1. Column with point loads.

{c1

hlMI l t i M3 h2V ^^

M1 M M3V

V

c b1

(c) U.+ i-dr r+–d2 = h2

{ d1

iB / B

Fig. 3.3. Beam with a recess.

80

a]IL l I 4 3 TTTTT r I- m

h

t t^

(C) d=h_,

rt+I h

tf1:!II!1IIIIle!!!:

Fig. 3.2. Beam with direct supports.

(a]

^`. D ^ ^ I^ d=b

fFig. 3.4. T-beam.

(a) real structure (c) self-equilibrating state of stress

(b) loads and reactions applied in (d) real structure with B- andaccordance with Bernoulli hypothesis D-regions

Fig. 3. Subdivision of four structures into their B- and D-regions, using SaintVenant's principle (Fig. 4).

PCI JOURNAL/May-June 1987 81

Q)

F a6=0\i0

6 ^ d=h

b) Fy4______

2y!_2 ?_iTIh

HId=h d

_' "Y x x

1. 0 h 10 h

dydy

1,0 h 1,0 h

Fig. 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 them-selves. This distance defines the rangeof the D-regions (d).

It should be mentioned that crackedconcrete members have different stiff-nesses in different directions. This situ-

ation may influence the extent of theD-regions but needs no further discus-sion since the principle of Saint-V€nantitself is not precise and the dividinglines between the B- and D-regionsproposed here only serve as a qualita-tive aid in developing the strut-and-

82

p h

B B t^ B

/a2h

c) }

B ^' B â

B Bf^4h

B 01"OkolhI h B B

I >Zh^•4h1

0 ^h

UIF^

w

r rf ► f

Fig. 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 B-

and D-regions is, however, already ofconsiderable value for the under-standing of the internal forces in thestructure. It also demonstrates, that sim-ple 1fh rules used today to classify

beams, deep beams, short/long/highcorbels and other special cases are mis-leading. For proper classification, bothgeometry and loads must be considered(see Figs. 3, 5 and 6).

If a structure is not plane or of con-stant width, it is for simplicity sub-

PCI JOURNALMay-June 1987 83

divided into its individual planes, whichare treated separately. Similarly, three-dimensional stress patterns in plane orrectangular elements may be looked atin different orthogonal planes. There-fore, in general, only two-dimensionalmodels need to he considered. How-ever, the interaction of models indifferent planes must be taken into ac-count by appropriate boundary condi-

tions.Slabs may also be divided into B-re-

gions, where the internal forces are eas-ily derived from the sectional forces,and D-regions which need further ex-planation, If the state of stress is notpredominantly plane, as for example inthe case with punching or concentratedloads, three-dimensional strut-and-tie-models should be developed.

3. GENERAL DESIGN PROCEDURE ANDMODELLING

3.1 ScopeFor the majority of structures it would

be unreasonable and too cumbersome tobegin immediately to model the entirestructure with struts and ties. Rather, itis more convenient (and common prac-tice) to first carry out a general structuralanalysis. However, prior to starting thisanalysis, it is advantageous to subdividethe given structure into its B- and D-re-gions. The overall analysis will, then,include not only the B-regions but alsothe D-regions.

If a structure contains to a substantial

part 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 mo-ments(Mr ) (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 structural

Fig. 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-re-gions of the same structure.

Slabs and shells consist predom-inantly of B-regions (plane strain dis-tribution) - Starting from the sectionaleffects of the structural analysis, imagi-nary strips of the structure can be mod-elled like linear members.

If a structure consists of one D-regiononly (e.g., a deep beam), the analysis of

sectional effects by a statical system maybe omitted and the inner forces or stress-es 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-

Table 1 Analysis leading to stresses or strut-and-tie-forces.

Structure consisting of:

B- and D-regions D-regions onlyStructure

Analysise.g., linear structures, slabs and shells e.g., deep beams

B-regions f3-regions D-regions

Overall structural analysis Sectional effectsBoundary forces:

(Table 2) gives: M, N, V. Mr Sectional effects Support reactions

Analysis of State I Via sectional values I Linear elastic analysis*inner forces (uncracked) A,Js,Jr (with redistributed stress peaks)or stressesin individual State II

(cracked)Strut-and-tie-models

and/or nonlinear stress analysis *region Usuall y truss

May Le cuuwbiuct[ with overall .uiak his.


p) simple span with cantilever

hrte,

9, reg an O-region

b) force in the bolt on chord

/ trussf

—multiple truss

Mlz

t t4.-- ttsingle truss — -multiple truss (steps) --^

C)

aIîbeam

E ocMx

Cclx-a1model Cclx)T -1 11

I) vix a}^^ Tw _

VIx al^ vx I T w

x o xb S Ts(x-a) o 8' x Tslx),h—a-z cot 8 —

Cc Ix } M—ll - V2x Cot o

Cw I x }_ s 8 — — cwlxl b z snV(x)

B Ismeo ed dogcrd stress)V(xJTwlx- l = V (x} – – t w o) = z rat 9 (pEr a nil length of beam)

TS (xl= MZx) • V- cot 8

V Ix) may nclude shear forces from torque according to fig 28

Fig - 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 statically

indeterminate structures.

Corresponding method of analysisLimit Overall of sectional effects and support reactionsstate structural behavior

Most adequate Acceptable

Essentially uncracked Linear elastic

Service- Considerably cracked, Linear elastic (or plasticability with steel stresses below Nonlinear if design is oriented at

yield elastic behavior)

Plastic with limited Linear elastic orUltimate Widely cracked, rotation capacity nonlinear or perfectlycapacity tbrming plastic hinges or elastic with plastic with

redistribution structural restrictions

nite element method analysis may beapplied. A follow-up check with a strut-and-tie-model is recommended, espe-cially if the major reinforcement is notmodelled realistically in the FEManalysis.

3.2 Comments on the OverallAnalysis

In order to be consistent, the overallanalysis of statically indeterminatestructures should reflect the realisticoverall behavior of the structure. Theintent of the following paragraph (sum-marized in Table 2) is to give someguidance for the design of statically in-determinate structures. Some of thisdiscussion can also be applied to stat-ically determinate structures especiallywith regard to determining deforma-tions.

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 struc-

tures 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 reinforce-ment) is oriented at the theory of elas-ticity.

3.3 Modelling of Individual B- andD-Regions

3.3.1 Principles and General DesignProcedure

After the sectional effects of the B-re-gions and the boundary forces of the D-regions 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 stress-es (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 stress-strain relation or stress block).

If the tensile stresses in individual B-or D-regions exceed the tensile strengthof the concrete, the inner forces of those

PCI JOURNALIMay-June 1987 87

regions are determined and are de-signed according to the following pro-cedure:

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 con-sideration of crack width limitations (seeSection 5).

This method implies that the structureis designed according to the lowerbound theorem of plasticity. Since con-crete permits only limited plastic de-formations, the internal structural system(the strut-and-tie-model) has to be cho-sen 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 struc-ture.

In highly stressed regions this ductil-ity 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 ar-ranged according to practical consid-erations. The structure adapts itself tothe assumed internal structural system.Of course, in every case an analysis andsafety check must be made using the fi-nally chosen model.

This method of orienting the strut-and-tie-model along the force paths in-dicated by the theory of elasticity obvi-ously neglects some ultimate loadcapacity which could be utilized by apure application of the theory of plastic-ity. On the other hand, it has the major

advantage 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 eas-ily be adapted to this stage of loading byshifting its struts and ties in order to in-crease the resistance of the structure. Inthis case, however, the inelastic rotationcapacity of the model has to be consid-ered. (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 frac-tion of the compressive strength. Caseslike those given in Fig. 7 would be un-safe even if both requirements of thelower bound theorem of the theory ofplasticity are fulfilled, namely, equilib-rium 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 in-clination of the diagonal struts orientedat the inclination of the diagonal cracksfrom elastic tensile stresses at the neu-tral axis. A reduction of the strut angle by10 to 15 degrees and the choice of verti-cal stirrups, i.e., a deviation from theprincipal tensile stresses by 45 degrees,usually (i.e., for normal strength con-crete and normal percentage of stirrupreinforcement) causes no distress. Sinceprestress decreases the inclination ofthe cracks and hence of the diagonalstruts, prestress permits savings of stir-rup reinforcement, whereas additionaltensile forces increase the inclination.

The distance z between the chordsshould usually be determined from theplane strain distribution at the points of

88

-10 l-a La lha)

I IH -k i T l I I

L//Cj t-/--. ^J C1 ^ l i l Z

T --- struttie

b)

a^- a

0,5 ____ A^ Mfr

1 t5 d r2FQG Z' a/1-01

70° 0,3

1/P I art/pt

Q

00.5 0,5 0.7 9.8 09 1.0 1.1 1,? 1,3 1,6 1,5 lbd/t

Fig. 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 be

discussed later in Section 5.1.For the D-regions it is necessary to

develop a strut-and-tie-model for eachcase individually. After some training,


A B A B

p i

loodpoth cjr

T 1 ti'–—`r

IA lBA B

Fig. 10. Load paths and strut-and-tie-model.

IF15'

S /^

A '

1

I II

PF B B

B B

F

Fig. 11. Load paths (including a "U-turn") andstrut-and-tie-model.

this can be done quite simply. De-veloping a strut-and-tie-model is com-parable to choosing an overall staticalsystem. Both procedures require somedesign experience and are of similar rel-evance for the structure.

Developing the model ofa D-region ismuch simplified if the elastic stressesand principal stress directions are avail-able 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 of

principal compressive stresses or themore important struts and ties can be lo-cated at the center of gravity of the cor-responding 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 strut-and-tie-models using so-called "loadpaths." This is demonstrated in moredetail by some examples in the nextsection.3.3.2 The Load Path Method

First, it must be ensured that the outer

90

Ce hi Clal

Ir I 1 ^`7^ - Q ^f

I+II I ^ T-l- i LV x^ t 1C7I ii Vic, :c. Ic_,

-r 7 drl Y1 I 1II

LI-iJ ^- - - } 1 _1III I k I II IIII I I I D 1 1 IIII I I D I I IIII

ILLY s p _ P L ^ ^

Fig. 12.1. A typical D-region: (a) elastic stress trajectories; (b) elastic stresses;(c) strut-and-tie-models.

a a

IF

" t r ,,

dI4d'r ICT jCr

1 t

8 t !

A I I

Fig. 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 reac-tions (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)

exists as in Figs, 10 and 11.The stress diagram is subdivided in

such 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 paths


H )

Ti

'

I I

I !

f/

B c

a } A B ^}P

Iv

II ^

Al Tcshear force'

A B c

moment

b) A B m P

C AT tc

strut

tie

_.^ load path

-mtea anchorage length of the bar

Fig. 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 near

stress concentrations (support reactionsor singular loads).

Obviously, there will be some caseswhere the stress diagram is not com-pletely used up with the load paths de-scribed; there remain resultants (equal

92

a) good b) bad

UIII i LnT TTY_ i P (L;1 lrlul .LU.11111 P

r I I I

11 d_I i t d=I

Fig. 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 illus-trated by forces B or Figs. 11 and 13a.

Until now, equilibrium only in the di-rection 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 ar-ranged with proper consideration ofpracticality of the reinforcement layout(generally parallel to the concrete sur-face) and of crack distribution require-me nts -

The resulting models are quite oftenkinematic which means that equilib-rium 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 compli-cated cases is the combination of anelastic finite element method analysiswith the load path method. This com-

bined 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-re-gion 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, respec-tively, on the left hand and right handside of the vertical plane which is de-terrnined 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 sec-tions 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 in Fig. 12.2 shows thatthe tie T 3 of Fig. 12.Ic disappears, if theload F moves toward the corner of theD-region.


a) p

RAIf^ f # 1i

b) Q

e) + 2l

f-;

I I`1r 1

r r

fhigh medium low long

tl)

^ I 1

Yr Ì I it t !_, I '-, 4

r ^ ly ^^ II I

L---4-- -r-+i I I

I r r I

4 i+ 7 T t i tf

Fig. 15. The two most frequent and most useful strut-and-tie models:(a) through (b), and some of their variations (c) through (e).

94

3.3.3 Model OptimizationIn Fig. 13 one load case has been

solved with two different models. Theleft part of Figs. 13a and 13h shows howthey were developed by the load pathmethod and how they are connected tothe overall sectional effects of the D-re-gions. The ties T 1 and Tx in Fig. 13awould require inclined reinforcement,which is undesirable from a practicalviewpoint.

Therefore, a tie arrangement has beenchosen in Fig. 13b which can be satis-fied by an orthogonal reinforcing netwith the bars parallel to the edges.

