Schlaich Strut and Tie Model

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Special Report

Toward a

Con s is ten t Des ign o f

S tru c tu ral Co n cre te

J o r g Sch lai chDr.-Ing.Professor at the Ins titute

of Reinforced Concrete

University of StuttgartWest Germany

Ku r t Scha ferDr.-Irtg.Professor at the Institute

of Reinforced ConcreteUniversity of StuttgartWest Germany

Matt ias Jennew e inDipl.-Ing.Research Asso ciateUniversity of StuttgartWest Germany

This report (which is being considered by Comite Euro-International du Bt ton

in connection w ith the revision of the Model Cod e) represents the latest and

most authoritative information in formulating a consistent design approach for

reinforced and prestressed concrete structures.

74



ONT NTS

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 .............. 84

3.1 Scope

3.2 Comm ents on the O verall Analysis

3.3 Modelling of Individual B- and D-R egions

4. Dimensioning the Struts, Ties and Nodes ............... 97

4.1 Definitions and G enera l Rule

4.2 Singular Nodes

4.3 Sm eared Nodes

4.4 Concrete Com pression Struts — Stress Fields C.

4.5 Concrete Tens ile Ties --- Stress Fields T,

4.6 Reinforced Ties T,

4.7 Se rviceability: Cracks and Deformations

4.8 Con cluding Re marks

5. Exampes of Appications ............................. 1105.1 The B-Regions5.2 Some D-Regions

5.3 Prestressed Con crete

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

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

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

PCI JOURNAL+May-June 1987 5



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

The truss model is today considered

by researchers an d practitioners to be

the rational and appropriate basis for the

design of cracked reinforced concrete

beams loaded in bending, shear and tor-

sion. However, a design based on the

standard truss model can cover only

certain parts of a structure.

At statical or geometrical discontinu-

ities such as point loads or frame cor-ners, corbels, recesses, holes and other

openings, the theory is not applicable.

Therefore, in practice, procedureswhich are based on test results, rules of

thum b and past experience are usually

applied to cover such cases.

Since all parts of a structure including

those mentioned above are of similar

importance, an acceptable design con-cept must be valid and consistent for

every part of any structure. Further-

m ore, since the function of the experi-ment in design should be restricted to

verify or dispute a theory but not to de-

rive it, such a concept m ust be based on

physical models which can be easily

understood and therefore are unlikely to

be misinterpreted.

For the d esign of structural concrete*

it is, therefore, proposed to generalize

the truss analogy in order to app ly it in

the form of strut-and-tie-models to every

part of any structure.

This proposal is justified by the fact

that reinforced concrete structures carry

loads through a set of compressive stress

fields wh ich are distributed and inter-

connected by tensile ties. The ties m ay

be reinforcing bars, prestressing ten-

dons, or concrete tensile stress fields.

For ana lytical purposes, the strut-and-

'Following a proposal by Dr. J. E. Breen and

Dr. A. S. C. Bruggeling, the term structural

concrete covers all loadbearing concrete, including

reinforced, prestressed and also plain (unrein-

forced concre te, if the latter is part of a re inforced

concrete structure.

tie-models condense all stresses in com-

pression and tension mem bers and join

them by nodes.

This paper describes how strut-and-

tie-models can be developed by fol-

lowing the path of the forces throughout

a structure. A consistent design ap-

proach for a structure is attained w hen

its tension and compression members

(including their nodes) are designed

with regard to safety and serviceab ility

using un iform design criteria.

The concept also incorporates them ajor elem ents of what is today called

detailing, and replaces em pirical pro-

cedures, rules of thumb and guess work

by a rational design m ethod. Strut-and-

tie-models could lead to a clearer under-

standing of the behavior of structural

concrete, and codes based on such a n

approach would lead to improved

structures,The authors are aware of the en-

couraging fact that, although they pub-

lished papers on this topic earlier,1.2.3

they are neither the first nor the only

ones thinking and work ing along these

lines. It was actually at the turn of the last

century, when Ritter*' and M cirsch in-

troduced the truss analogy. This method

was later refined and expanded by

Leonhardt, ° Rusch, 7 Kupfer, 8 and others

until Thurlimann's Zurich school, a withMarti l u and Mu eller, created its scien-

tific basis for a rational application in

tracing the concept back to the theory of

plasticity.Collins and Mitchell further consid-

ered the deformations of the truss model

and derived a rational design method for

shear and torsion.In various applications, Bay, Franz,

Leonhardt and Thurliman n had shown

that strut-and-tie-models could be use-

fully applied to deep beams and corbels.

From that point, the present authors

began their efforts to system atically ex-

pand su ch m odels to entire structures

76



Synops is

Certain parts of structures are de-

signed with almost exagg erated accu-

racy while other parts are designed

using rules of thumb or judgment

based on past experience. How-

ever, all parts of a structure are of

similar importance.

A unified design concept, which is

consistent for all types of structures

and all their parts, is required. To be

satisfactory, this concept must be

based on realistic physical models.

Strut-and-tie-mode Is, a generalizationof the well known truss analogy

method for beams, are proposed as

the appropriate approach for design-

ing structural concrete, which includes

both reinforced and prestressed con-

crete structures.

This report shows how suitable

models are developed and proposes

criteria according to which the mode l's

elements can be dimensioned uni-

formly for all possible cases. The con -

cept is explained using numerous de

sign examples, many of which treat

the effect of prestress.

This report wa s init ial ly prepared for

discussion within CEB (ComitdEuro-International du Beton) in con-

nection with the revision of the Model

Code,

and all structures.

The approaches of the various authors

cited above differ in the treatmen t of theprediction of ultimate load and the

satisfaction of serviceability require-

ments. From a practical viewpoint, true

simplicity can only be achieved if so-

lutions are accepted with sufficient

(hat not p erfect) accuracy. Therefore, it

is proposed h ere to treat in gene ral the

ultimate limit state and serviceability in

the cracked state by using one and thesam e m odel for both. As will be shown

later, this is done by orienting the

geom etry of the strut-and-tie-mo del at

the elastic stress fields and designing

the model structure following the theoryof plasticity.

The proposed p rocedure also perm its

the dem onstration that reinforced and

prestressed concrete follow the same

principles although their behavior

under w orking loads is qu ite distinct.It should be m entioned that only the

essential steps of the proposed m ethod

are given here. Further support of thetheory and other information may be

found in Ref. 3.

2. THE STRUCTURE S B- AND D-REGIONS

Those regions of a structure, in which

the Bernou lli hypothesis of plane strain

distribution is assum ed valid, are usu-

ally designed with almost exaggerated

care and accuracy. These regions are

referred to as B-regions (where B stands

for beam or Bernoulli). Their internal

state of stress is easily derived from the

sectional forces (bend ing and torsional

mom ents, shear and axial forces).

As long as the section is uncracked ,

these stresses are calculated with the

help of section properties like cross-

sectional areas and mom ents of inertia.

If the tens ile stresses exceed the tensile

strength of the concrete, the truss model

or its variations apply.

The B-regions are designed on the

basis of truss models as d iscussed later

on in Section 5.1.

PCI JOURNAL May-June 1987 7



h a1 1 ^ — h 2 .—h-4 h-4

t ^ b

IIFH2. h—L I

Fig. 1 D-regions (shaded areas) with nonlinear strain distribution due to (a) geometrical

discontinuities; (b) statical and/or geo me trical discontinuities.

The above standard methods are notapplicable to all the other regions and

details of a structure where the strain

distribution is significantly nonlinear,

e.g., near concentrated loads, corners,

bends, openings and other discon-tinuities (see Fig. 1). Such regions are

called D-regions (where D stands for

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

cracked, they can be readily analyzed by

the linear elastic stress method, i.e., ap-

plying Hooke s Law. However, if the

sections are cracked, accepted design

approaches exist for only a few cases

such 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 even

these approaches usually only lead to

the design of the required amount of

reinforcement; they do not involve

a clear check of the concrete

stresses.

The inadequate (and inconsistent)

treatment of D-regions using so-called

detailing, past experience or good

practice has been one of the main rea-

sons for the poor performan ce and even

failures of structures. It is apparent,

then, that a consistent design

philosophy m ust comprise both B- and

D-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 com plex

D-region design will need to be sim-

plified with some loss of accuracy.

H owever, even a simplified methodical

concept of D-region design will be pref-

erable to today's practice. The preferredconcept is to use the strut-and-tie-model

approach. This method includes the

B-regions with the truss model as a spe-

cial case ofa strut-and-tie model.

