Toward a Consistent Design of Structural Concrete

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 ^a

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.

PCI JOURNAL/May-June 1987 85

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)

aI^ibeam

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,

PCI JOURNALIMay-June 1987 89

A B A B

pi

loodpothcjr

T 1 ti'–—`r

IA lBA B

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

IF15'

S /^

A '

1

I II

P

F B BB 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

PCI JOURNAL/May-June 1987 91

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