Thereby, special methods such as thosegiven in Ref. 17, which consider devia-tions of reinforcement directions fromthe principal stress directions, are notneeded.

Doubts could arise as to whether thecorrect model has been chosen out ofseveral possible ones. In selecting themodel, it is helpful to realize that loadstry to use the path with the least forcesand deformations. Since reinforced tiesare much more deformable than con-crete struts, the model with the least andshortest ties is the best. This simplecriterion for optimizing a model may be

b)

' I

B

ON 055 1.011 1511

i ------ ^`-----' -----

c) srbT

a - o

rr,

I II 5rr

d)^

r r+ r

(see also fig. 35)

Fig. 16. One single type of a D-mode& appears in many different structures: here fourexamples are given.


a) } } îoad^ b) anchor plate tendon

t I s __

columnT 1

- -------H

Fig. 17. (a) Deep beam on three supports; (b) end of a beam or slabwith anchorages of three prestressing tendons. Both cases areidentical if the reactions in (a) and the prestressing forces in (b) areequal.

formulated as follows:

!.F ;l ; e„,; = Minimum

whereF, = force in striitortiei1, = length of memberie ms = mean strain of member i

This equation is derived from theprinciple of minimum strain energy forlinear elastic behavior of the struts andties after cracking. The contribution ofthe concrete struts can generally heomitted because the strains of the strutsare usually much smaller than those ofthe steel ties. This criterion is alsohelpful in eliminating less desirablemodels (see Fig. 14).

Of course, it should be understoodthat there are no unique or absolute op-timum solutions, Replacing a con-tinuous set of smooth curves by indi-vidual polygonal lines is an approxima-tion in itself and leaves ample room forsubjective decisions, Furthermore, in-dividual input such as the size of the re-gion or reinforcement layout are alwaysdifferent. But an engineer with someexperience in strut-and-tie-modellingwill always find a satisfactory solution.

3.3.4 The Pedagogical Value of Mod-elling

Anyone who spends time developingstrut-and-tie-models will observe thatsome types of D-regions appear overand over again even in apparently verydifferent structures. The two most fre-quent D-regions, which are even relatedto one another because they havethe same characteristic stress distri-bution along their centerline, are giv-en in Fig. 15 with some of theirvariations.

Fig. 16 shows applications of the firsttype of model (Fig. 15a) to four differentstructures: The distribution of cableforces in a bridge deck (Fig. 16a); a wallwith big openings (Fig. 16b); a box gird-er with anchor loads from prestressingtendons (Fig. 16c); and a detail of inter-nal forces in a rectangular beam whichshows that stirrups need to he closed.(Fig. 16d). In all of these cases the pat-tern of internal forces is basically identi-cal.

To recognize such common features ofstructures is of considerable pedagogi-cal value and very helpful to the designengineer. On the other hand, it is con-fusing if the same facts are given dif

96

ferent designations only because theyappear under different circumstances.For example, the deep beam in Fig. 17"does not care" whether its identicalloads result from supports or from pre-stressing tendons. It, therefore, does not

react with "bending" in the one caseand with "splitting tension" in the othercase, but simply with tension and com-pression.

Numerous other strut-and-tie-modelsare found in Ref. 3.

4. DIMENSIONING THE STRUTS, TIES AND NODES

4.1 Definitions and General RuleFig. 18 shows some typical examples

of strut-and-tie models, the correspond-ing stress trajectories and reinforcementlayout. Node regions are indicated byshading. Looking closely at these ex-amples as well as those in the previousand following sections, the followingconclusions may be drawn:

Dimensioning not only means sizingand reinforcing the individual struts andties for the forces they carry, but alsoensuring the load transfer between themby checking the node regions. There is aclose relation between the detailing ofthe nodes and the strength of the strutsbearing on them and of the ties an-chored in them because the detail of thenode chosen by the design engineer af-fects the flow of forces, Therefore, it isnecessary to check whether the strut-and-tie-model initially chosen is stillvalid after detailing or needs correction.Thus, modelling and dimensioning is inprinciple an iterative process.

There are basically three types ofstruts and ties to be dimensioned:

C,: Concrete struts in compressionTT : Concrete ties in tension without

reinforcementTies in tension with reinforce-ment (mild steel reinforcement orprestressing steel)

There are essentially four types ofnodes depending on the combination ofstruts C and ties T (see Fig. 19):

CCC-nodeCCT-nodeCTT-node

TTT-nodeThe principle remains the same if

more than three struts and ties meet.

Struts and TiesWhereas the T, are essentially linear

or one-dimensional elements betweentwo nodes, the C. and T,, are two- (orthree-) dimensional stress fields, tend-ing to spread in between two adjacentnodes. This spreading, indicated by thebulging of the struts in Figs. 18 and 19a,can result in transverse tensile andcompressive stresses which then musthe considered either by introducingthese stresses into the failure criterion ofthe C struts and the Tc ties or by againapplying a strut-and-tie-model to them(see Figs, 18c and 18d). Both approacheslead to the same result.

The struts in the model are resul-tants of the stress fields. Since by defini-tion the curvatures or deviations of theforces are concentrated in the nodes, thestruts are straight. This is, of course, anidealization of reality. If doubts arisewhether by doing so in a highly stressedstructure some tensile forces are notsufficiently accounted for, the straightlengths of the struts can be reducedeither by refining the model itself or bysmearing (or spreading) the node over asubstantial length of the strut (see forexample Figs. 18a2 and 18b2).

To cover all cases of compressionfields including those of the B-regions,three typical configurations are suffi-cient:

(a) The "fan" (see Fig.24a).e


all a2) o3)

I _ F1_{!.) IGIlGla]{)1III1iTI[JIIJi]11111

I ^_ CC C

C

1c c `

bll b2) b3)

s ^C T T C

C C^

C1) c2) not reinforced c3) reinforced

1C^ O

EE

` f

Fig. 18. Some typical examples of strut-and-tie-models, their stress fields, nodes andcorresponding reinforcement (if any is provided).

98

(b) The "bottle" (see Fig. 20b).(c) The "prism" or parallel stress

field (see Fig. 20c), being the limitcase of both a = 0 and hla = 1.

NodesThe nodes of the model are a simpli-

fied idealization of reality. They areformally derived as the intersectionpoints of three or more straight struts orties, which themselves represent eitherstraight or curved stress fields or rein-forcing bars or tendons. A node as intro-duced into the model implies an abruptchange of direction of forces. In the ac-tual reinforced concrete structure thisdeviation usually occurs over a certain

length and width.If one of the struts or ties represents a

concentrated stress field, the deviationof forces tends to he locally concen-trated also. On the other hand, for wideconcrete stress fields joining each otheror with tensile ties, which consist ofmany closely distributed reinforcingbars, the deviation of forces may besmeared (or spread) over some length.Therefore, in the former case the nodesare called singular (or concentrated)nodes, whereas in the latter case theyare called smeared (or continuous)nodes. Nodes A and B in Fig. 18a1 serveas typical examples of both types ofnodes.

d1)

d 2) struts not reinforced d 3) struts reinforced

e)

Fig. 18 (cont). Some typical examples of strut-and-tie-models, their stress fields, nodesand corresponding reinforcement (if any is provided).

PCI JOURNALIMay -June 1987 99

ai02

HHae`\ anchor pla4e t

I^-he12^1

tanch rOge length

b 3 b4

withart with ainchomge lengthcrosspressure with loop

nnchort^ge length

Fig. 19. Examples of the basic types of nodes: (a) CCC-nodes. Idealized "hydrostatic"singular nodes transfer the concentrated loads from an anchor plate (a,) or bearing plate(a2 ) into (bottle shaped) compression fields; (b) CCT-nodes. A diagonal compressionstrut and the vertical support reaction are balanced by reinforcement which is anchoredby an anchor plate behind the node (b,), bond within the node (b2), bond within andbehind the node (b,), bond and radial pressure (b,).

IPISI

Failure Criteria for ConcreteThe strength of the concrete in com-

pression fields or within nodes dependsto a large extent on its multiaxial state ofstress and on disturbances from cracksand reinforcement.

(a) Transverse compression is favor-able especially if it acts in both trans-verse directions, as for example in con-fined regions. Confinement may be pro-vided by transverse reinforcement or bybulk concrete surrounding a relativelysmall compression field (see Fig. 21).

(b) Transverse tensile stresses and thecracks caused by them are detrimental.rsThe concrete may fail considerablybelow its cylinder strength if the trans-verse tension causes closely spacedcracks approximately parallel to theprincipal compression stresses such that

the prisms between those cracks areragged and narrow. The reduction ofcompressive strength is small or nomi-nal if the tensile forces are carried by thereinforcement and the cracks are wideenough apart.

(c) In particular, cracks which are notparallel to the compressive stresses aredetrimental.

In 1982, an empirical formula for cal-culating the strength of parallel concretecompression fields with transverse ten-sion was published by Collins et al.,2aand 2 years later a similar formula wasintroduced into the new CanadianCSA-Standard A 23.3-M 84. 24 Theseformulas summarize the influence ofsuch significant parameters as crackwidth, crack distance and crack direc-tion by the transverse tensile strain E,

c1 ) C2)

d 1 ) dz)

l It f

Fig. 19 (cont.). Examples of the basic types of nodes: (c) CTT-nodes. A compressionstrur is supported by two bonded reinforcing bars (c,), respectively, by radial pressurefrom a bent-up bar (c1); (d) TTT-nodes, as above with the compression strut replaced bya bonded tie.


a} b)

` 1 1; J j r 1 I J 1 I 1 1 1 G 1411111J4

I I iI I I rI I Ii I Ir ^ I

^tllfft/f fffcd ffittfll^ g^f d 1fi11t11^ a^fcd

— aV^-I

Fig. 20. The basic compression fields: (a) the "fan"; (b) the "bottle"; (c) the "prism".

which, however, is not readily availablein the analysis.

For practical purposes, the followingsimplified strength values of f , areproposed for dimensioning all types ofstruts and nodes:f *! = 1.0 f,: for an undisturbed and

uniaxial state of compressivestress as shown in Fig. 20c;

fir = 0.8 f': if tensile strains in thecross direction or transverse ten-sile reinforcement may causecracking parallel to the strut withnormal crack width; this appliesalso to node regions where ten-sion steel bars are anchored orcrossing (see Fig. 19b);

f, = 0.6 f,: as above for skew crackingor skew reinforcement;

= 0.4 fCd: for skew cracks with ex-traordinary crack width. Suchcracks must be expected, if mod-elling of the struts departs signifi-cantly from the theory of elastic-ity's flow of internal Forces (e.g.,due to redistribution of internalforces in order to exploit amaximum ultimate capacity).

Note that f , denotes the concretecompressive design strength, which isrelated to the specified compressivestrength f, and which in turn dependson the safety factor of the designatedcode of practice. According to the CEBCode,f^d is determined by:

0.85fcye

where y, = 1.5 is the partial safety factorfor the concrete in compression and thecoefficient 0.85 accounts for sustainedloading, In the CEB Code, 2 0 = 1.0 inall cases and the load factors for deadand live loads are 1.35 and 1.5, respec-tively.

The increase in strength due to two-or three-dimensional states of compres-sive stresses may be taken into accountif the simultaneously acting transversecompressive stresses are considered re-liable.

Skew cracks are not expected, if thetheory of elasticity is followed suffi-ciently closely during modelling. Thismeans that the angle between struts andties entering a singular node should not

102

a^ --- b 7

-- - ,confiningreinforcement

F#g. 21. Confinement (a) by the surrounding concrete; (b) byreinforcement increases the design compression strengthf . *. d.

be too small. However, skew cracks mayalso be left over from a previous loadingcase with a different stress situation.

Before deciding on one of the givenstrength values, both transverse direc-tions must always he considered.

General ruleSince singular nodes are bottlenecks

of the stresses, it can be assumed that anentire D-region is safe, if the pressureunder the most heavily loaded bearingplate or anchor plate is less than 0.6 ff,,(or in unusual cases 0.4 f,.) and if allsignificant tensile forces are resisted byreinforcement and further if sufficientdevelopment lengths are provided forthe reinforcement. Only refinementswill be discussed in the following Sec-tions 4.2 through 4.5.

4.2 Singular NodesSingular nodes equilibrate the forces

of the ties and struts acting on them rel-atively abruptly as compared to thesmeared nodes. The deviation of theforces occurs over a short length or smallarea around the theoretical nodal point.Such nodes originate mainly from singleloads or support reactions and from con-centrated forces introduced by thereinforcement through anchor plates,bond, or radial pressure inside bent

reinforcing bars such as loops. Fur-therrnore, geometrical discontinuities(e.g., reentrant corners) can cause stressconcentrations which are representedby a singular node.