In using the strut-and-tie-m odel ap-

proach, it is helpful and inform ative to

first subd ivide the structure into its B-

and D-regions. The truss m odel and the

design procedu re for the B-regions are

then readily available and only the

strut-and-tie-m odels for the D-regions

remain to be developed and added.

Stresses and stress trajectories are

quite smooth in B-regions as compared

to their turbulent pattern n ear discon-tinuities see Fig. 2). Stress intensities

decrease rapidly with the distance from

the origin of the stress concentration.

This behavior allows the identification

of B- and D-regions in a structure.

In order to find roughly the division

lines between B- and D-regions, the

following proced ure is proposed, wh ich

is graphically explained by four exam -ples as shown in F ig. 3:

1. Replace the real structure (a) by the

fictitious structure (b) wh ich is loaded in

such a way that it complies with the

Bernoulli hypothesis and satisfies

equilibrium with the sectional forces.

Thu s, (b) cons ists entirely of one or sev-

eral B-regions. It usually violates the

actual boundary conditions.2. Select a self-equilibrating state of

stress (c) which, if superimposed on (b),

satisfies the real b oundary conditions of

(a).

3. App ly the principle of Saint-Wnan t

(Fig. 4) to (c) and find that the stresses

are negligible at a distance a from the

equilibrating forces, which is approxi-

PCI JOURNALMay-June 1987 9



{ a 1 b) c) d)

hFh1 h F

dh

+ d= h

Fig. 3.1. Column with point loads.

{c

h l l t M3 h2

V ^M1

M

M

V

cb

c) Ui-dr r+–d2 = h2

{ d1

i B

Fig. 3.3. Beam with a recess.

80



a]ILl 43TTTTTr m

h

t

t

C) d=h_, t+

tf : II IIIIle :

Fig. 3.2. Beam with direct supports.

(a]

`. D ^ I d=b

Fig. 3.4. T-beam.

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

(b) loads and reactions applied in d) real structure with B- and

accordance with Bernoulli hypothesis -regions

Fig. 3. Subdivision of four structures into their B- and D-regions, using Saint

Venant s principle (Fig. 4).

PCI JOURNAL/May-June 1987 1



Q

Fa6=0i0

d=h

b Fy ______

2y_2 _ iT h

d=h d

_ " x x

1 . 0 h 10 h

ddy

1,0 h 1,0 hFig. 4. The principle of Saint-Venant: (a) zone of a b ody

affected by self-equilibrating forces at th e surface; (b)

application to a prismatic bar (beam ) loaded at one face.

m ately equal to the m aximum distance

between the equilibrating forces them -

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

It should be m entioned that cracked

concrete members have different stiff-

nesses in different directions. Th is situ-

ation may influence the extent of the

D-regions but needs no further discus-

sion since the principle of Saint-V€nant

itself is not precise and the dividing

lines between the B- and D-regions

proposed here only serve as a qualita-

tive aid in developing the strut-and-

82



p h

^

a2h

c

B B a

B B

f 4h

B

1 O k o lhI h B

I >Zh^•4h

U I F ^

w

r r

f Fig. 5. The identification of their B- and D-regions (acco rding to Fig. 3) is

a rational metho d to classify structures or parts thereof with respect to

their Ioadbearing behavior: a) deep beam; b) through d) rectangular

beams; (e) T-beam,

tie-models.

The subdivision of a structure into B-

and D-regions is, however, already of

considerable value for the under-

standing of the internal forces in the

structure. It also demonstrates, that sim-

ple fh rules used today to classify

beams, deep beams, short/long/high

corbels and other special cases are mis-

leading. For p roper classification, bo th

geometry and loads m ust 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 JOURN ALMay-June 1987

8 3



divided into its individual planes, which

are treated sep arately. Similarly, three-

dimen sional stress patterns in plane or

rectangular elem ents may be looked at

in different orthogonal planes. There-

fore, in general, only two-dimensional

models need to he considered. How-ever, the interaction of models in

different planes must be taken into ac-

count by app ropriate boundary con di-

tions.

Slabs m ay 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 not

predom inantly plane, as for examp le inthe case with punch ing or concentrated

loads, three-dimensional strut-and-tie-

mod els should be developed.

3. GENERAL DESIGN PROCEDURE ANDMODELLING

3.1 Scope

For the m ajority of structures it would

be unreasonable and too cumbersome to

begin immediately to model the entire

structure with struts an d ties. Rather, it

is more convenient (and com mon prac-tice) to first carry out a general structural

analysis. H owever, prior to starting this

analysis, it is advantageous to subd ividethe given structure into its B- and D-re-

gions. The overall analysis will, then,

include not on ly the B-regions but also

the D-regions.

If a structure contains to a substantial

part B-region s, it is represented by its

statical system (see Fig. 6 ). The general

analysis of linear structures (e.g., beams,

frames and a rches) results in the support

reactions and sectional effects, the

bending moments (M), normal forces

N), shear forces V) and torsional mo-ments(M r ) (see Table 1).

The B -regions of these structures canthen be easily dimensioned by a pplying

standard B-region models (e.g., the truss

model, Fig. 8) or standard methods

using handbooks or advanced codes of

practice. 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 stresses

due to the spreading of forces in the concrete) are

unsafe for plain concrete.

analysis and B-region design provide

also the boundary forces for the re

gions of the same structure.

Slabs and shells consist predom-inantly of B-regions (plane strain dis-

tribution) Starting from the sectional

effects of the structural analysis, imagi-

nary strips of the structure can be mod-

elled like linear members.

If a structure consists of one D-region

only (e.g., a deep beam), the analysis of

sectional effects by a statical system may

be omitted and the inner forces or stress-

es can be determined directly from the

applied loads following the principlesoutlined for D-regions in Section 3.3.

However, for structures with redundant

supports, the support reactions have to

be determined by an overall analysis

before strut-and-tie-models can be

properly 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 . Sectional effects Support reactions

Analysis of State I Via sectional values Linear elastic analysis*

inner forces (uncracked) A,Js,Jr (with redistributed stress peaks)

or stresses

in individualState II

(cracked)

Strut-and-tie-models

and/or nonlinear stress analysis *region

Usuall y truss

May Le cuuwbiuct[ with overall uiak his.

PCI JOURN AL/May-June 1987 5



p) simple span with cantilever

hte

reg an

O-region

for e in the bolt on hor

russ

f—multiple truss

Mlz

t tsingle truss

— -

multiple truss (steps) --C)

a

beam E oc

M x Cclx-a1 model Cc l x

T 111

I vix a} Tw _

VIx a l x I

w

x o

xbS Ts(x-a) o

8 x Tslx

,h—a-z cot 8 —

C c Ix } M l l V 2xCt o

C w I x }_ 8 — — cwlxl b z snV(x)

B Ismeo ed d ogcrd stress

V x JTwlx- l = V (x}

– – t w o ) =zrat 9 (pEr a nil length of beam

T (x= x • V-ot 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 analysis

m t Overall of sectional effects and support reactions

s t a t e structural behavior

Most adequate Acceptable

Essentially uncracked Linear elastic

Service Considerably cracked, Linear elastic (or plastic

ability with steel stresses below Nonlinear if design is oriented at

yield elastic behavior)

Plastic with limited Linear elastic or

Ultimate Widely cracked, rotation capacity nonlinear or perfectly

capacity tbrming plastic hinges or elastic with plastic with

redistribution structural restrictions

nite element method analysis may be

applied. A follow-up check with a strut-

and-tie-model is recommended, espe-

cially if the major reinforcement is notmodelled realistically in the FEM

analysis.

3.2 Comments on the Overall

Analysis

In order to be cons istent, the overa ll

analysis of statically indeterminate

structures should reflect the realistic

overall behavior of the structure. The

intent of the following paragra ph (sum -

marized in Table 2) is to give someguidance for the design o f statically in-

determinate structures. Some of this

discussion can a lso be app lied to stat-

ically determ inate structures esp ecially

with regard to determining deforma-

tions.

Plastic methods of analysis usually

the static method ) are suitable primarily

for a realistic determina tion of ultimateload capacity, while elastic m ethods are

more appropriate under serviceability

conditions. According to the theory of

plasticity, a safe solution for the ultimate

load is also obtained, if a plastic analysis

is replaced by a linear or nonlinear

analysis. Experience further shows thatthe design of cracked concrete struc-

tures for the sectional effects using a

linear elastic analysis is conservative.Vice versa, the distribution of sectional

effects derived from plastic methods

may for simplification purposes also be

used for serviceability checks, if the

structural design (layout of reinforce-

m ent) is oriented at the theory of elas-

ticity.