Although numerous possibilities existfor detailing nodes (and despite the factthat they all behave somewhat differ-ently), in most cases their forces balanceeach other in the interior of the nodethrough direct concrete compressivestresses, which is a helpful observation.The ideal tie anchor (with a plate)transfers the load "from behind" andthus causes compression in the node(Fig. 19h,). Also, bond is essentially aload transfer via concrete compressivestresses which are supported by the ribsof the steel bar (Fig. 19h 2 and b 3 ) and byradial pressure in bent bars (Fig. 19b,).

Even in these cases the flow of forcescan be visualized by strut-and-tie-models with singular nodes at the ribs ofthe bar. If the se models require concretetensile ties, this is valuable informationfor the design engineer. However, forpractical purposes, the anchorage andlap lengths of the applicable codes ofpractice should be used.

In summary, then, dimensioning sin-gular nodes means:

(a) Tuning the geometry of the nodewith the applied forces.

For CCC-nodes it is helpful, thoughnot at all mandatory, to assume the bor-


derline to be perpendicular to the re-sultant of the stress field and the state ofstress within the interior of the node tobe plane hydrostatic. In this particularcase the unequivocaI geometrical rela-tion a,: a 2 : as - C,: C 2 : C 3 results (Fig.19a,), which may be used for dimen-sioning the length of the support or thewidth of an anchor plate. However, ar-rangements of forces which lead tostress ratios down to 0.5 on adjacent

edges of a node are satisfactory. De-parting even more from hydrostaticstress and still disregarding thenonuniform stress distribution in thenode may lead to compatibility stresses,which are not covered by the strengthvalues given above.

When designing a singular CCT-node, the design engineer must beaware that the curvature of the load pathand the corresponding compression

p al- fcd B

ao.2[0 j0 Hp B iC

*a O 1' fcd Cr 022 1• tanplcns2 ....-lo_--__._

^p C—

1^ 1

Q } " i b,net 1h,net1

aZ Q2

a3 _C o3,o1 tcnp

b) at°aZccs2W Q1=ap

C) 01_.

op f*

±t + o t y f ^

ci) —at— 01' `b,net

Fig, 22. A proposal for the dimensioning of typical singular CCT -nodes with differentreinforcement layouts: (a) multilayered tie (a 3 relatively large, o < a,); (b) single layer tie(a 3 small a-z>v,); (c) special case in between (o-2 =o 3 ); (d) same type as (b) with pressurefrom a compression field.

104

field is largest at the origin of the con-centrated load, i.e., next to the bearingplate or anchor plate (see Figs. 9 and10). The ties and plates should be ar-ranged accordingly.

(b) Checking whether the concretepressures within the node are within thelimits given in Section 4.1.

This condition is automatically satis-fied for the entire node region, if thestresses along the horderlines of thenode do not exceed those limits and ifthe anchorage of the reinforcement inthe node is safe. If the reinforcement isanchored in the node region, cracks arepossible and correspondingly the con-crete strength for cracked concreteapplies.

For CCT-nodes with bonded rein-forcement arranged according to Fig. 22,a check of concrete stresses a, respec-tively, o in the adjacent compressionstruts is sufficient. Since in most cases itis apparent from the geometry of thenode which pressure out of the twostruts controls, only one of them needsto be analyzed. An analysis on the basisof Fig. 22 rewards reasonably the ar-rangement of multilayered reinforce-ment distributed over the width a g (Fig.22a) compared with a tie consisting ofone layer only (Fig. 22b and d).

(c) Ensuring a safe anchorage of ties inthe nodes (except for CCC-nodes).

In the case of anchor plates, this in-volves a check of the bending strengthof the anchor plate and of the weldedconnection with the tie. In this case asmooth surface of the tie where it cross-es the node is better than good bondquality because strain compatibilitywith the bonded bar will tend to crackthe node's concrete.

In the case of directly anchored rein-forcing bars, hook or loop anchorages(with cross-pressure in CCT-nodes) arepreferred. Generally, the minimumradius allowed by the applicable code isselected.

For straight bar anchorages, the

length of the anchorage is selected fol-lowing the designated code. The designengineer must ensure that it is locatedwithin and "behind" the node (Fig. 19b2and b 3). Anchorage begins where thetransverse compression stress trajec-tories of the struts meet the bar and aredeviated; the bar must extend to theother end of the node region in order tocatch the outermost fibers of the de-viated compression stress field (Fig.18b 3 , c3).

4.3 Smeared NodesSince D-regions usually contain both

smeared and singular nodes, the latterwill be critical and a check of concretestresses in smeared nodes is unneces-sary. However, if a smeared CCT-nodeis assumed to remain uncracked, thetensile stresses of the correspondingconcrete stress field need to be checked(Section 4.5). An example of this case isNode 0 in Fig. 18c1 and the stress fieldin Fig. 18c2.

Safe anchorage of reinforcement barsin smeared nodes must be ensured fol-lowing the rules for singular nodes(Section 4.2).

4.4 Concrete Compression Struts—Stress Fields C,

The fan-shaped and prismatic stressfields do not develop transverse stressesand accordingly the uniaxial concretestrenth fed applies. If transverse stresses,cracks or tension bars cross the strut, thestrength may be based on the valuesgiven in Section 4.1.

The bottle-shaped compression stressfield (Fig. 20b) applies to the frequentcase of compressive forces being intro-duced into concrete which is unrein-forced in the transverse direction.Spreading of the forces causes biaxial ortriaxial compression under the load andtransverse tensions farther away. Thetransverse tension (combined with lon-gitudinal compressive stresses) can re-

PCI JOURNALJMay-June 1987 105

m

— —biaxial compression failurein bottle neckcrocked, but withtransverse reinforcement(degree w) in thebelly regionuncrocked, plain concrete

nfinement

b) * o*pa

la

+—b1 2 3 4 5 6 7 BY

IF--po- a-t

^^^ 0,21

II*ate ^1,31a^

Fig. 23. Dimensioning plane bottle shaped stress fields: (a) diagrams giving safepressure values P d with regard to cracking and crushing of plain unreinforced concretestress fields, yielding of transverse reinforcement and biaxial compression failure in thebottle neck region; (b) geometry of the stress field; (c) model and reinforcement layout ofstress field with transverse reinforcement w.

106

suit in early failure. Again, the failurecriteria from Section 4.1 apply.

The general rule given at the end ofSection 4.1 usually makes the calcula-tion of stresses within the stress fieldunnecessary . However, for questionablecases, computational aids should beprovided to facilitate the safety check.

As an example, Fig. 23a shows dia-grams for checking plane bottle shapedstress fields in D-regions. 12 '3• " Thisstress field can he characterized by thewidth a of the anchor plate, themaximum width b available in thestructure for the stress field and the dis-tance 1 of the anchor plate from the sec-tion where the stress trajectories areagain parallel (see Fig. 23b and also Fig.20b). The diagram for compressionfields without transverse reinforcement(bold line) is based on an elasticanalysis, a concrete tensile strength off= f115 and a biaxial compressive tensilefailure criterion as given in Fig. 26b.

It can be seen, that for certain geomet-rical relations a pressure at the anchorplate as low as 06f could cause crack-ing. However, the failure load of thestrut is usually higher than its crackingload. A comparison of test results showsthat the diagram generally appears to beconsiderably on the safe side andfurther research in this area is required.Better knowledge is also needed onsubstantially nonsymmetrical stressfields which originate from singularnodes with tension ties crossing or an-chored there. Comparisons with test re-sults suggest that checking the singularnode (Section 4.2) and applying thediagrams of Fig. 23 is also safe for thosecases.

The bottle shaped stress field pro-vides a safe lower bound for unrein-forced compression struts, whereas anindiscriminate application of the theoryof plasticity to cases such as those shownin Fig. 7 (mainly Fig. 7a) would permitprismatic stress fields between two op-posite anchor plates with 1.0 ff as a fail-ure stress and could lead to a premature

failure.For compression struts with trans-

verse reinforcement the failure loadsanalyzed with the model in Fig. 23c arealso given in Fig. 23a. It can be seen thata reinforcement ratio:

w = aJ9y--0.06tfd

(where a, is the cross section of rein-forcement per unit length) approxi-mately compensates for the tensilestrength of the concrete.

If it is desired not to rely so heavily onthe concrete tensile strength, lowerreinforcement ratios may be used withreduced values of g aff f, as shown in Fig.23a.

The compressive strength of compres-sion reinforcement may be added to theconcrete strength if the reinforcement isprevented from buckling.

4.5 Concrete Tensile Ties — StressFields T^

In the case of uncracked tensile stressfields, the tensile strength of concrete isused. Although it is difficult to developdesign criteria for this case, it would beeven worse to maintain the formalisticview that the tensile strength of con-crete cannot and therefore must not beutilized. Following the flow of forcesgap free and consistently with strut-and-tie-models will inevitably show thatequilibrium can frequently only besatisfied if ties or tensile forces can beaccepted in places where, for practicalreasons, reinforcement cannot be pro-vided, i.e., if the tensile strength of con-crete is utilized.", It should be apparentthat no anchorage, no lap, no framecomer, no slab without stirrups and (asshown) no unreinforced strut or com-pression member can work withoutusing the tensile strength of concrete.Unfortunately, most codes of practice donot recognize this fact and, therefore,surrogates such as bond, shear

PCI JOURNALiMay-June 1987 107

and other misnomers have been intro-duced. As a result, codes have becomeunduly imprecise and complicated.

Until further research work is avail-able in this field, the following simpleguidelines are proposed, which appearto yield safe results when compared totests:

The tensile strength of concreteshould only be utilized for equilibriumforces where no progressive failure isexpected. Thereby, restraint forces andmicrocracks have to be taken into ac-count, even in "uncracked" and un-loaded concrete. As shown in Fig. 24,redistribution of stresses which avoidsprogressive cracking may be assumed tobe possible if at any part of the stressfield a cracked failure zone with an areaJA, can be assumed, without the in-creased tensile stresses in the remainingsection exceeding the tensile strengthJet.

As a preliminary proposal, it is sug-gested that:

and ?A,,110

whereA, = area of tensile zone anddo = diameter of largest aggregateProgressive failure of a section or

member generally starts from theperiphery of structures in the case ofsteep stress gradients, as for example inthe bending tensile zone of beams (Fig.25).

The tensile stresses may be analyzedwith a linear elastic materials law. Stresspeaks in the outer fibers or at failurezones may be distributed over a width of5 cm (2 in.) but not more than 3 d, a rulewhich finds its justification in fracturemechanics of concrete. 13

The design engineer will have to de-cide case by case which fraction of thetensile strength can he used for carryingloads and which fraction has been usedup by restraint stresses. The latterstresses are usually large in the longitu-dinal direction of a structural member

and at its surface, but are smaller in thetransverse direction and in greaterdepth.

If the tensile stress field is crossed bya compression field, the reduced biaxialstrength must be considered. The graph(see Fig. 26c) provides a safe assump-tion.

4.6 Reinforced Ties T.

Usually, reinforcing steel should beprovided to resist tensile forces. Theaxis of the steel reinforcement mustcoincide with the axis of the tie in themodel. The dimensioning of these ties isquite straightforward; it follows directlyfrom the cross section A. (reinforcingsteel) or A, (prestressing steel) and theyield strength f,,, and f, of the respectivesteels:

Ta _-AJr,+A,Af,,

Since it is proposed here (see Section5.3) to introduce prestress as an externalload into the analysis and dimensioning,the acting tie force Tx is the result of allexternal loads (including prestress).Then, however, part of the strength ofthe prestressed steel is already utilizedby prestressing and only the rest, Af,, isavailable to resistT,.

4.7 Serviceability: Cracks andDeflections

If the forces in the reinforced tiesunder working loads are used and theireffective concrete area A,,ef as definedin the CEB Model Code or in Ref. 19 isattributed to them, the known relationsfor crack control can be applied direct-ly.'' •on In principle, it is proposed thatthe same model be used at the ultimatelimit state and the serviceability limitstate.

In very critical cases it may be ad-vantageous to select a model very closeto the theory of elasticity, i.e., to providereinforcement that follows the path ofthe elastic stresses. However, proper

108

failure zone AAc

ilure zonF

Fig. 24. Assumption of a failure zone for the check of the tensilestrength of a concrete tension tie T.

concrete stresses

with torture zcnefe without

failure zone A A, (crack)

Fig, 25. Progressive failure of a beam because of a local failure zone, whichincreases maximum tensile stresses a., to U., f^.

detailing (provision of minimum rein-forcement, adequate selection of bar di-ameters and bar spacing) is usually bet-ter than sophisticated crack calcula-tions.

Having determined the forces of themodel, the analysis for the deformations

is straightforward. Since the contrihu-tion of the concrete struts is usuallysmall, it is sufficient to use a mean valueof their cross section even though thisvaries over their length. For the ties,tension stiffening follows from theabove crack analysis.

PCI JOURNAL;May-June 1987 109

a) cr f

fct

^c fc

b) c) }

-1 J fct fct

tc, 0.5fc' fCf

Fig. 26. The biaxial compressive-tensile strength of concrete and twosimplified assumptions for analytical application,

4.8 Concluding RemarksDespite the fact that the major princi-

ples have been set forth in this chapter,much research work remains to be donewith respect to the more accurate di-mensioning of the concrete ties andstruts. This, however, should not pre-clude application of the proposed pro-cedure as a whole. Current designmethods of D-regions are, in fact, worsebecause they simply ignore such un-solved questions,

Following the flow of forces by strut-and-tie-models is of considerable value

even if used only to find out if, andwhere, reinforcement is needed. In astructure with reasonable dimensionswhich is not over-reinforced, the con-crete compressive stresses are usuallynot the main concern. Furthermore, it ismuch more important to determinewhere the tensile strength of the con-crete is utilized, and then to react withreinforcement if possible, than to quan-tify the strength of the concrete ties.

In the next chapter the application ofthe foregoing principles is elaboratedupon with many design examples.

5. EXAMPLES OF APPLICATION

With an unlimited number of exam-ples it might be shown, that trackingdown the internal forces by strut-and-tie-models results in safe structures andquite often provides simple solutions forproblems which appear to be rathercomplicated. It should, however, also headmitted that it sometimes takes someeffort to find the appropriate model.However, the strut-and-tie-model is al-ways worthwhile because it can often

reveal weak points in a structure whichotherwise could remain hidden m thedesign engineer if he approaches themby standard procedures.

The major advantage of the method isto improve the design of the critical D-regions. However, the authors also be-lieve that the concept will lead to morerealistic and workable codes of practicealso for the B-regions. Therefore, theB-regions are first discussed here by ap-

am

L1LIFH]Ui I

Fig. 27. Various cross sections of beams and their webs.

plying truss or strut-and-tie-models tothem. Thereafter, some U-regions aretreated (see Refs. 1 and 3 for additionaldesign examples).

Finally, the basic approach to pre-stressed concrete is discussed and illus-trated with several examples.

5.1 The B-RegionsThe plane rectangular webs consid-

ered here apply not only to beams withrectangular cross section but may alsobe part of a T, 1, double tee or box beam(see Fig. 27). The shear loading may re-sult from shear forces or from torsion(see Fig. 28), the axial forces from ex-ternal loads or prestress (see Section5.3).

As discussed above, the web of a B-region modelled with the same criteriaas proposed for the strut-and-tie modelsof the D-regions would in most caseslead to a standard truss (Fig. 8), with theinclination (f of the struts oriented at theinclination a of the principal compres-sive stresses according to the theory ofelasticity.

The design of beams for bending,shear and torsion is then nothing more

Vf

_ 0.5M1I vw + CT 1 Vw bm

Vf hm–bm—#-

Fig. 28. Shear forces as a resultof torsion.

than the well known analysis of the trussforces and the check of the compressivestresses of the concrete and the tensilestresses of the reinforcement. Since thisanalysis of the truss includes the chords,possible problems like the staggeringeffect or the question why both chordsare simultaneously in tension at thepoints of inflection (moments – 0, shearforces 4 0) are solved automatically.

PCI JOURNAUMay-June 1987 111

O Reqaoq krefeld, Thurstonq Bresler, Scorde/ s® Rujagopalan, Ferguson

Petterson• Leorhardt, Walther

0,2 • ,Yaddadrn et a!.Taylor

a Braestnup et al.^- Lyngberg, Orden, Sorensene Rodriguez et al.

Guralnick A I/ ZL G •

fs =

of I •

•

a o

• 1

,

G

0, 7 — refute ^__ • • L_i^1o^ s^^^ a / / V E ^ I a

y Q q

5' / \'gym ^'8^ A^`r • 1 R

• V0 b Z.tc

9 005 010 0,15 d ZO OZ5 030 Q35

Fig. 29. Comparison of the required amount of vertical stirrups in a beam according to experiments, to different codes and to astrut-and-tie-model analysis, corresponding to Fig. 33, Ref. 15.

It should be mentioned also, that incontrast to the bending theory, trussmodels are capable of dealing withstresses perpendicular to the beam'saxis and variable shear forces along theaxis:

(Note that these shear forces are notcompatible with Bernoulli's assumptionof plane strains as defined for theB-regions.)

Why then, if everything appears to beso simple, is there a "shear riddle" andwhy has the "shear battle" been wagedfor so many decades. In fact, the endlessdiscussions on shear could be "put tobed" if design engineers contentedthemselves with the same level of accu-racy (and simplicity) in the B-regions asthey do for the D-regions.

Since in principle it makes no sense todesign B- and D-regions, which areparts of the same structure, at differentlevels of sophistication, there would bevery good reasons for such a comparableaccuracy in every aspect of the analysis.However, at present this seems not to beappropriate mainly because of historicalreasons, i.e., since so many researchershave invested so much time investigat-ing the B-regions, they have found that(under certain conditions) savings instirrup reinforcement are possible com-pared to simple truss design.

Fig. 29 shows the well known plot (indimensionless coordinates) of the ulti-mate shear forces V. versus the amountof stirrups a„ (cm 2/m) required to carry V.for beams in pure bending and shear. Ifthe straight line according to the trussmodel with 45 degree struts (MOrsch) iscompared with the compiled test re-suIts, it is found that a large discrepancyresults, mainly in the region of low tomedium values of V,,.

The discrepancy is reduced if smallerangles of inclination of the struts thanthose taken from the elastic stresses atthe neutral axis are assumed. In fact,from test results it is observed that theangle a of the initial crack from pureshear can be up to 10 degrees less than

45 degrees, depending on the amount ofstirrup reinforcement and the width ofthe web expressed by b,,,lb. If, in addi-tion, axial compressive forces such asprestress act, a is a priori smaller than 45degrees but deviates less, that is, thesmaller a, the closer it is to the anglegiven by the theory of elasticity. 11

However, the fact remains that theexplanation of the real stirrup stressesonly by the standard truss would implythe assumption of struts with less thanrealistic inclinations, in extreme casesdown to d = 15 degrees. Other knownexplanations are also not satisfactory: Aninclined compression chord would si-multaneously reduce the inner lever armz, which is not compatible with the B-region assumption of plane strains andcan, therefore, develop only in the D-regions which extend from the supports;the pure arch or direct struts with tie ac-tion can only be applied if the tie is notbonded with the surrounding concreteor to "short" beams, i.e., beams withoutB-regions; the dowel action of the rein-forcement, though leading in the rightdirection, can only to a small degree beresponsible for the effect under discus-s ion.

What does really happen in the web?On the following pages it will be shown,that it is the concrete's tensile strengthand aggregate interlock in the webwhich really causes the reduction of thestirrup stresses. Readers who are satis-fied with this explanation or with thepure truss design of B-regions may nowjump ahead to Section 5.2. The authorswrote what follows not with the inten-tion of adding one more paper to theshear dispute, but only to show the ef-fectiveness of the strut-and-tie-modelapproach even for such cases.

After the principal tensile stresseshave reached the tensile strength of theconcrete, the web cracks at angles asdiscussed above. Consequently, fol-lowing the direction of the load, indi-vidual pieces of the web, only con-trolled in their movement by the

PCI JOURNAL.iMay-June 1987 113

al

T ^Fi C '

Q

,tr. fps;, stirrup

AFF

bI load path 1

stirrup tie-canerete strut

c) food path 2

fl r

concrete tie - concrete strut

Fig. 30. Internal forces in the web due to shear: (a) kinematics of load path 1 if actingalone (8 = a); (b) through (c) load paths 1 and 2 in the web if acting combined (0 < a).

flanges, try to fall down. There, they arecaught by the stirrups which hang upthe load via T into the adjacent pieceevoking C in the struts for vertical equi-librium (Fig. 30a). The chords (orflanges) provide horizontal equilibriumwith additional tensile forces F. This isthe principal load path 1, if the con-crete's tensile strength is disregarded(Fig. 30b).

Looking closer, it is recognized thatthe kinematics as described evoke anadditional load path 2 (Fig. 30c) whichcombines with the load path 1 (Fig. 30b)but which is usually neglected: Thevertical movement v has two compo-nents, the crack opening w perpendic-ular to the crack and a sliding A parallelto the crack (Fig. 31). The sliding A isobviously resisted by aggregate inter-lock in the crack and it appears reason-able to assume, that the resisting force Racts in the direction of ', i.e., parallel tothe crack. The force R has two compo-nents, a compressive force C, with aninclination B < a and a concrete tensileforce 'I perpendicular to it (Fig. 32).Again, the chords are activated

for equilibrium.Both load paths jointly carry the load

and therefore their combined compres-sive struts together assume the inclina-tion 0 -_ a, As long as it can be sustainedby the concrete, the concrete tensileforce perpendicular to the struts is re-sponsible for the fact that the stirrupsneed to carry only part of the shearloads. However, it also causes the con-crete of the struts to be biaxially loaded,thus either reducing their compressivestrength or resulting in a second array ofcracks with inclinations less than a, de-pending on the load case. Only if 0 = adoes load path 2 disappear (Fig. 30a).When this occurs the compressive strutsare uniaxially loaded and can thereforedevelop their maximum strength.Therefore, the maximum capacity of abeam for shear forces is achieved if thestruts are parallel to the cracks and if thecorresponding large amount of stirrupsis provided.

What can so simply be described inwords must also be accessible to a rela-tively transparent analysis. This is pos-sible, if the compatibility between load

114

paths 1 and 2 is solved by plastic su-perposition of both the stirrup and theinterlock forces, making use of the factthat aggregate interlock is sufficientlyductile. However, since a biaxial failurecriterion for the concrete has to be ap-plied, a reproduction of the analyticaldescription (Ref. 15) would go beyondthe scope of this paper. The result isplotted in Fig. 33 and compared therewith results from experiments.

It is important to note that the beam"remembers" the initial inclination a ofthe diagonal cracks. This permits thelogical introduction of the effect of pre-stress into web design, resulting in sav-ings of stirrup reinforcement but also inan earlier compressive strut failure.

The analytical curve in Fig. 33, whichgives also the actual inclinations B - a asa function of V, must be cut off before itintersects the abscissa (where 0 = d2),because a minimum of stirrup reinforce-ment is necessary in order to guaranteethat a truss model can develop at all. Asshown in Fig. 34, it is necessary to avoidthat the flanges separate from the webafter a diagonal crack has formed.

Of course, the longitudinal and trans-verse spacing of the stirrups must furtherbe limited to ensure a parallel diagonalcompression field. If the spacing is toolarge, the smeared diagonal compres-sive strength may be less than that takenfor Fig. 33, because the stresses con-centrate in the nodes with the stirrups(Fig. 35).

To use load path 2 in the practical de-sign of the webs of B-regions — if it isconsidered desirable to be more sophis-ticated there than in the D-regions,where aggregate interlock or the con-crete tensile strength is neglected -simple diagrams may be derived fromFig. 33. Even easier (and correspondingto the present CEB Model Code), it mayhe sufficient to reduce in the "low shearrange" the acting shear force for stirrupdesign by an amount which is attributedto load path 2.

For that purpose, it may be sufficient

v =vertical movementw=crock openingti =sliding

Fig. 31. Displacements in the web becauseof the crack.

crack face

Tc R cL1C TcR

Tc = resultant of the tension field

Cc= resultant of the compression field

R = force of the aggregate interlock

/

R

Fig. 32. Aggregate interlock force R andcorresponding compression C. and tensionT, in the concrete.

to replace the curved branch of the dia-gram in Fig. 33 by a straight line. It isonly important that the design engineeris aware that reliance is- placed on theconcrete tensile strength in the web ifthis method is used to reduce the re-quired amount of stirrups. Fig. 29 corn-pares the amount of stirrup reinforcerment derived in this way from the strut-and-tie-model with that from differentcodes and tests.

One might also ask whether the fulldesign strength f,d can be exploited inthe compression chord of a beam, be-


pv fms s , bfc

0,40

4 Broestrup MW . Nelsen M P . Bach F, Jensen B.0• Bresler B , Scabelis AC.

0,35 — • Gurnlnik $4• Hoddodm M J , Hong ST . Mattock A H

Krefeld WJ , Thurston CW0 . 30 --- A Leonhordt F Walther R .--

° Lynberg BS , Ozden K , Sorensen HC °Petterson T / °

0 . 25Ramapo lnn KS , Ferguson P M ,

-- -- o Regan PE -- f^_-0 RodrquesJJ,BionchmiAC,Viest1M,

Kesler C E // /020 _ o Taylor R.