3 .3 Mod e ll i ng o f I nd i v i dua l B - and

D-Reg ions

3.3.1 Principles and General Design

Procedure

After the sectiona l effects of the B-re-gions and the boundary forces of the D-

regions have been determined by the

overall structural analysis, dimensioning

follows, for which the internal flow of

forces has to be searched and quantified:

For uncracked B- and D-regions,

standard m ethods are available for the

analysis of the concrete and steel stress-

es (see Table 1). In the case of highcom pressive stresses, the linear stress

distribution may have to b e m odified by

replacing H ooke's Law with a nonlinear

materials law (e.g., parabolic stress-

strain relation or stress block ).

If the tensile stresses in individual B-

or D-regions exceed the tensile strength

of the concrete, the inner forces of those

PCI JOURNALIMay J u n e 1987 7



regions are determined and are de-

signed according to the following pro-

cedure:

1. Develop the strut-and-tie-model as

explained in Section 3.3. The struts and

ties condense the real stress fields by

resultant straight lines and concentrate

their curvature in nodes.

2. Calculate the strut and tie forces,

which satisfy equilibrium. These are theinner forces.

3. Dimension the struts, ties and

nodes for the inner forces with due con-

sideration of crack w idth limitations (see

Section 5).

This method implies that the structure

is designed according to the lower

bound theorem of plasticity. Since con-

crete permits only limited plastic de-

formations, the internal structural system

(the strut-and-tie-model) has to b e cho-

sen in a way that the deform ation lim it

(capacity of rotation) is not ex ceeded a t

any point before the assumed state of

stress is reached in the rest of the struc-ture.

In highly stressed regions th is ductil-

ity requirem ent is fulfilled by ad apting

the struts and ties of the model to the

direction a nd size of the internal forces

as they would appear from the theory of

elasticity.

In norma lly or lightly stressed regions

the direction of the struts and ties in themodel may deviate considerably from

the elastic pattern without exceeding

the structure's ductility. The ties and

hence the reinforcement may be ar-

ranged according to practical consid-

erations. The struc ture adapts itself to

the assumed internal structural system.

Of course, in every case an analysis and

safety check mu st be made using the fi-

nally chosen model.

This method of orienting the strut-

and-tie-mod el along the force paths in-

dicated by the theory o f elasticity obvi-

ously neglects some ultimate load

capacity which could be utilized by a

pure application of the theory of plastic-

ity. On the other hand, it has the major

advantage that the sam e m odel can be

used for both the ultimate load and the

serviceability check . If for some reason

the purpose of the analysis is to find the

actual 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. In

this case, however, the inelastic rotation

capacity of the m odel has to be consid-

ered. (Note that the optimization of

models is discussed in Section 3.3.3.)

Orienting the geometry of the model

to the elastic stress distribution is also a

safety requiremen t because the tensile

strength of concrete is only a sm all frac-

tion of the compressive strength. Cases

l ike those given in Fig. 7 w ould be un-

safe even if both requirements of the

lower bound theorem of the theory of

plasticity are fulfilled, na m ely, equ ilib-

rium and F IA --f,. Compatibility evokestensile forces, usually transverse to the

direction of the loads wh ich may cause

premature cracking and failure. The bottle-shaped compressive stress

field, which is introduced in Section

4.1, further eliminates such hidden

dangers when occasionally the model

chosen is too simple.

For cracked B-regions, the proposedprocedure obviously leads to a truss

model as shown in Fig. 8, with the in-

clination of the diagonal struts orientedat the inclination of the diagonal cracks

from elastic tensile stresses at the neu-

tral axis. A reduction of the strut angle by

10 to 15 degrees and the choice of verti-

cal stirrups, i.e., a deviation from the

principal tensile stresses by 45 degrees,

usually (i.e., for normal strength con-

crete and norm al percentage of stirrup

reinforcement) causes no d istress. Since

prestress decreases the inclination of

the cracks and hence of the diagonal

struts, prestress permits savings of stir-

rup reinforcem ent, whereas a dditional

tensile forces increase the inclination.

The distance z between the chords

should usually be determined from the

plane strain distribution at the points of



10l a La lh

a)

IIH k T lI

L//

Cj

t-/--. 1 ^ ` i

l T -- strut

tie

b)

a- 0,5____ ^ fr1t5 d r2F

QGZa/1-01

70° 0,3

1

/PI art/pt

Q0.5 0,5 0.7 9.8 09 1.01.11,?1,31,61,5lbd/t

Fig. 9. A typical D-region: (a) elastic stress trajectories, elastic

stresses and strut-and-tie-model; (b) diagram of internal forces,internal lever arm z and strut angle 0.

maximum moments and zero shear and

for simplicity be kept constant between

two adjacent points of zero moments.

Refinemen ts 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-m odel for each

case individually. After some training,

PCI JOURNALIMay-June 1987 9



A p ioodpoth rT –—

Fig. 10. Load pa ths and strut-and-tie-mode l.

1 1

P

F

F

Fig. 11. Load paths (including a U-turn ) and

strut-and-tie-model.

this can be done quite simply. De-

veloping a strut-and-tie-model is com -

parable to choosing an overall statical

system. Both procedures require som edesign experience an d are of similar rel-

evance for the structure.

Developing the m odel ofa D-region is

much simplified if the elastic stressesand principal stress directions are avail-

able as in the case of the example shown

in Fig. 9. Such an elastic analysis is

readily facilitated by the w ide variety of

computer programs available today. The

direction of struts can then be taken in

accordance with the mean direction of

principal compressive stresses or the

more important struts and ties can be lo-

cated at the center of gravity of the cor-

responding stress diagrams, C and T in

Fig. 9a, using the y diagram given

there.

H owever, even if no elastic analysis is

available and there is no time to prepareone, it is easy to learn to develop s trut-

and-tie-models using so-called load

paths. This is demonstrated in more

detail by some examples in the next

section.

3.3.2 The Load P ath MethodFirst, it must be ensured that the outer

90



C

e i lar 1 ^

- Q

I+II I ^ T--

i LV x t 1C7

I Vc, :c. c_

r7 drl

Y

II

L iJ ^ - -}

III I I

III I IIII I I D III

ILL Y p _ P LFig. 12.1. A typical D-region: (a) elastic stress trajectories; (b) elastic stresses;

(c) strut-and-tie-models.

a aIF

,

dI4

drI T jCr

t

8 t

I

Fig. 12.2. Special case of the D-region in Fig. 12.1 with the load at the

corner; (b) elastic stresses; (c) strut-and-tie-models.

equ ilibrium of the D-region is satisfied

by determining all the loads and reac-

tions (supp ort forces) acting on it. In a

boundary adjacent to a B-region the

loads on the D-region are taken from the

B-region design, assuming for example

that a linear distribution of stresses (p)

exists as in Figs, 10 and 11.

The stress diagram is subdivided in

such a w ay, that the loads on one side of

the structure find their counterpart on

the other, considering that the load

paths conn ecting the opposite sides will

not cross each other. The load paths

PCI JOURN AL/May-June 1987 1



H

Ti

f/

B a} B

v

Al Tc

shear force'

A B

c

moment

b) B m P

C T cstrut

tie

_ load path

-mtea nchorage length of the bar

Fig. 13. Two models for the same case: (a) requiring oblique reinforcement;

(b) for orthogonal reinforcement.

begin and en d at the center of gravity of

the corresponding stress diagrams and

have there the direction of the applied

Ioads or reactions. They tend to take the

shortest possible streamlined way in

between. Curvatures concentrate near

stress concentrations (support reactions

or singular loads).

Obviously, there will be some cases

where the stress diagram is not com-

pletely used up with the load paths de-

scribed; there rem ain resultants (equa l

92



a good b bad

UII i LnTTTY_ P L1ru .LU11111P

r

I

I

1 d_ 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 it

again on U-turn o r form a whirl as illus-

trated by forces B or Figs. 11 and 13 a.

Un til now , equilibrium on ly in the di-

rection of the applied loads has been

con sidered. After plotting all load paths

with smoo th curves and replacing them

by polygons, further struts and ties m ust

be added for transverse equilibrium

acting between the nodes, including

those of the U-turn.

While doing so, the ties must be ar-

ranged with proper consideration of

practicality of the reinforcem ent layout(generally parallel to the c onc rete sur-face) and of crack distribution require-me nts -

The resu lting m odels are quite often

kinematic which means that equilib-

rium in a given model is possible only

for the specific Ioad case. The refore, the

geom etry of the appropriate m odel has

to be adapted to the load c ase and is in

m ost cases determined by e quil ibrium

conditions after only a few struts or ties

have been chosen.A very powerful m eans of developing

new strut-and-tie-models for compli-

cated cases is the combination of an

elastic finite element method analysis

with the load path method. This com-

bined approach is applied in Fig. 12 and

the num erical exam ple in Sec tion 5.2.1.