6 0

0,15 — — — ^0l 7• 5 df _ off'

010 — a

o0/// - 250 0 • -O,OS^/o. o - A

19. b z f c

° 0,25 0,0 0,35 0,400,05 010 015 020

Fig. 33. Required amount of vertical stirrups in a web according to the strut- and-tie-model(for crack inclination a = 38 degrees), compared with simple truss analogy and tests.

cause the chord is subject also to tensilestrains in both transverse directions. Inthe horizontal direction such stressesare quite obvious in the flange of T-beams with transverse bending andtransverse tension from the flange con-nection. In addition, there is verticaltension from compatibility with stirruptensile strains. Some cross tension alsoexists in both directions in rectangularbeams (Fig, 35).

The neglect of this fact in practical de-sign is to some extent balanced by theusual neglect of the staggering effect inthe compression chord, which reduces

the forces in the compression chord asmuch as it increases the tension chordforces. However, if the chord forces arecorrectly derived from the truss model,the compression strength should be as-sumed to be only 0.8 ff, according toSection 4.1.

Consider, in addition, a special butvery frequent B-region, namely, beam orcolumn with rectangular cross sectionloaded by an axial force in addition toshear forces and bending moments. Thiscase is typical also for prestressedbeams. If the axial force is large enoughto keep the resultant of the normal force

compression chord

^— stirrup yweb c—

T s 1s

stirrup r webC

tension chordT5 • CC 20

Fig. 34, A minimum amount of stirrups is necessary to tietogether the web and the flanges.

a) .........__— ^_........ _ b

^ / iJI

Fig. 35. The compression strut in the web with stirrups.

and of the moment within the kern ofthe cross section, the standard trussmodel no longer applies. Instead, allinternal forces, including the shear forceV, may be represented by a single in-clined compression strut as in Fig. 36b.

Looking closer at the transition be-tween such special B-regions and thoserepresented by the truss model, it isfound that the compression stresses dis-tributed over the whole section have toconverge into the narrow compression

chord of the adjacent B-region, therebycreating transverse tensile stresses.These stresses can be assessed by themore refined model in Fig. 36c. Stirrupsmay be used to cover them; however,the model shows that tensile forces areinherently quite different from the shearforces in other B-regions but similar tothose in the bottle shaped compressionstrut.

Speaking of B-region design, thereremains the issue of "beams without


a) moment

shear force

compression force

a

^^ nA I

resultant withinkern-zone

resultant within lever arm z

model compatible with the standard truss model

C)

± ,^^ IkernzoneLi1'[JEforces of stirrups

d).

forces of stirrups

Fig. 36. Beam with compressive axial load: (a) sectional forces; (b) simplified model witha compression strut between the trusses: (c) refined model for a rectangular crosssection with vertical tensile forces; (d) refined model for a I-beam.

118

shear reinforcement." Subdividing suchbeams into B- and D-regions first of allreveals that an arch-and-tie explanationonly applies if the two D-regions of theopposite supports touch, leaving no sig-nificant B-region in between (see alsoFig. 38a). Unbonded tensile chord re-inforcement also stimulates arch-and-tieaction, in other words it increases thelength of the D-regions. The web of areal B-region, however, can carry shearloads only with the help of the con-crete's tensile strength. There are sev-eral approaches based on rational mod-els to exploit this finding.12.16

Again, as above, aggregate interlock isinvolved and, therefore, an analyticalsolution is rather complicated. On theother hand, the design engineer mustknow and understand what actuallyhappens. Thus, it would realistically re-flect the present situation to replace thepermissible shear stresses of the currentcodes by the equivalent permissibleconcrete tensile stresses. Unfortunately,the stresses in the web are not uniformlydistributed. Therefore, calibrated aver-age admissible tensile stresses must begiven in codes.

For completeness, it should be men-tioned that a B-region contains severalmicro-D-regions, which again can beunderstood and designed with strut-and-tie-models. Fig. 35b shows why astirrup must he closed on the top andbottom in a rectangular beam in order totake the cross tension from the strutsupport.

5.2 Some D-Regions5.2.1 Deep Beam With a Large Hole(Numerical Example)Given:Dimensions (see Fig. 37a)Factored loadF, F = 3 MNConcrete compression design strengthff,1 = 17 MPaReinforcement design yield strengthf„d = 434 MPaRequired:

Strut-and-tie-modelForces in the struts and tiesDimensioning of ties: reinforcementCheck of stresses of critical struts andnodesReinforcement layoutSolution:

(1) External equilibrium: reactions

A=3x --= 1.07 MN7.0

B=3x 4=5 = 1.93 MN7.0

F=A +B– 3.00 MN

(2) Elastic stress analysisThis is a rather complicated structure.

If the design engineer is not yet suffi-ciently experienced with modelling, hewill first employ an elastic finite ele-ment program and plot the elastic stress-es (see Fig. 37b) for orientation of thestrut-and-tie-model.(3) Modelling

The whole structure is essentially oneD-region. Two short B-regions are dis-covered in the linear parts to the left andbelow the hole (see Fig. 37c).

The load path connecting the reactionB and its counterpart within F is readilyplotted (see Fig. 37c). The positions ofNodes 1 and 2 are typical (see Fig. 15)and the forces C and T balancing thesenodes in the horizontal direction arethus also given.

Before continuing modelling on theleft side, the right side may be finished:The strut between Nodes 1 and 2 willspread and cause transverse tensileforces as sketched in Fig. 37d; option-ally, it can be treated as a bottle shapedstress field as sketched in Fig. 371i anddescribed later on.

Now the left side's boundary forcesare clearly defined (Fig. 37e) and it can,therefore, be modelled independentlyfrom the right side. (In passing, it maybe mentioned that the fictitious separa-tion line between the two sides is wherethe overall shear force of the deep beamis zero and the bending moment is

PCI JOURNAUMay-June 1987 119

• o 7

I FF-3MN

O i

A 8

41.5t ^5 25 X05

70

a) dimensions [m] and load

i i Ii' \ ti•\ \ \\\

tiJI__

compression stressestens on stresses

b) elastic stresses

120

B

c) load path, right side

I1 \

^ t

d) complete model, right side

Fig. 37. Deep beam with a large hole.


A

C

e) boundary forces, left side

Bi

IBZ-rNôn

A

f) model 1, left side

).5A

105E

0.5 T

A2=05A

CZ=05C

I ` it

I 'I

I jl

I T4 ^

4 50 T =05T

ArO5A

g) model 2, left side

122

node 2

arpa

- d

'(b.net

bottle

h) complete strut-and-tie-model

I-2 t5#4-J i---4 2x7#5

L^LTI I L^`L 11

i) reinforcement

Fig. 37 (cont.). Deep beam with a large hole.


moaei

ref ined sump€ohed

n1

Tt -s fig 74 tiÂ h

1 1 Bi

tl

cl

♦- t

d)

et ^?

e er e ^^-

stirrup forces

main reinforcement

C-- 1î

.rlHH!

N

Fig. 38. Load near the support; transition from beam to beam.

maximum. Therefore, only the horizon-tal forces 177 =1 CI connect the twosides.)

Forces C and A meet at Node 3 andtheir resultant is given. From the bottomthe reactionA enters the structure verti-cally and it is assumed that it remains inthis direction until it has passed thohole. The B,-region is thus a centricallyloaded column.

In fact, part of the reaction A couldalso he transferred via the B 2-region bybending moments and shear forces.Comparing the axial stiffness of B, withthe bending stiffness of B 2 , this part isobviously negligible. This fact is alsoconfirmed by the elastic stresses, whichare very small there (Fig. 37b). Ofcourse, some nominal reinforcementwill be provided in the member belowthe hole for distribution of cracks due toimposed deformations.

Figs. 37f and 37g give two differentstrut-and-tie-models for the Ieft side ofthe deep beam. It is seen that there issome similarity with a beam having adapped end. The model in Fig. 37f isbased on 45 degree struts and on tiesalso at 45 degrees from the struts. Thisgives a reinforcement layout, which forpractical reasons is orthogonal.

The model in Fig. 37g assumes a 45degree tie at the corner of the holewhich is known to be effective in similarcases like opening frame corners ordapped beams. Each model in itselfwould be sufficient but - looking at theelastic stresses of Fig. 37b - a combi-nation of both appears to be better thaneither of them. Therefore, it is assumed,that each model carries half the load.

Finally, Fig. 37h shows the superpo-sition of both models of the left side in-cluding the right side as described be-fore. When comparing model and elasticstresses one finds a satisfactory coinci-dence. The geometry of the model is in-deed oriented at the elastic stress fields.

(4) Design of the tiesThe reinforcement requirement for

the respective tie forces is given in thetable below.

TieForce[MN I

Req'd.A, [cm 2 l Use (c? Fm I

T 1.07 24.6 8#7

T, 0.535 12.3 4#7

T., 0.535 12.3 4#7

T, 0.535 12.3 2x7#5 0.08_-T 0.535 12.3 2x5#4 0.3__-T2 1.07 24.6 2x7#5 0.08_-'I' 1.07 24.6 2x7#5 0.08_-1 7 0.535 12.3 2x5#4 0.3__-1', 0.535 12.3 2x7#5 0.08_-T9 0.663 15.2 4#7

T, ° 0.402 9.2 or bottle check

(see below)T„ 0.402 9.2

Note:! cm 0.155 in.

(5) Check of the concrete stressesStresses under the bearing plates:

3.0 - = 10.7 MPa < 1.2f.40.7 x 0.4

(biaxial compression)CrA_ 1.07-=.).4MPa<0.8fd

0.5 x 0.41.93

0 8 = ----=9.7MPa<0.8ff40.5 x 0.4

(compression with transverse tensionfrom tie)

The most heavily loaded node is thesupport B. Following Fig. 22a and thedetail of Node 2 in Fig. 37h:

9.7Pa = v^ =

[ 1 + 0=5 tan 29° l cost 29°l 0.5 1

= 8.1MPa<0.8f4

Bottle shaped stress field at right side:pa=8.1MPapa= 8.1

=0.48fed 17From Fig. 23a for a = 0.68 m and b

2.0 m and hla = 3.4, the permissible


p . 1ffa for uncracked plain concrete is0.62 > 0.48.

The stress field is safe without trans-verse reinforcement. If no dependenceis placed on the concrete's tensilestrength, the reinforcement as given byFig. 23a or, equivalent, as calculatedfrom T,,, and T„ may be provided.Chord C under the loaded plate:

C–Ir1-1.07 MNRequired depth of compression zone:

d 1.070.4x1.2f,,,

1.07

0.4 x 1.2 x 17= 0.13 m < 0.4 m

Nodes I and 3 were taken 0.2 m belowthe upper surface and are the centroidsof the rectangular compression zonewhich has therefore an available depthof0.4 m.(6) Check of the anchorage lengths ofthe reinforcing bars

The anchorage length lb , ef of the tie Tin Node 2 must comply with the anchor-age length given in the code, consider-ing the favorable transverse compres-sion from Reaction B. Anchorage beginsat the left end ofthe bearing and extendsover the whole length of the bearing.Accordingly, T3 is anchored with crosspressure over the column type B1-regionto the left of the hole, Less favorablestress conditions with transverse tensionexist in the node where T, and T, meet.Therefore, the T,, bars should be loopanchored whereas the T, bars passthrough those loops and find sufficientanchor length in the B,-region below thehole. Anchorage in the other nodes isless critical.

(7) Reinforcement layoutIn Fig. 37i only the main reinforce-

ment resulting from the above design isshown. The design engineer should pro-vide further reinforcement, such as amesh on either surface of the wall, nomi-nal column reinforcement at the left ofthe hole and stirrups below the hole.

5.2.2 Transition from Beam to DeepBeam —Variation of Span-to-DepthRatio

If a single load is applied at a distancea C h near the support (Fig. 38a), theload is carried directly to the support bya compression stress field; therefore, noshear" reinforcement is required be-

tween the load and the support. How-ever, the transverse tensile stresses inthe compression strut may call for diag-onal reinforcement or vertical rein-forcement instead.

With increasing a, this single strutmodel gradually blends into the trussmodel as shown in Fig. .38b, c. Follow-ing the principle of minimum strainenergy, the compression member C, andthe tensile member T, (Fig. 38a) of thechord combine (Fig. 38b) and therebycancel the compression force C, and anequivalent part of the tension force T1.

Simultaneously, the transverse tensileforces of the strut (Fig. 38a) blend intothe vertical ties of the truss model (Fig.38b), which are now also needed forhanging up the shear forces.

In the typical truss models on theright sides of Fig. 38c and d and of Fig. 8the inclination B, of the support strut isshown to be much steeper than 8 of theinterior struts. This can be proven by theprinciple of minimum potential energyif models with different strut inclina-tions are compared. Only the geometri-cal relation:

cot H, =a + c B2z 2

provides constant vertical web tensiont#, or equal distances e between all verti-cal ties and thereby distributed stirrupforces which are proportional to theshear forces (Fig. 39). The force in thetensile chord would have to increasenear the support, if 9 were constantthroughout, or in other words, thesupplementary chord force from a con-stant shear force is constant along the

126

i,i Tw=V w i

1az•cof9

0) Tw=V

rcote ----

T ry TW=V I^ z^ w

T

111111' 4hhhlhhhl Iw=V

b^ Tw=V

cot 9t

cot"^= z

Fig. 39. Detailed geometry of D 2-region from Fig. 38:(a) direct support; (b) indirect support.

chord only if B, satisfies the above givenrelation.

5.2.3 CorbelsSome of the so-called problems of

corbel design result simply from thefact that the individually differentboundary conditions are not con-sequently considered. Each case leadsto a different strut-and-tie -model (Figs.40 and 41). Furthermore, from Fig. 40dit becomes evident that the compressionstrengths of the struts, which accordingto Fig. 23 are a function of alb, dependvia a, ,, on the detailing and via a, s and a,Ron the depth x of the bending compres-sion zone of the column.

The modelling of a deep corbel isshown in detail in Fig. 41: The stressesin the column of Fig, 41a may beevaluated according to standardmethods or the theory of plasticity andare then applied to the D-regions to-gether with the corbel load F 2 . The factthat the vertical struts at both sides of

the B 2-region in Fig. 41b do not coincide(the stress diagrams, of course, do) isdue to the different grouping of stressesfor the struts with respect to modellingof the D-regions D2 and Dg.

Balancing the applied forces betweenthe two boundaries of each D-region(Fig. 41b) shows that two tensionchords develop, the first one as us-ual, right below the upper edge of thecorbel, and a second one in the lowerpart of the corbel. Their forces are in thiscase T, = 0.23 F2 and TZ = 0.32 F2 , re-spectively.

The compression struts of the samecorbel are not critical because the sing-ular nodes in the D, and D, regions withmaximum compressive failure strengthsof 1.2f according to Fig. 23 keep thecompressive stresses in the rest of thestructure relatively low.

5.2.4 Details of a Box Girder BridgeThe critical D-regions of a box girder

bridge are described by strut-and-tie-

PCI JOURNAL+'May-June 1987 127

0 ] 1 i1 i

1 ,I ;

\ I,1 '^

i $L

b) $ fr,

• i

^^ t r

. .

ssary

f•

1 ^

f /

1

• I. 1,' I

it/ 1

_J. 1

t

3

I p

m

128

e)

♦ R45I,\

4 ti

♦ i r

T.Tr .

Fig. 40. Different support conditions lead to different strut-and-tie-models and differentreinforcement arrangements of corbels.


models as shown in fig. 42.In Fig. 43 the diaphragm is pre-

stressed: The shear forces V, carried bythe diagonal struts C, in the webs of thebox girder (Fig. 43a) are in this casetransferred to the central support via aprestressed web reinforcement de-signed for T, = V,/sin/3 = 1.01 V, andcurved tendons designed for:

T2 = 0.82 (2 V,) in the diaphragm.

5.2.5 Beam With OpeningFig. 44a shows the sectional forces

and the subdivision of the beam into B-and D-regions. Following the proposedprocedure, the boundary forces of the

B-regions have to be applied as loads tothe D-regions.

The B 1-region will be represented bythe standard truss, which yields for 9 =31 degrees:

C, M,,= – cot (z 2

_ V,C^ sin B

T, = M' + 21 cot 9z

_ 3,55F F,1.15 + --1.66

= 3.92 F4

850KN

B^

F2=425 KN Fz

1 LJTy r110,

, o It ° 3 Tz

04 b) c)'

t

Fig. 41. Deep corbel projecting from a column: (a) B- and D-regions; (b) boundary forcesof the D-regions and their models; (c) reinforcement layout.

130

The B.1-region is designed for a con-stant tensile force:

TZ ,6M1 15 ^ – 5.35 FaThe lever arm z = 1.15 m is deter-

mined from the assumption of planestrain in the cross section in the middleof the opening by standard methods.Therefore, B., has to carry the axial com-pressive force C = T,, (eccentrically withrespect to the axis of the B 2-region) plusthe differential moments AM = M – M ,plus the total shear force V.

Under the combined action of theseforces, the B, region shows the transi-tion from the column type B-region onits left end (resultant C 3 ) to the trusstype B-region at the other end (C g , C5,T3 ) (see Fig. 36). For simplicity, themodel of the B,-region is extendedsomewhat into the D2 and D3-regions,leaving over for modelling only the D,'and D3 regions in Fig. 44b.

With known boundary conditionsfrom B1 , B, and k, the model for the D,-region can be developed (Fig. 44c):Looking for the counterparts of C„ C2and T, at the opposite side of the D,-region it helps to split up C3 into threeforces C , C' and C'.,' in order to estab-tablish the load paths, which balancetheir horizontal components: (C 3'5 = C1;

,_Vertical equilibrium in the D2-region

is established by a vertical tension tie TKand by a vertical compressive strut C.Their forces depend on the choice oftheir position. Knowing that they repre-sent transverse stresses which are insidethe D 2-region and that these stressestend to fill the available space, the re-sultant tension TS is chosen in the mid-dle of the D2-region and the resultantcompression C. at the right end of D,,(which is inside D2 ). Then:

Ts = (T2 – T1 ) tan B,

= (5.35 Fd – 3.92 F,,) tan 46°= 1.48F4

The tie force T5 may be interpreted asthe transverse tension necessary to an-chor the differential force T, – T l of the

beam's tension chord.In a similar way, the D;-region at the

other end of the opening may be treated.The transverse tension forces are:

T,=V =FdandTg =(T,–T.) tan H2

=(T,–T2 –V cot 0)tan H2

= (8.79 F4 – 5.35 F, – 1.66 F4) tan 50'= 2.12F4

A striking result of this example arethe stirrup forces which in some placesconsiderably exceed the "normal shear"reinforcement for a beam without open-ings (Fig. 44d).

This example also shows, however,that it sometimes takes some effort todevelop a realistic strut-and-tie-model.But if it is considered how many testshave been carried out and how manypapers have been written on beams withopenings without finishing the case, itwas worth this effort.

5.2.6 Overlap of Prestressing Tendons

In Fig. 45a the flow of forces in thegiven cross section is investigated withits B- and D-regions. In a longitudinalsection through the blister, the B-regionsof the cross section may also appear asD-regions, because the curvature forcesof the tendons are line loads.(a) Prestress only

Loads acting on the D-regions are theanchor forces of the tendons P. theknown concrete stresses along theboundaries hetween B- and D-regionsand the curvature forces Pa l from thetendons (Fig. 45b). The horizontal com-ponents of all forces follow two loadpaths, whose position and direction isgiven at all D-region boundaries.

The relatively small shift of the loadpath in between must be accomplishedby deviative forces within the D-regions. This is a rather strict condition,which limits the freedom of choice forthe bends of the load paths. A furtherhelp in finding appropriate nodal pointsof the model is again the tendency of thedeviative forces to fill the available


compress ionTrz3tensron

WE

!III.IIII! b) compression

rT r T r7 p

^ I I I ^ D

I I I

I I I

1 1 !, SI k

I •ft 1

I I 1^ I ^1'

I I I I It BI ^ I I

B

C^

d)

.'v

^ 5

r ^

II I!

"T ^r

I I

1 t

132

'►C 1 a^_ I

t F I 71

IIu I I I.

e)

BI II I I

k T I

1 11 l

1 1t ^ B

Fig. 42. Strut-and-tie-models of typical D-regions of a box girder bridge: (a) tensileflange with opening; (b) compression flange with opening; (c) web supported bydiaphragm; (d) pier and diaphragm with single support; (e) other model for diaphragm;(f) pier and diaphragm with two supports; (g) pier on a pile cap.


a}

b) 2V; T2 =Tp • ltrn, 2v.

L1 T±J

[)

IC

Fig. 43. Diaphragm of a box girder bridge: (a) D-regions andmodel of the web near the diaphragm: (b) diaphragm andmodel: (c) prestressing of the web and the diaphragm.

space of the D-region.Then, T, = P tan a2 = 0.18 P (see Fig.

45b).The reinforcement for Ti should be

chosen with due consideration of ac-ceptable crack widths in this service-ability state of stress, i.e., the tensilestrength of the reinforcement should notbe fully utilized and its slipfree anchor-age should be accounted for.

(b) Prestress plus other loadsTensile forces are assumed in the B,-

regions, which are considerably largerthan the prestressing force, as is the casein an overloaded or partially prestressedstructure. In case there is no additionalreinforcement provided, and the tendonhas to take all tension forces the excessforce (T h0 ,( according to Section 5.3.6)was computed to be:

134

a) _ 107 °

Z c5 1375kN

mC, B' 0Z 03 CC 04

115;0 Fd 6.15 â2 +

moments M

lvi

y z

shear forces V

O

b}

C ^ C^ C4 C6C^^ DZ C3 C3

^ B^ - --^ ^ a3 C7'^ T3

T, TZ Tz T

C)

f EF---T:::;4 i:j: iz T,

T8

d) stirrupsfl T5=2.1F,

__ __T5 05Fd

_I n IDF

—stondird design B-regon

Fig. 44. Beam with opening: (a) B- and D-regions, sectional forces: (b) reducedD-regions at both ends of the opening with boundary loads from the B-regions:(c) strut-and-tie-models of the 0 2-and D,-regions; (d) distributed forces for thedesign of the stirrups; (e) reinforcement layout.

e)


T,=T-P=0.33PThe load path for T R and the resulting

model (Fig. 45c) is now quite differentfrom that for prestress alone, becausethe TL on both sides do not meet, butrather try to equilibrate on the shortestpossible way with the nearby anchor

loads of the other tendons. Thereby,transverse tension T, develops.

The position of T, may have to beshifted toward the center of the blister,if the bond of the tendon is very poor. Itis the tension from these bond forceswhich now calls for transverse rein-

0)g2 â1 0I ^-

P- P

curvature forces P a j Zn, T nag

c)

TzzT-P

crock

T1

Tz

r7m:

INNNI!!!h oil- iii.Fig. 45. Overlap of prestressing tendons: (a) layout, B- and D-regions; (b) model forprestress only; (c) model for prestress and additional load T d ; (d) layout of the transversereinforcement.

136

forceinent T, = 0.53 T2 , not the splittingaction as for prestress alone.

The overlap of the prestressing ten-dons results in twice the prestressingforce within the blister, which thereforeremains basically uncracked and rela-tively stiff. The strains of the tendon inexcess of the initial prestress willtherefore accumulate within the bondlength of the tendon in the blister andcause a crack at the jump of the wallthickness.

For large bond lengths this crack may

open several millimeters wide, even ifthe B 1-region is reinforced for crack dis-tribution in the usual way. Therefore, inthis case not very large tendons withgood bond properties have to be appliedor additional parallel wall reinforcementmust be provided, which takes overmuch of the tendon forces, before thetendons enter the blister.

5.2.7 Beam With Dapped EndIt is common practice to suspend the

reaction F of the beam in Fig. 46 besides

tII Lf -T =F +]

II

3-'ice

F s^ ^^

11r-11— 13=zcot*—+144414 1444111 lUrlill

r t3=1z^

ILL!1Fig. 46. Beam with dapped end.

al Tff] T______ fll+ifr

^c a^

bl

Fig. 47. Girder with bent top flange.

PCI JOUJRNALMay -June 1987 137

rig. 48. 1 he stepped beam.

the dapped end (T, = F). But the com-plete strut-and-tie-model clearly re-veals that it is not sufficient to simplyadd T, to the regular "shear reinforce-ment" which amounts to the vertical tieforces t 3 = F/1 3. In fact, there are addi-tional vertical tensile forces T 2 = F be-cause the horizontal tie force T at therecess needs to be anchored. The tieforce T2 is distributed over a length 1 2 <l 3 and therefore t 2 is clearly considerablylarger than t 3. If, as usual, an additionalhorizontal force H acts at the recess, thenecessary amount of vertical stirrupsfurther increases.

5.2.8 Tapered Beam With Bent TopFlange

The girder in Fig. 47a obviously pro-duces a vertical tension force T at thebend of the compression chord. Butwhere does it go? The straight horizon-tal tension chord cannot equilibrate it.The model shows that stirrups in theweb are necessary throughout this webeven in regions without shear forces.