In Fig. 12.1 the vertical struts and ties

are found by the load path method as

explained in the previous examples:

The structu re is divided into a B-regionand a D-region. The bottom of the D -re-

gion is acted on by the stresses (p) as

derived for the adjace nt B -region.

These stresses are then resolved into

four compone nts: The two com pressive

forces Cs + C. = F, which leaves two

equal forces T Z and C Y . The forces C,

and C, are the components, respec-

tively, on the left hand an d right handside of the vertical plane which is de-

terrnined by the load F. By laterally

shifting the load components into the

given positions, transve rse stresses are

generated.

The corresponding horizontal strutsand ties are located at the center of

gravity o f stress diagram s in typical sec-

tions which are de rived from an elastic

analysis (Fig. 1 2.1b). Their nodes with

the vertical struts also determine the

position o f the diagonal struts (see Fig.

12.lc).The example in F ig. 12.2 show s that

the tie T 3 of Fig. 12.Ic disappears, if the

load F moves toward the corner of the

D-region.




a p

AI 1 i

b Q

e + 2 l

f-;

`r 1

f

high medium low long

tl

^ I 1

Yr _, I '-, 4

r ly ^^ L 4r-+

i II r r

4 i+ 7 Tt f

Fig. 15. The two m ost frequen t and most useful strut-and-tie mo dels:

(a) through (b), and som e of the ir variations (c) throug h (e).

94



3.3.3 Model Optimization

In Fig. 13 one load case has been

solved with two different models. The

left part of Figs. 13a and 1 3h sh ows ho w

they were developed by the load path

m ethod and how they are connected to

the ove rall sectional effects of the D -re-gions. The ties T and T x in Fig. 13a

would require inclined reinforcement,

which is undesirable from a practical

viewpoint.

Therefore, a tie arrangem ent has been

chosen in F ig . 13 b which can be sat is-

fied by an orthogonal reinforcing net

with the bars parallel to the edges.

Thereby, special m ethods such as those

given in Ref. 17, which con sider devia-

tions of reinforcement directions from

the principal stress directions, are notneeded.

Doubts could ar ise as to w hether the

correct model has been chosen out ofseveral possible ones. In selecting the

m odel, it is h elpful to realize that loads

try to use the path w ith the least forces

and deformations. Since re inforced ties

are much more deformable than con-

crete struts, the m odel with the least and

shortest ties is the best. This simple

criterion for optimizing a model m ay be

b)

I

B

ON 055 1 011 1511

i----- ----- -----

c) ra - o

rr

I I I 5 r r

d

r r r

(see also fig. 35)

Fig. 16. One s ingle type of a D-m ode& appears in m any different structures: here four

examples are given.




a) } ioad b anchor plate tendon t I _

column

T 1 - -------HFig. 17. (a) Deep beam on three supports; (b) end of a beam or slab

with anchorages of three prestressing tendons. Both cases are

identical if the reactions in (a) and the prestressing forces in (b) are

equal.

formulated as follows:

.F ; l ; e„,; = M inimu m

whereF, = force in striitortiei

1, = length of memberie m s = me an strain of me mbe r i

This equation is derived from the

principle of minimu m s train ene rgy for

linear elastic behavior of the struts and

ties after cracking. The c ontribution of

the concrete struts can generally he

om itted bec ause the strains of the strutsare usually m uch smaller than those of

the steel ties. This criterion is also

helpful in eliminating less desirable

m odels (see F ig. 14).

Of course, it should be understood

that there are no unique or absolute op-

timum solutions, Replacing a con-

tinuous set of smooth curves by indi-

vidual polygonal line s is an approxim a-tion in itself and leaves am ple room for

subject ive decisions, Fu rthermore, in-

dividual input such as the size of the re-

gion or reinforcem ent layout are always

different. But an engineer with some

experience in strut-and-tie-modelling

will always find a satisfactory solution.

3.3.4 The Pedagogical Value of Mod-

elling

Anyone who spends t im e deve loping

strut-and-tie-models will observe thatsome types of D-regions appear over

and over again even in apparently very

different structures. The two m ost fre-quent D-regions, which are even related

to one another because they have

the same characteristic stress distri-

bution along their centerline, are giv-

en in Fig. 15 with some of their

variations.

Fig. 16 show s applications of the firsttype of m odel (Fig. 15a) to four different

structures: The distribution of cable

forces in a bridge dec k (Fig. 16a); a wall

with big open ings (Fig. 16b); a box gird-

er with anchor loads from prestressing

tendon s (Fig. 16c); and a detail of inter-

nal forces in a rectangular beam wh ich

shows that stirrups n eed to he c losed.

(Fig. 16d). In all of these c ases the pat-tern of internal forces is basically identi-

cal.

To recognize such com mon features of

structures is of considerable pedagogi-

cal value and veryhe lpful to the design

engine er. On the othe r hand, it is con-

fusing if the same facts are given dif

96



ferent designations only because they

appear under different circumstances.

For exam ple, the deep beam in Fig. 17

"does not care" whether its identical

loads result from supports or from pre-

stressing tendons. It, therefore, does not

react with "bending" in the one case

and with splitt ing tension in the other

case, but sim ply with tension and com -

pression.

Num erous other strut-and-tie-models

are found in Ref. 3.

4. DIMENSIONING THE STRUTS TIES AND NODES

4.1 Def in i t ion s and Genera l Ru le

Fig. 18 shows som e typical examples

of strut-and-tie mode ls, the co rrespond-

ing stress trajectories and reinforcem entlayout. Node regions are indicated by

shading. Looking closely at these ex-

amples as w ell as those in the previous

and following sections, the following

conclusions may be drawn:

Dim ensioning not only m eans siz ing

and reinforcing the ind ividual struts and

ties for the forces they carry, but also

ensu ring the load transfer betwee n them

by che cking the node regions. There is a

close relation between the detailing of

the nodes and the strength of the struts

bearing on them and of the ties an-

chored in them because the detail of the

node chose n by the design engineer af-

fects the flow of forces, The refore, it is

necessary to check whether the strut-

and-tie-model initially chosen is still

valid after detailing or nee ds correc tion.Thus, m odelling and dim ensioning is in

principle an iterative process.

There are basically three types of

struts and ties to be dimensioned:

C,: Conc rete struts in com pression

TT : Concrete ties in tension without

reinforcement

Ties in tension with reinforce-

me nt (mild steel reinforcem ent orprestressing steel)

There are essentially four types of

nodes depen ding on the combination of

struts C and ties T (see F ig. 19 ):

CCC-node

CCT-node

CTT-node

TTT-node

The principle remains the same if

more than three struts and ties mee t.

Stru ts and T ies

Wh ereas the T, are es sentially l inear

or one-dimensional elements between

two nodes, the C. and T,, are two- (or

three-) dimen sional stress fields, tend-

ing to spread in between two adjacent

nodes. This spreading, indicated by the

bulging of the struts in Figs. 18 and 19 a,

can result in transverse tensile and

compressive stresses which then m ust

he considered either by introducing

these stresse s into the failure criterion of

the C struts and the Tc ties or by again

applying a strut-and-tie-model to them

(see Figs, 18c and 18d). Both approaches

lead to the same re sult.

The struts in the model are resul-

tants of the stress fields. Since by de fini-

t ion the curvatures o r deviations o f theforces are conce ntrated in the n odes, the

struts are straight. This is, of course, an

idealization of reality. If doubts arise

whe ther by doing so in a highly stressed

structure some tensile forces are not

sufficiently accounted for, the straight

lengths of the struts can be reduced

either by refining the m odel itself or by

sme aring (or spreading) the node over asubstantial length of the strut (see for

example F igs. 18a2 and 18b2).

To cover all cases of compression

f ields including those of the B -regions,

three typical configurations are suffi-

cient:

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

PCI JOURNAL/May-June 1987

7



all 2) 3)

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

C

bll 2) 3)

s C C

C

C1) 2) not reinforced 3) reinforced

C E E

` Fig. 18. Some typical examples of strut-and-tie-models, their stress fields, nodes and

corresponding reinforcement (if any is provided).

98



(b ) The bottle (see Fig. 20b).

(c ) The "prism" or parallel stress

field (see F ig. 20c), being the limit

case of both a = 0 and hla = 1.