Looking at Fig. 47b, it is apparent thatthe compression chord is narrowed bythe stirrups, resulting in a concentrationof compression stresses over the web.Furthermore, unfavorable tensilestresses in the transverse direction ofthe flange appear.

5.2.9 Stepped BeamThe stepped beam in Fig. 48 is fre-

quently used and is usually detailed by

overlapping the reinforcement comingfrom both sides with an elegant loop.The strut-and-tie-model supplies factsfor a rational reinforcement layout.

5.2.10 Frame CornerThe frame corner with opening mo-

ment, more often discussed by re-searchers than actually occurring inpractice, can be modelled quite differ-ently (Fig. 49). Obviously, the designengineer has to choose between a rela-tively simple reinforcement combinedwith a reduced moment capacity (Fig.49a,b) or a more sophisticated solution(Fig. 49c,d,e). The consequent applica-tion of strut-and-tie-models makes thedesigner aware of what is occurringwhile offering a rational choice.

5.3 Prestressed ConcreteAs a last example, it will be shown

that looking at prestressed concretebeams through strut-and-tie-modelshelps to understand their behaviorwhich today gets hidden behind somany black box rules. There is a commondenominator of all types of prestress:post-tensioning, pretensioning and un-bonded prestress can be understood asreinforced concrete which is loaded byan artificial loading case, i.e., prestress.

As any other Ioading case, it simplyhas to be introduced into the analysis ofthe structure according to the actualhistory, e.g., for post-tensioning: con-

138

b1 _ >H fti__ ri $ILJP

d? ^

J l

Fig. 49. Different strut-and-tie-models and the corresponding reinforcement for a framecorner with positive moment.


creting and hardening of the reinforcedconcrete, applying the prestress(thereby activating dead load), provid-ing bond and imposing external loads.After bond is activated, the prestressingsteel acts as reinforcement, like regularreinforcement does; only its preloading,its different surface with respect to thestrength of bond and its sensitivityneeds to be taken into account.

5.3.1 Prestress as a LoadBy prestressing, forces are artificially

created with the help of hydraulic jacks;on the one hand, these forces act asloads on the prestressing steel, on theother as loads on the reinforced concretestructure (Fig. 50). The loads acting onthe prestressing steel and on the rein-forced concrete are inversely equal. Thedesign engineer chooses the tendonprofile, the type and magnitude of theprestressing force in such a manner thatthese artificial loads influence the loadpaths and sectional effects or stressesdue to the actual loads (dead and liveloads and other loads) favorably and asefficiently as possible.

It is proposed to treat these pre-stressing loads like permanent loadswhich never change after the prestress-ing jack has been removed. All thechanges of stress in the prestressingsteel, which occur after removal of thejack, ought to be attributed to those loadcases which cause them. In those loadcases the prestressing steel adds to theresistance of the section or member likenonprestressed steel.

The view sometimes expressed thatthe prestressing itself increases underlive loads because the stress in pre-stressing steel increases or decreases asa result of the bond with the concrete, ismisleading. The stress also changes innormal reinforcing steel due to theseeffects and one would never regard thisas a change of prestressing.

Accordingly, the changes of stress inthe prestressing steel due to creep,shrinkage and steel relaxation (often

referred to as prestress losses) are inreality simply stress redistributionsas in any reinforced compressedmember and can be treated accordingly.

If prestressing is introduced in thisway into the design of reinforced con-crete, all types of prestressed structures(linear members, plates, deep beams,shells) can he designed, analyzed anddimensioned like reinforced concretestructures: The sectional forces of a pre-stressed beam are determined and com-bined for the load cases prestress, deadload, live loads, etc., and the resistanceof a cross section is derived as for re-inforced concrete with prestressingsteel as additional (passive) reinforce-ment.

Thereby, the same method of analysisand dimensioning can be applied to allload combinations in serviceability andultimate Iimit states. Consequently, inthe ultimate limit state the reinforcedconcrete sections (with reinforcementA,andA,) of a prestressed beam have tobe dimensioned for the following (ac-tive) sectional effects:

Normal forces: N = –P + NfShear forces: V = V,, + VLMoments: M = M„ + ML

where

P = prestressing force immediatelyafter prestressing

A1, Vp = corresponding moment andshear force due to pre-stressing

N,,, VL , M , = sectional effects fromother loads

The sectional effects given here aremeant to include the (partial) safetyfactors according to the chosen safetyconcept.

The proposed treatment of prestress-ing leads to the same results as does theusual method with all stresses in thetendons regarded as being passive inthe ultimate limit state. However,the method proposed here is more gen-eral.

7

140

e} E'i :iFig. 50. Loads due to prestressing (anchor forces, friction forces,deviation forces due to the curvature of the tendon) acting (a) on theprestressing steel; (b) on the reinforced concrete member.

Also, the different degrees of pre-stressing (full prestressing, partial pre-stressing, no prestressing) and differentapplications Iike pretensioning, post-tensioning and unbonded cables can allbe treated alike with these principles:Forces due to prestressing as permanentactive loads which never change andprestressing steel contributing to the re-sistance after the cable is anchored.

5.3.2 Prestressing of Staticallyindeterminate Structures

In structures with statically indeter-minate supports the properties of thematerials and the geometry of thestructure have to he considered whendetermining the sectional effects. Thosefrom prestressing can be split up in:

V„= P sin 6+V,M,= –Ve +M,12

where S denotes the inclination of theprestressing cable and e denotes the ca-ble's eccentricity. V., and M me , the stat-ical indeterminate portion of the pre-stressing effects resulting from supportreactions due to prestressing, are of thesame kind as the statically indetermi-nate moments which result from deadloads or live loads. M, is not the resultof restraints such as those due to

changes in temperature or settlements;these diminish or even disappear in thewhole structure if the stiffness decreasesdue to cracking, those from prestressingare only redistributed.

If it can be assumed that the stiff-nesses are not altered by loading, mo-ments resulting from different loadingcases may be superimposed. However,because part of the reinforced concretegirder passes from the untracked to thecracked state and because plastic de-formation of the concrete and the steelin the reinforced concrete girder is pos-sible, the local stiffnesses change withvariations of the load and, therefore, thedistribution of moments also changes.When this occurs the individual mo-ments can no longer be superimposed.Nevertheless, if the theory of linearelasticity is taken as a basis for calculat-ing the forces, the resulting overall mo-ments (including M1 ) can be adaptedto the real loadbearing behavior by re-distribution.5.3.3 The Prestressed Concrete BeamWith Rectangular Cross Section

In Fig. 52 the strut-and-tie or the trussmodel of a simple prestressed concretebeam with a straight eccentric tendon is


shown and in the preceding Fig, 51, forcomparison drawn in the same manner,of a reinforced concrete beam, The de-sign engineer selects the prestress P andcombines it with the support force A = Fto a resultant, R, entering the beam atthe support.

If the resultant meets the action lineof the load F within the kern of the sec-tion, the condition is full prestress, if Fis the working load F. Then the modelhas no tensile chord (Fig. 53). If the re-sultant meets the compression chord ofthe beam before it meets F. such thatvertical ties (stirrups) and, therefore,also a truss with a tension chord isnecessary to transport F. into the in-clined resultant, the condition is partialprestress (Fig, 52).

It is apparent that the vertical tie andinclined strut forces of this truss are (al-most, see below) the same as for thereinforced concrete beam, because theshear forces are the same. The span ofthe truss is only shorter and, therefore,the forces in the tensile chord are less,whereas the compression chord also in-cludes the additional prestressing force.

If the load is increased from F,,, to theultimate F,,, of course, the inclination ofthe resultant R increases. The fully pre-stressed beam now becomes "partially"prestressed, the initially partially pre-stressed beam gradually approaches thereinforced concrete beams. (This, inci-dentally, shows that manipulating safetycompletely via a factored load is mis-leading in contrast to partial safety fac-tors being put on the material and theload.)

Full prestressing transforms the gird-er under service loads into a "horizon-tal column" (Fig. 36); its eccentric nor-mal force is generated artificially andthe external load which actually has tobe carried by it is relatively small.Whatever amount of prestress P is cho-sen, it shortens that part of the girder,where a truss must form to carry the loadand replaces it by a direct and shorterload path. It is, thus, directly apparent

that prestressing improves the load-bearing behavior as compared to thenonprestressed reinforced concretegirder.

Since prestress does not fully utilizethe strength of the high tensile steelused for tendons, it can be used asT.I ,wY i to cover a part of the tensile forceof the chord T,.,. If the prestressingsteel is not bonded with the concrete, itis unsuitable to serve as reinforcementfor T orn. It only affects the sectionalforces in the reinforced concrete via ad-ditional anchor and deviation forces,and all tensile forces of the chord haveto he taken by regular reinforcing steelto satisfy equilibrium.

The resultant entering the beam fromthe supported end, as discussed, has thetendency to spread in the web of thebeam as in the bottle shaped compres-sion field (Fig. 52b). Transverse tensionforces develop as well as a "force whirl"in the corner (Fig. 53) which causes hightensile stresses near the anchor plate.22These tension forces have to bechecked. If they cannot be covered bythe tensile strength of the concrete (seeSection 4.5), stirrups have to be pro-vided (see Section 4.6).

Tensile edge forces, splitting tensileforces, tensile end forces, etc., in thezone of introduction of anchoring forcesare thus simply part of the strut-and-tie-model. In fact, they require no spe-cial names suggesting that they aresomething special or specific to pre-stressed concrete. The "problem" ofsuperposition of the reinforcement forthe shear force and the splitting tensileforce is resolved by the model.

5.3.4 The Prestressed Concrete I-GirderIf a beam is not plane (rectangular)

but has a distinct profile like the T, I orbox girders, with relatively large cross-sectional areas of the flanges, the resul-tant entering the beam as discussedabove, will spread on a path differentfrom that in the rectangular beam;though of course for the same applied

142

-- -- compressiontension

Fig. 51. Strut-and-tie-model of a reinforced concrete beam loadedwith two single loads.

a) result t R F edge cut

L :J r:a —T chord

moment -

shear force

normal force

b1

Detail l-1 ::::

Fig. 52. (a) Strut-and-tie-model of a partially prestressed beam with rectangularcross section: (b) detailed strut-and-tie-model of the beam area, where theresultant is within the beam section.

forces P and A the simplified model isthe same for any type of cross section(compare Fig. 54a with Fig. 53a).

The detailed strut-and-tie-model ofthe prestressed I-girder (Fig. 54) showsthat a truss already develops in the area

where the total resultant force remainswithin the kern zone of the girder sec-tion (Fig. 54b). This is because the lon-gitudinal forces here are mainly con-centrated in the flanges. Therefore, onlya part of the prestressing force can join

PCI JOURNAL, May-June 1987 143

L.1it4t _J.:k4 Hb) force whirl 1F

it pQq^^^

A

Fig. 53. (a) Simplified strut-model of a beam with its rectangular cross section"fully prestressed"; (b) detailed strut-and-tie-model.

01 F

A

blweb

c) top f[ange

.-.

centroid of the forces coming from the web

d) bottom flange

Fig. 54. Strut-and-tie-models of an I-girder with full prestress: (a) simplified model;(b) through (d) detailed models of the web, top flange and bottom flange, respectively.

144

with the support force to flow into theweb. The effective resultant force in theweb is, therefore, smaller and at agreater angle than in the prestressedgirder with a rectangular cross section.It is, however, still at a smaller anglethan in the nonprestressed reinforcedconcrete girder.

The chords in the two flanges arelinked to one another via the web strutsand ties. In this way, compression forcesare introduced into the flanges (Fig. 54c,d). The strut-and-tie-model shows thatthe spreading of the forces from thewidth of the web to the width of theflange generates transverse forces in theflange. The transverse reinforcementmust be distributed in accordance withthe length and intensity of the introduc-tion of forces.

5.3.5 The Loadbearing Behavior of theWeb

The strut-and-tie-model of the pre-stressed girder with rectangular crosssection shows, that the stirrup forces inthe part of the girder where the total re-sultant force remains in the kern zone ofthe section result from the spreading ofthe compression forces (Fig. 52b, 53b).The "shear reinforcement" in that areais in reality a "tensile splitting rein-forcement." As soon as the truss modeldevelops in the web (which as shownhappens for a rectangular beam furtheraway from the support than for a beamwith a profiled section), the dimen-sioning of its struts and ties follows asdiscussed for the B-regions of reinforcedconcrete.