Nodes

The nodes of the m odel are a sim pli-

fied idealization of reality. They are

formally derived as the intersection

points of three or m ore straight struts or

t ies, which them selves represent either

straight or c urved stress fields or rein-

forcing bars or tendon s. A node as intro-

duced into the m odel implies an abrupt

chan ge of direction of forces. In the ac-

tual re inforced con crete structure thisdeviation u sually occu rs over a certain

length and width.

If one of the struts or ties represen ts a

conce ntrated stress field, the deviation

of forces tends to he locally concen-

trated also. On the other h and, for wide

concrete stress f ields joining each other

or with tensile ties, which consist of

many closely distributed reinforcing

bars, the deviation of forces may be

sm eared (or spread) over some length.

Therefore, in the former case the n odes

are called singular (or concentrated)

nodes, whereas in the latter case they

are called smeared (or continuous)

nodes. Nodes A and B in F ig. 18a1 serve

as typical examples of both types ofnodes.

d1) 2) struts not reinforced 3) struts reinforced

e)

Fig. 18 (cont). Some typical examples of strut-and-tie-models, their stress fields, nodes

and corresponding reinforcement (if any is provided).

PCI JOURNALIMay -June 1987 9



ai

anchor pla4e t

I -he12 1

anch rOge length

b 3 b4withart with inchomge length

crosspressure ith loop

nnc hort^ge length

Fig. 19. Examples of the basic types of nodes: (a) CCC-nodes. Idealized "hydrostatic"

singular nodes transfer the con cen trated loads from an anch or plate (a,) or bearing plate

(a 2 ) into (bottle shaped) compression fields; (b) CCT-nodes. A diagonal compression

strut and the vertical support reaction are balanced by reinforcemen t wh ich is anchored

by an anchor plate behind the n ode (b,) , bond w ithin the n ode (b ), bond within and

behind the n ode (b,), bond and radial pressu re (b,).

S



Failure Criteria for Concrete

The strength of the conc rete in com -

pression fields or within nodes depe nds

to a large ex tent on its m ultiaxial state of

stress and on disturbances from cracks

and reinforcem ent.(a ) Transverse com pression is favor-

able especially if it acts in both trans-verse directions, as for exam ple in con-

f ined regions. Confinem ent m ay be pro-

vided by transverse reinforcem ent or by

bulk conc rete surroun ding a relatively

small com pression field (see F ig. 21).

(b ) Transverse tensile stresses and the

cracks caused by them are detrime ntal.rsThe concrete may fail considerably

below its cy linder strength if the trans-

verse tension causes closely spaced

cracks approximately parallel to the

principal compression stresses such that

the prisms between those cracks are

ragged and narrow. The reduction of

com pressive strength is small or nom i-

nal if the tensile forces are carried by the

reinforcem ent and the cracks are wide

enough apart.(c ) In particular, cracks which are not

parallel to the com pressive stresses are

detrimental.

In 19 82, an em pirical formula for cal-

culating the strength of parallel concrete

com pression fields with transverse ten-

sion was pu blished by Collins et al.,2a

and 2 years later a similar formula was

introduced into the new Canadian

CSA-Standard A 23.3-M 84. These

formulas summarize the influence of

such significant parameters as crack

width, crack distance an d crack direc-

tion by the transverse tensile strain

c C 2 )

d dz

It f

Fig. 19 (cont.). Examples of the basic types of nodes: (c) CTT-nodes. A compression

strur is supported by two bo nde d reinforcing bars (c, ), respectively, by radial pressu re

from a bent-up bar (c1); (d) TTT-nodes, as above with the compression strut replaced by

a bonded tie.

PCI JOU RNAL/May-June 1987 101



a} b` 1; J j r I J I 1 G 1411111J4

i

I r

I r tllfft/f fffc d fittfll g f d fi11t11^ a^fcd

— aV^-I

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

wh ich, how ever, is not readily availablein the analysis.

For practical purposes, the following

simplified strength values of f re

proposed for dimensioning all types ofstruts and nodes:

f * = 1.0 f,: or an undisturbed and

uniaxial state of compressive

stress as shown in Fig. 20c;

fir 0.8 f : f tensile strains in the

cross direction or transverse ten-

sile reinforcement may cause

cracking parallel to the strut with

norm al crack width; this applies

also to node regions where ten-

sion steel bars are anchored or

crossing (see F ig. 19b);

f, 0.6 f :as above for skew cracking

or skew reinforcement;

= 0 4 d or skew cracks with ex-

traordinary crack width. Suchcracks must be expected, i f mod-

elling of the struts departs signifi-

cantly from the theory of elastic-

ity's f low of internal F orces (e.g.,

due to redistribution of internal

forces in order to exploit a

maxim um ultimate c apacity).

Note that f denotes the concrete

compressive design strength, which is

related to the specified compressive

strength f , and wh ich in turn depends

on the safety factor of the designated

code of practice. According to the CEB

Code,f^ d is determ ined by:

0.85fc

ye

whe re y, = 1 .5 is the partial safety factor

for the con crete in compression and the

coefficient 0.85 acco unts for sustained

loading, In the CEB Code, 2 0 = 1.0 in

all 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 com pres-

sive stresses may be taken into account

i f the simultaneou sly acting transversecom pressive stresses are considered re-

liable.

Skew cracks are not expected, i f the

theory of elasticity is followed suffi-

ciently closely during modelling. This

me ans that the angle between struts and

ties entering a singu lar node sh ould not

102



a^ -- 7-- - ,confining

reinforcement

F#g. 21. Confinement (a) by the surrounding concrete; (b) by

reinforcement increases the design compression strengthf . d.

be too small. How ever, skew cracks may

also be left over from a previous loading

case w ith a different stress situation.

Before deciding on one of the given

strength values, both transverse direc-

tions m ust always he considered.

General rule

Since singu lar nodes are bottlene cks

of the stresses, it can be assume d that an

entire D-region is safe, if the pressure

under the m ost heavily loaded bearing

plate or ancho r plate is less than 0.6 ff

(or in unusual cases 0.4 f,.) and if all

significan t tensile forces are resisted by

reinforcement and further if sufficientdevelopment lengths are provided for

the reinforcement. Only refinements

will be discussed in the following Sec-

tions 4.2 through 4.5.

4.2 Singular Nodes

Singu lar node s equilibrate the forces

of the ties and struts acting on the m rel-

atively abruptly as compared to the

smeared nodes. The deviation of theforces occurs over a short length or small

area around the theoretical nodal point.

Suc h nodes originate m ainly from single

loads or support reactions and from co n-

centrated forces introduced by the

reinforcement through anchor plates,

bond, or radial pressure inside bent

reinforcing bars such as loops. Fur-

therrnore, geometrical discontinuities

(e.g., reentrant corners) can c ause stress

concentrations which are represented

by a singular node.Although numerous possibilities exist

for detailing nodes (and despite the fact

that they all behave somewhat differ-

ently), in most c ases their forces balanceeach other in the interior of the node

through direct concrete compressive

stresses, which is a helpful observation.

The ideal tie anchor (with a plate)

transfers the load "from behind" and

thus causes compression in the node

(Fig. 19h,). Also, bond is essentially a

load transfer via concrete com pressive

stresses which are su pported by the ribs

of the steel bar (Fig. 19h 2 and b 3 ) and by

radial pressure in bent bars (Fig. 19b,).

Even in these cases the f low of forces

can be visualized by strut-and-tie-

m odels with singular nodes at the ribs of

the bar. If the se mode ls require conc rete

tensile ties, this is valuable information

for the design engineer. However, for

practical purposes, the anch orage and

lap lengths of the applicable codes ofpractice shou ld be used.

In sum m ary, then, dimensioning sin-

gular nodes means:

(a) Tuning the geometry of the node

with the applied forces.For CCC-nodes it is helpful, though

not at all mandatory, to assume the bor-




derline to be perpendicular to the re-

sultant of the stress field and the state of

stress within the interior of the node to

be plane hydrostatic. In this particular

case the u nequ ivocaI geom etrical re la-

tion a,: a 2 : a - C,: C2 : C 3 resu lts (Fig.

19a,), which may be used for dimen-sioning the length of the su pport or the

width of an anchor plate. Howe ver, ar-

rangements of forces which lead to

stress ratios down to 0.5 on adjacent

edges of a node are satisfactory. De-

parting even more from hydrostatic

stress and still disregarding the

nonuniform stress distribution in the

node m ay lead to com patibility stresses,

which are not covered by the strength

values given above.When designing a singular CCT-

node, the design engineer must be

aware that the curvature of the load path

and the corresponding compression

al c d B

ao.2[0 0 Hp iC a 1' f C 02 1 • tanplcns2 ...-lo_--__._

p —1

Q } ib,net

1h,net1

a Z Q 2

a 3_C o3,o1 tcnp

baaZccs WQ1=ap

C) 1_.

op f*

±t + oy

f

ci ——b,net

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

104



f ield is largest at the origin of the c on-

centrated load, i.e., next to the bearing

plate or anchor plate (see Figs. 9 and

10). The ties and plates should be ar-

ranged ac cordingly.

(b ) Checking whether the concrete

pressures within the node are within the

limits given in S ection 4.1 .

This con dition is au tomatically satis-

fied for the entire node region, if the

stresses along the horderlines of the

node do not exceed those limits and if

the anchorage of the reinforcement in

the node is safe. If the reinforcem ent is

anchored in the n ode region, cracks arepossible and c orrespondingly the con-

crete strength for cracked concrete

applies.

For CCT-nodes with bonded rein-

forcement arranged according to Fig. 22,

a check of conc rete stresses a respec-

tively, o in the adjacent compression

struts is sufficient. Since in m ost cases it

is apparent from the geometry of thenode which pressure out of the two

struts controls, only one of them needs

to be analyzed. An analysis on the basis

of Fig. 22 rewards reasonably the ar-rangement of multilayered reinforce-

m ent distributed over the width a g (Fig.

22a) compared with a tie consisting of

one layer on ly (Fig. 22b and d).

(c ) Ensuring a safe anch orage of ties inthe nodes (exce pt for CCC-nodes).

In the case of anc hor plates, this in-

volves a check of the ben ding strength

of the anchor plate and of the welded

connection with the tie. In this case a

smo oth surface of the tie wh ere it cross-

es the node is better than good bond

quality because strain compatibility

with the bonded bar wil l tend to crackthe node's concrete.In the case of directly anch ored rein-

forcing bars, hook or loop anchorages

(with cross-pressure in CCT-node s) are

preferred. Generally, the minimum

radius allowed by the applicable code is

selected.For straight bar anchorages, the

length of the anch orage is selected fol-

lowing the designated code. The design

engine er mu st ensure that it is located

within and behind the node (Fig. 19 b2

and b 3 ). Anchorage begins where the

transverse compression stress trajec-tories of the struts m eet the bar and are

deviated; the bar must extend to the

other end of the n ode region in order to

catch the outermost fibers of the de-

viated compression stress field (Fig.

18b 3 , c3) .

4.3 Smeared Nod es

Since D-regions usually contain bothsm eared and singular nodes, the latter

will be critical and a check of con crete

stresses in smeared nodes is unnec es-

sary. Howe ver, if a smeared CC T-node

is assumed to remain uncracked, the

tensile stresses of the corresponding

concrete stress f ield need to be che cked

(Section 4.5 ). An ex am ple of this case is

Node 0 in F ig. 18c1 and the stress fieldin Fig. 18c2.

Safe anchorage of reinforcem ent bars

in smeared nodes m ust be ensured fol-

lowing the rules for singular nodes

(Section 4.2).

4 .4 Conc re te Com press ion S t ru ts

—Stress Fields C

The fan-shaped and prism atic stress

fields do not evelop transverse stresses

and accordingly the uniaxial concrete

strenth applies. If transverse stresses,

cracks or tension bars c ross the strut, the

strength may be based on the values

given in Sec tion 4.1 .

The bottle-shaped com pression stress

field (Fig. 20b) applies to the frequent

case of com pressive forces being intro-

duced into concrete which is unrein-

forced in the transverse direction.

Spreading of the forces c auses biaxial or

triaxial compression unde r the load and

transverse tensions farther away. The

transverse tension (com bined with lon-

gitudinal com pressive stresses) can re-

PCI JOURNALJMay-June 1987 05



m — — biaxial com pression failure

in bottle neck

crocked, but w ith

transverse reinforceme nt

de gree w) in the

belly region

uncrocked, plain conc rete

nfinement

b o

pa

la

—b YI F p o a t 2 1

II ate 1,31a

Fig. 23. Dimensioning plane bottle shaped stress fields: (a) diagrams giving safe

pressure values P with regard to cracking and crushing of plain unreinforced concrete

stress fields, yielding of transverse reinforcement and biaxial compression failure in the

bottle neck region; (b) geometry of the stress field; (c) model and reinforcement layout of

stress field with transverse reinforcement w

106



suit in early failure. Again, the failure

criteria from Sec tion 4 .1 apply.

The general rule given at the end of

Sect ion 4.1 usually m akes the calcula-

tion of stresses within the stress field

unnecessary . However, for questionable

cases, computational aids should be

provided to facilitate the safet y check.

As an example, Fig. 23a shows dia-

grams for chec king plane bottle shaped

stress fields in D-regions. ' 3 • " This

stress field can he c haracterized by the

width a of the anchor plate, the

maximum width b available in the

structure for the stress field and the dis-

tance 1 of the anch or plate from the sec-tion where the stress trajectories are

again parallel (see F ig. 23b and also Fig.

20b). The diagram for compression

f ields without transverse reinforcem ent

(bold line) is based on an elastic

analysis, a conc rete tensile strength off

= f115 and a biaxial com pressive tensile

failure criterion as given in F ig. 26b.

It can be see n, that for certain geome t-rical relations a p ressure at the an chor

plate as low as 06f cou ld cause crack-

ing. However, the failure load of the

strut is usually higher than its cracking

load. A comparison of test results shows

that the diagram gene rally appears to be

considerably on the safe side and

further research in this area is required.

Better knowledge is also needed on

substantially nonsymmetrical stress

fields which originate from singular

nodes w ith tension ties crossing or an-

chored there. Com parisons with test re-

sults suggest that checking the singular

node (Section 4.2) and applying the

diagrams of Fig. 23 is also safe for those

cases.

The bottle shaped stress field pro-

vides a safe lower bound for unrein-forced compre ssion struts, wh ereas an

indiscriminate application of the theory

of plasticity to cases such as those shown

in Fig. 7 (mainly Fig. 7a) would perm it

prismatic stress fields between two op-

posite anchor plates with 1.0 ff as a fail-

ure stress and cou ld lead to a premature

failure.

For compression struts with trans-

verse reinforcement the failure loads

analyzed with the mode l in F ig. 23c are

also given in F ig. 23a. It can be see n that

a reinforcemen t ratio:

w =y

0.06tfd

(where a, is the cross section of rein-

forcement per unit length) approxi-

mately compensates for the tensile

strength of the c oncrete.

If it is desired not to rely so h eavily on

the concrete tensile strength, lowerre inforcement ratios m ay be used w ith

reduce d values of g a ff f, as shown in Fig.

23a.The com pressive strength of compres-

sion reinforcemen t may be added to the

concre te strength if the reinforcem ent is

prevented from buc kling.

4.5 Concrete Tensile Ties — Stress

Fie lds T^

In the case of unc racked tensile stress

f ields, the tensile strength o f concre te is

used. Although it is diff icult to deve lop

design c riteria for this case, it wou ld be

even w orse to m aintain the formalistic

view that the tensile strength of con-

crete cannot and therefore m ust not be

utilized. Following the flow of forcesgap free and consistently with strut-

and-tie-mod els will inevitably show that

equilibrium can frequently only be

satisfied if t ies or ten sile forces can beaccepted in places where, for practical

reasons, re inforceme nt cannot be pro-

vided, i.e., if the tensile strength of con-

crete is u tilized. , It shou ld be apparent

that no anchorage, no lap, no framecom er, no slab w ithout stirrups an d (as

shown) no unreinforced strut or com-

pression member can work without

using the tensile strength of concrete.

Unfortunately, most code s of practice do

not recognize this fact and, therefore,

surrogates such as bond, shear

PCI JOURNALiMay-June 1987 07



and other misnom ers have been intro-

duced. As a result , codes have becom e

unduly imprecise and complicated.

Un til further rese arch w ork is avail-

able in this field, the following simple

guidelines are proposed, which appear

to yield safe results when com pared totests:

The tensile strength of concrete

shou ld only be u tilized for equilibrium

forces where no progressive failure is

expe cted. Thereby, restraint forces and

microcracks have to be taken into ac-

count, even in "uncracked" and un-

loaded concrete. As shown in Fig. 24,

redistribution of stresses w hich avoids

progressive cracking may be assum ed to

be possible if at any part of the stress

f ield a cracked failure zon e w ith an area

JA, can be assumed, without the in-

creased ten sile stresses in the rem aining

section exceeding the tensile strength

Jet .