Whether the web crack develops fromthe bending crack or begins in the webitself (as specified for example by theGerman Code DIN 4227 as "zones a andb") has no effect on the loadhearing be-havior of the members in the crackedweb. After the web has started to crackthe prestressing normal force is onlypresent in the compression chord, likeany normal compression force. In thetruss (or B-region) the web itself is in-

iluenced by the axial force due to pre-stressing only via the inclination a of theweb crack, which is shallower than for anonprestressed beam, corresponding tothe shallower inclination of the princi-pal tensile stresses in the concrete whencracking begins.

This effect of a can be considered inthe web design as discussed in Section5.1 (B-regions). From there it is shownthat the greatest possible inclination ofthe diagonal strut is parallel to the crackin the web. The ultimate load capacityof the diagonal struts in a web of a pre-stressed concrete girder is thereforesomewhat smaller than that of the webof a girder without prestressing, how-ever, it requires less stirrup reinforce-ment for a similar beam and load.

5.3.6 Dimensioning the PrestressingSteel for the Different Types ofPrestressing

As already mentioned above, the pre-stressing steel can and will serve as reg-ular reinforcement, if it is bonded withthe concrete, in other words it acts as thetensile chord of the truss, developed forthe structure loaded with prestress andother loads. If the capacity of the pre-stressing steel still available after pre-stressing Tp,r , r,i cannot alone cover theforce of the chord, reinforcement mustbe supplemented in such a way that thetotal chord force T rd can be taken bythe prestressing steel (p) and the rein-forcing steel (s):

Tchord = TA,ehorrt + T..char,i

In this equation To"r, represents thetotal chord force from the various loads(including prestress). The loads are tobe multiplied by the partial safety fac-tors y which are, of course, different forvarious kinds of load. The right handside of the equation stands for the re-sisting chord forces, divided by the ap-propriate partial safety factors.

The force T 5 of the prestressingsteel which is still available for thechord after prestressing is equal to its

PCI JOURNALMay -June 1987 145

permissible total force T,,,,0, minus theprestressing force yAP,

T.., rd = Tn,tat – Y"PAs a rule, under ultimate load the pre-

stressing steel is strained beyond itsyield point and, therefore, T.,,, = A ,, fnu.Then, the following simple equationapplies:

Tp, iwr,r = yPPYr

If this is not the case, fp,, in the aboveequation has to be replaced by the stressQ,, of the prestressing steel correspond-ing to its total strain

€ p,rot = €:, p + Ep.(ttorri

where= strain from prestressing (active

strain)e,, , , wrr = strain of chord after

bonding (passive strain)

__ T,,.cpw,_.z

(A,+A8)EsSome codes restrict the strain of the

chord after decompression of the con-crete to 0.005 (DIN 4227) or 0.010 (CEBModel Code). Considering the arbitrari-ness of these quantities, the small de-compression strain including the strainsfrom creep and shrinkage can be ne-glected, which means €v chord is limitedfrom 0.005 to 0,010, respectively.

II n„ hand is provided after pre-stressing, the prestressing steel cannotbe considered as reinforcement in thisway. Rather, it acts as a tic member. Itsstress increment can only be deter-mined from an internally statically in-

determinate analysis. If the truss modelcontains a tension chord, supplementarybonded reinforcing steel must be pro-vided.

If pretensioning is used and the ten-don profile is straight, the prestressingforce acts as an external normal force onthe reinforced concrete girder. The pre-stressing steel is then part of the rein-forcement.

5.3.7 ResultConsidering prestressing forces as

external loads is not only an advantagewith regard to service load design, butalso for checking ultimate load designand all other checks, because the load-bearing behavior of the entire pre-stressed concrete girder can then besimply explained in terms of a strut-and-tie-model.

Prestressing steel is used for two dis-tinct purposes. On the other hand, itsprestress applies favorable loads to thereinforced concrete girder, while on theother hand, it works as passive rein-forcement when it is bonded with theconcrete. In this latter respect it is notdifferent from reinforcing steel. Other-wise, if it is not bonded it acts as a tiemember.

As a result of this approach, the task ofdesigning any type of prestressed con-crete girder becomes the task of de-signing a reinforced concrete girderwith regard to bending, shear forces,and normal forces, which among othersinclude the additional loading case ofprestress.

146

ACKNOWLEDGMENT

This paper is a progress report of thework in this field at the University ofStuttgart. The authors want to acknowl-edge the contributions of several formerand present members of their Institute,mainly K. H. Reineck, D. Weischede,H. G. Reinke and P. Baumann.

The authors further received valuablecontributions and encouraging supportthrough many critical discussions withcolleagues; thanks go mainly toB. Thiirlimann (Zurich) and M. P. Col-lins (Toronto), who also promote theidea of a consistent design of concretestructures and to the members of theCEB Commissions concerned, in par-ticular T. P. Tassios (Athens), G. Macchi

(Pavia), P. Regan (London) and J. Per-chat (Paris).

Finally, the authors wish to thank thereviewers of the PCI JOURNAL whooffered us critical but constructive help.In particular, we wish to thank J. E.Breen (Austin) and J. G. MacGregor(Edmonton). It is by no means their faultif the paper is still a burden to thereader.

Lastly, the authors hope that thispaper will generate fruitful discussionsin the interest of producing quality con-crete structures. To this end, we wish tothank the Editor of the PCI JOUR-NAL for offering us such a prominentforum.

NOTE: Discussion of this report is invited. Please submityour comments to PCI Headquarters by February 1, 1988.

PCI JOURNAt1May-June 1987 147

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praktisches Verfahren zum methodi-schen Bemessen and Konstruicren imStahlbetonhau" (A Practical Method forthe Design and Detailing of StructuralConcrete), Bulletin d'Information No.150, Comite Euro-International duBeton, Paris, March 1982.

2. Schlaich, J., Schafer, K., "Towards aConsistent Design of Reinforced Con-crete Structures," 12th Congress ofIABSE, Vancouver, British Columbia,September 1984.

3. Schlaich, J., and Schafer, K., "Kon-struieren im Stahlbetonbau" (Design andDetailing of Structural Concrete), Beton-kalender 1984, Part II, W. Ernst & Sohn,Berlin-Munchen, pp. 787-1005.

4. Bitter, W., "Die Bauweise Hennebique"(The Hennebique System), Schweize-rische Bauzeitung, Bd. XXXIII, No. 7,January 1899.

5- Mbrsch, E., "Der Eisenbetonbau, seineTheorie and Anwendung" (ReinforcedConcrete, Theory and Application), Ver-lag Konrad Wittwer, Stuttgart, 1912.

6. Leonhardt, F., "Reducing the Shear Re-inforcement in Reinforced ConcreteBeams and Slabs," Magazine of ConcreteResearch, V. 17, No. 53, December 1965,p. 187.

7. Rusch, H., "Ober die Grenzen der An-wendbarkeit der Fachwerkanalogie beider Berechnung der Schubfestigkeit vonStahlhetonbalken" (On the Limitationsof Applicability of the Truss Analogy forthe Shear Design of Reinforced ConcreteBeams), Festschrift F. Campus "Ainici etAlumni," Universite de Liege, 1964,

8. Kupfer, H., "Erweiterung der Morsch'-schen Fachwerkanalogie mit Hilfe desPrinzips vom Minimum der For-manderungsarbeit" (Expansion ofMarsch's Truss Analogy by Applicationof the Principle of Minimum StrainEnergy), CEB-Bulletin 40, Paris, 1964.

9. Thiirlimann, B., Marti, P., Pralong, J.,Ritz, P., and Zimmerli, B., "Vorlesungrum Forthildungskurs fur Bauinge-nieure" (Advanced Lecture for Civil En-gineers), Institut far Baustatik and Kon-stniktion, ETH Zurich, 1983 (see furtherreferences here).

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Concrete Beam Design," ACl f ournal, V.82, No. 1, January-February 1985, pp.46-56 (see also Ref. 25).

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12. Weischede, D., "Untersuchungen rummethodischen Konstruieren imStahlbetonbau" (Investigations on theMethodical Detailing of Structural Con-crete), Thesis, Institut fir Massivhau,Stuttgart, 1983.

13. Reinke, H. G., -Zum Ansatz der Beton-zugfestigkeit bei der Stahlbeton-bemessung" (On the Assessment of theConcrete Tensile Strength in the Designof Structural Concrete), Thesis, Institutfor Massivbau, Stuttgart, 1986.

14. Kupfer, H., and Moosecker, W., "Bean-spnichung and Verformung der Schub-zone des schlanken profiliertenStahlbetonbalkens" (Stresses and De-formations of the Shear Zone of SlenderProfiled Reinforced Concrete Beams),Forschungsheitrage for dir Baitpraxis(Kordina-Festschrift), W. Ernst & Sohn,Berlin, 1979, pp. 225-236.

15, Jennewein, M. F., "Zum Verstandnis derLastahtragung and des Tragverhaltensvon Stahlbetontragwerken mittels Stab-wekmodellen" (Explanation of the LoadBearing Behavior of Structural Concreteby Strut-and-Tie-Models). Thesis being

prepared, Institut fur Massivbau, Stutt-gart.

16. Reineck, K. H., "Model for Beams With-out Shear Reinforcement," Work inpreparation, Institut ftir Massivbau,Stuttgart.

17. Baumann, P., "Die Beton-Druckfelderbei der Stahlbetonbemessung mittelsStabwerkmodellen" (Concrete Compres-sion Fields for the Design of StructuralConcrete by Strut-and-Tic-Models).Thesis in preparation, Stuttgart.

18. Jahn, M., "Zum Ansatz der Betonzugfes-tigkeit bei den Nachweisen zur Trag-und Cebrauchsfahigkeit von unbe-wehrten and bewehrten Betonbauteilen"(On the Assessment of Concrete TensileStrength for the Ultimate Capacity andthe Serviceability of Concrete Members

148

With and Without Reinforcement),DAfStb.-Heft341, Berlin, 1983.

19. Konig, G., "Control of Cracks in Rein-forced Concrete and Prestressed Con-crete," Proceedings 1 of the Tenth Inter-national Congress of the FIP, NewDelhi, 1986, pp. 259-268.

20. Niyogi, S. K., "Concrete BearingStrength — Support, Mix, Size Effect,"Journal of the Structural Division,ASCE, V. 100, No. ST8, August 1974, pp.1685-1702.

21. Schober, H., "Ein Modell zur Be-rechnung des Verbundes and der Risseim Stahl- and Spannbeton" (A Model forthe Assessment of Bond and Cracks inReinforced and Prestressed Concrete),Thesis, Stuttgart, 1984.

22. Stone, W. C., and Breen, J. E., "Designof Post-Tensioned Girder AnchorageZones," PCI JOURNAL, V. 29. No. 1,January-February 1984, pp. 64-109, and V.

29, No.2, March-April 1984, pp. 28-61.23. Collins, M. P., and Vecchio, F., "The Re-

sponse of Reinforced Concrete to In-plane Shear and Normal Stresses," Pub-lication No. 82-03, University of Toronto,March 1982.

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26. CEB-FIP Model Code for ConcreteStructures, Comite Euro-Internationaldu Btton (CEB), 1978.

27. Mueller, P., "Plastische Berechnung vonStahlbetonscheiben and Balken" (PlasticAnalysis of Reinforced Concrete DeepBeams and Beams), Bericht No. 83, In-stitut fur Baustatik and Konstruktion,ETH Zurich, July 1978.


APPENDIX -- NOTATION

Geometrya = width of anchor plate1, = width of compression field or

plated = length of D-regiond,, = diameter of largest aggregatee = eccentricityh = depth of beamI = span, lengtht = thicknessV = vertical movementw = crack widthx = depth of bending compression

zonez = lever arm of internal forcesA = sliding parallel to cracka = crack inclination (see Fig. 30)H = diagonal compression strut

angle= area of concrete tensile zone

AA, = area of assumed failure zone= effective concrete area for ten-

sion stiffeningA, = cross section of reinforcing steela, = cross section of reinforcing steel

per unit length_ mechanical degree of

t f, , , reinforcement

Forces and MomentsC = compression force, compression

strutF = loadM = bending momentM T = torque

M„2 = statical indeterminate portion ofmoment from support reactionsdue to prestressing

P = prestressing forcep = load per unit Iengthp Q = pressure under an anchor plateR = resultant forceT = tensile force, tensile tieV = shear force

Strengthf, = specified compressive strength

of concretef = concrete compressive strengthfor design of undisturbed timi-axial stress fields

f,, = average concrete cylinderstrength

= specified tensile strength ofconcrete

fa = specified yield strength of rein-forcing steel

fp„ = specified yield strength of pre-stressing steel

y = partial safety factor

SubscriptsC = concrete or compressived = designp = prestressing, prestressing steels = steelt = tensile, tensionu = ultimatew = working, web

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