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

and ?A,,110

where

A, = area of tensile zone and

o diameter o f largest aggregate

Progressive failure of a section or

member generally starts from the

periphery of structures in the case of

steep stress gradients, as for example inthe ben ding tensile zone of beam s (Fig.

25).

The tensile stresses m ay be analyzed

with a linear elastic materials law. Stress

peaks in the outer fibers or at failure

zones m ay be distributed over a width of

5 cm (2 in.) but not more than 3 d a rule

wh ich finds its justif ication in fractureme chanics of concrete. 13

The design engineer will have to de-cide case by c ase wh ich fraction of the

tensile strength can he use d for carrying

loads and w hich fraction has been u sed

up by restraint stresses. The latter

stresses are u sually large in the longitu-

dinal direct ion of a structural m em ber

and at its surface, but are sm aller in the

transverse direction and in greater

depth.

If the tensile stress field is crossed by

a com pression field, the reduced biaxial

strength mu st be considered. The graph

(see Fig. 26c) provides a safe assump-tion.

4.6 Reinforced Ties T.

Usually, reinforcing steel should be

provided to resist tensile forces. The

axis of the steel reinforcement must

coincide with the axis of the tie in the

mo del. The dimensioning of these ties is

quite straightforward; it follows direc tly

from the cross section A. (reinforcing

steel) or A, (prestressing steel) and the

yield strength f,,, and f of the respective

steels:

_-AJr,+A,Af,,

Since it is proposed here (see S ection

5.3) to introduce prestress as an external

load into the analysis and dime nsioning,

the acting tie force T is the resu lt of all

external loads (including prestress).

Then, however, part of the strength of

the pre stressed steel is already u til ized

by prestressing and on ly the rest, Af,, is

available to resistT,.

4.7 Serv iceab i l i t y : Cracks and

De f lec t i ons

If the forces in the reinforced ties

under w orking loads are used and their

effective concrete area A , ,e f as defined

in the CEB M odel Code or in Ref . 19 is

attributed to them, the known relations

for crack control can be applied direct-

ly. o n In principle, it is propose d that

the same m odel be used at the ultim atelimit state and the serviceability limit

state.

In very critical cases it may be ad-

vantageous to select a mode l very close

to the the ory of elasticity, i.e., to provide

reinforcement that follows the path of

the elastic stresses. However, proper

108



failure zone AAc

ilure zonF

Fig. 24 . Assum ption of a failure zon e for the chec k of the ten sile

strength of a concrete tension tie

concrete stresses

with torture zcnefe without

falure zone AA

cack

Fig, 25. Progressive failure of a beam because of a local fai lure zon e, which

increases maximum tensile stresses a., to U., f^.

detailing (provision of minimum rein-

forceme nt, adequate selection of bar di-

am eters and bar spacing) is usually bet-

ter than sophisticated crack calcula-

tions.Having determined the forces of the

m odel, the analysis for the de formations

is straightforward. Sinc e the co ntrihu-tion of the concrete struts is usually

small, it is sufficient to use a mean value

of their cross sect ion even though this

varies over their length. For the ties,

tension stiffening follows from the

above crack analysis.

PCI JOURNAL;May-June 1987 09



a) c r f

fct

cfc

b) c) -1

ct fcttc 0.5fc fC f

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

4.8 Concluding Remarks

Des pite the fact that the m ajor princi-

ples have been set forth in this chapter,

mu ch research work remains to be done

with respect to the more accurate di-mensioning of the concrete ties and

struts. This, however, should not pre-

clude application of the proposed pro-

cedure as a whole. Current design

m ethods of D-regions are, in fact, worse

because they simply ignore such un-

solved questions,

Fo llow ing the f low of forces by strut-

and-tie-m odels is of con siderable value

even if used only to find out if, and

where, reinforcement is needed. In a

structure with reasonable dimensions

which is not over-reinforced, the con-

crete com pressive stresses are u suallynot the main conc ern. Furtherm ore, it is

much more important to determine

where the tensile strength of the con-

crete is util ized, and the n to reac t with

reinforcem ent i f possible, than to qu an-

tify the strength of the concrete ties.In the ne xt chapter the application of

the foregoing principles is elaborated

upon with many design exam ples.

5 EXAMPLES OF APPLICATION

With an unlim ited num ber of exam -ples it might be shown, that tracking

down the internal forces by strut-and-

tie-models results in safe structures and

quite often provides simple solutions for

problems which appear to be rather

com plicated. It should, how ever, also he

admitted that it som etimes takes som eeffort to find the appropriate model.

How ever, the strut-and-tie-m odel is al-

ways worthwhile because it can often

reveal weak points in a structure w hich

otherwise could remain hidden m the

design engineer i f he approache s them

by standard procedures.The m ajor advantage of the m ethod is

to improve the d esign of the critical D-

regions. Howe ver, the au thors also be-

l ieve that the con cep t will lead to more

realistic and workable code s of practice

also for the B-regions. Therefore, the

B-regions are first discussed he re by ap-

am



L L I HUi IFig. 27. Various cross sections of beams and their webs.

plying truss or strut-and-tie-mod els tothem. Thereafter, some U-regions are

treated (see Refs. 1 an d 3 for additional

design exam ples).Finally, the basic approach to pre-

stressed con crete is discussed and illus-

trated with several exam ples.

5.1 The B Reg io ns

The plane rectangular webs consid-

ered here apply not only to beam s with

rectangular cross sect ion bu t may alsobe part of a T, 1, double tee or box be am

(see Fig. 27). The she ar loading may re-

sult from shear forces or from torsion

(see Fig. 28), the axial forces from ex-

ternal loads or prestress (see Section

5.3).

As discussed above, the web of a B-

region m odelled w ith the same criter ia

as proposed for the strut-and-tie m odels

of the D-regions would in most cases

lead to a standard truss (Fig. 8), with the

inclination (f of the struts oriented at the

inclination a of the principal com pres-

sive stresses according to the theory of

elasticity.

The design of beams for bending,

shear and torsion is then nothing m ore

V f

_ 0 5M1 vw+CT Vwbm

Vhm–bm— -

Fig. 28. Shear forces as a result

of torsion.

than the we ll known analysis of the truss

forces and the chec k of the com pressive

stresses of the c oncrete an d the tensile

stresses of the reinforcem ent. Since this

analysis of the truss includes the cho rds,

possible problems like the staggering

ef fect or the quest ion why both chords

are simultaneously in tension at the

points of inf lection (mom ents – 0, shear

forces 4 0) are solved au tomatically.

PCI JOURNAUMay-June 1987 111



O eqao

refeld, Thurston

resler, Scorde/ s

® Rujagopalan, Ferguson

Petterson

• eorhardt, W alther

2 • ,Y addadrn et a .

Taylora Braestnup et al

Lyngberg, Orden, Sorensen

e Rodriguez et al

Guralnick ZL fs

of •

•

a

•

0, —r fut ^__ • L_ i^1o^ s ^ a V E

Q 5 gym8 ` 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 in

contrast to the bending theory, truss

models are capable of dealing with

stresses perpendicular to the beam's

axis and variable shear forces along the

axis:(Note that these sh ear forces are not

com patible with Bernou lli's assum ption

of plane strains as defined for the

B-regions.)

Why then, if everything appea rs to be

so simple, is there a shear riddle and

wh y has the shear battle been waged

for so many decades . In fact, the endless

discussions on shear could be put tobed if design engineers contented

themselves with the sam e level of accu-

racy (and simplicity) in the B-regions as

they do for the D -regions.

Since in principle it makes no sense to

design B- and D-regions, which are

parts of the same structure, at different

levels of s ophistication, there wou ld be

very good reasons for such a comparable

accuracy in every aspect of the ana lysis.

How ever, at present this seems not to be

appropriate mainly b ecause of h istorical

reasons, i.e., since so m any researchers

have invested so m uch time investigat-

ing the B-regions, they have foun d that

(under certain conditions) savings in

stirrup reinforcem ent are possible com -

pared to simple truss design.

Fig. 29 shows the we ll kno wn plot (indim ensionless coordinates) of the ulti-mate shear forces V. versus the am ount

of stirrups a„ (cm 2 /m ) required to carry V.

for beams in pure bending and shear. If

the straight l ine according to the truss

model with 45 degree struts (MOrsch) is

compared with the compiled test re-

suIts, it is found that a large discrepancy

results, mainly in the region of low tomed ium values of V,,.

The discrepancy is reduced if sm allerang les of inclination of the struts tha n

those taken from the elastic stresses at

the neutral axis are assumed. In fact,

from test results it is observed that the

angle a of the initial crack from pure

shear can be u p to 10 degrees less than

45 degrees, depending on the am ount of

stirrup reinforcement an d the w idth of

the web expressed by b lb. If, in addi-

tion, axial compressive forces such as

prestress act, a is a priori smaller than 45

degrees but deviates less, that is, thesmaller a, the closer it is to the angle

given by the theory of elasticity. 11

However, the fact remains that the

explanation of th e real stirrup stresses

only by the standard truss wo uld imply

the assump tion of struts with less than

realistic inclinations, in ex trem e cases

down to d = 15 degrees. Other kn own

explanations are also not satisfactory: Aninclined compression chord would si-

multaneously reduce the inner lever arm

z, which is not compatible with the B-

region assu mption of plane strains and

can, therefore, develop only in the D-

regions wh ich extend from the supports;

the pure arch or direct struts with tie ac-

tion can on ly be applied if the tie is not

bonded w ith the surrounding concrete

or to short beam s, i.e., beam s without

B-regions; the dowel action of the rein-

forcem ent, thou gh leading in the right

direction, can on ly to a sm all degree be

responsible for the effect und er discus-

s i o n .

Wh at does really happen in the web?

On the following pa ges it will be shown ,

that it is the con crete's tensile strength

and aggregate interlock in the webwhich really causes the reduction of the

stirrup stresses. Read ers who a re satis-

fied with this explanation or with the

pure truss design of B-regions m ay now

jump ah ead to Section 5.2. The authors

wrote what follows not with the inten-

tion of adding one more paper to the

shear dispute, but only to show the ef-

fectiveness of the strut-and-tie-modelapproach even for such cases.

After the principal tensile stresses

have reach ed the tensile strength of the

concrete, the web cracks at angles as

discussed 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 13



al Fi C

Q

t r fps

stirrup

b I load path 1

stirrup tie-canerete strut

c food path 2fl r

concrete tie - concrete strut

Fig. 30. Internal forces in the web due to shear: a) kinematics of load path 1 if acting

alone 8 = a); b) through c) load paths 1 and 2 in the web if acting combined 0 < a).

flanges, try to fall down. T here, they are

caught by the stirrups which hang upthe load via T into the adjacent piece

evoking C in the struts for vertical equi-

librium (Fig. 30a). The chords (or

flanges) provide horizon tal equilibrium

with ad ditional 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 that

the kinematics as described evoke an

additional load pa th 2 (Fig. 30c) w hich

comb ines with the load path 1 (Fig. 30b)

but which is usually neglected: The

vertical movement v has two compo-

nents, the crack opening w perpendic-

ular to the crack and a sliding A parallel

to the crack (Fig. 31). The sliding A is

obviously resisted by aggregate inter-

lock in the crack and it appears reason -able to assum e, that the resisting force R

acts in the d irection of ', i.e., parallel to

the crack. The force R has two com po-nents, a compressive force C, with an

inclination B < a an d a concrete tensile

force 'I perpendicular to it (Fig. 32).

Again, the chords are activated

for equ ilibrium .

Both load paths jointly carry the loadand therefore their comb ined comp res-

sive struts together assu m e the inclina-

tion 0 -_ a, As long as it can be sustained

by the concrete, the concrete tensile

force perpend icular to the struts is re-spon sible for the fact that th e stirrups

need to carry only part of the shear

loads. However, it also causes the con -

crete of the struts to be biaxially loaded,

thus either reducing their comp ressivestrength or resulting in a second array of

cracks w ith inclinations less than a , de-

pending on the load case. Only if 0 = a

does load path 2 disappear (Fig. 30a).Wh en this occurs the com pressive struts

are uniaxially loaded and can the refore

develop their maximum strength.

Therefore, the maximum capacity of a

beam for shear forces is achieved if thestruts are parallel to the cracks and if the

corresponding large am ount of stirrups

is provided.

What can so simply be described in

words mu st also be accessible to a rela-

tively transparen t analysis. This is pos-

sible, if the com patibility between load

114



paths 1 and 2 is solved by plastic su-

perposition of b oth the stirrup and the

interlock forces, making u se of the fact

that aggregate interlock is sufficiently

ductile. How ever, since a biaxial failure

criterion for the concrete ha s to be ap -

plied, a reproduction of the analytical

description (Ref. 15) would go beyond

the scope of this paper. The result is

plotted in Fig. 33 and compared there

with results from experiments.

It is imp ortant to note that the beam

rem em bers the initial inclination a of

the diagonal cracks. This permits the

logical introduction of the effect of pre-

stress into web design, resulting in sav-ings of stirrup reinforcement but also in

an earlier compressive strut failure.

The analytical curve in Fig. 33, which

gives also the actual inclinations B a as

a function of V, m ust be cut off before it

intersects the abscissa (where 0 = d2),

because a m inimum of stirrup reinforce-

m ent is necessary in order to guarantee

that a truss model can develop at all. Asshow n in Fig. 34, it is necessary to avoid

that the flanges separate from the web

after a diagonal crack has formed.

Of course, the longitudinal and trans-

verse spacing of the stirrups m ust further

be limited to ensure a pa rallel diagonal

com pression field. If the spacing is too

large, the smeared diagonal compres-sive strength may be less than that taken

for 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 is

considered desirable to be more soph is-

ticated there than in the D-regions,

where aggregate interlock or the con-

crete tensile strength is neglected -

simple diagrams may be derived from

Fig. 33. Even easier (and correspon ding

to the present CEB M odel Code), it may

he sufficient to reduce in the low shear

range the acting shear force for stirrup

design by an am ount wh ich is attributed

to load path 2.

For that purpose, it may be su fficient

v = v e r t i c a l m o v e m e n t

w=crock opening

ti=sliding

Fig. 31. Displacements in the web because

of the crack.

crack f a c e

TcRcL cR

Tc = r e s u l ta n t of the tension field

Cc= r e s u l t a n t of the compression fieldR = force of the aggregate interlock

Fig. 32. Aggregate interlock force R and

corresponding compression C. and tension

T, in the concrete.

to replace the curved bran ch of the dia-

gram in Fig. 33 b y a straight l ine. It is

only important that the design engineer

is aware that reliance is- placed on the

concrete tensile strength in the web if

this method is used to reduce the re-

quired am ount of stirrups. Fig. 29 corn-pares the amount of stirrup reinforcer

ment derived in this way from the strut-

and-tie-model with that from different

codes and tests.

One might also ask whether the full

design strength f can be exploited in

the compression chord of a beam, be-




pv fm s s bc

0 , 4 0

4 Broestrup MW . Nelsen M P . Bach F , Jensen B.0

• resler B , Scabelis AC

0,35 —• urnlnik 4

• oddodm M J , Hong ST . Mattock A H

Krefeld WJ , Thurs ton CW

0 30 --- A eonhordt F alther R

° ynberg BS , Ozd en K Sorensen H C Petterson T 0 25Ram apo lnn KS , Ferguson P M o Regan PE _-

0 odrquesJJ,BionchmiAC,Viest1M,

Kesler C E / 02 0 _ o aylor R.

6

0,15 — — — 0 df _ off'

010 — a

o 250 0 • ,OS^

/oo

1 9 . c

° 0,25 ,0 ,35 ,4 0,05 10 15 20

Fig. 33. Req uired amount of vertical stirrups in awebaccording 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 tensile

strains in both transverse directions. In

the horizontal direction such stresses

are quite obvious in the flange of T-

beams with transverse bending and

transverse tension from the flange con-

nection. In addition, there is vertical

tension from compatibility with stirruptensile strains. Some cross tension also

exists in both directions in rectangular

beams (Fig, 35).

The neglect of this fact in practical de-

sign is to some extent balanced by the

usual neglect of the staggering effect in

the compression chord, which reduces

the forces in the compression chord as

much as it increases the tension chord

forces. However, if the chord forces are

correctly derived from the truss model,

the compression strength should be as-

sumed to be only 0.8 ff, according to

Section 4.1.

Consider, in addition, a special butvery frequent B-region, namely, beam or

column with rectangular cross section

loaded by an axial force in addition to

shear forces and bending moments. This

case is typical also for prestressed

beams. If the axial force is large enough

to keep the resultant of the normal force



compression chord

^— tirrup web —T

s

stirrup

eb

C

tension chord

T 5 • C C 20

Fig. 34, A minimum amount of stirrups is necessary to tie

together the web and the flanges.

a) .........__—

_........

/ i

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

and of the moment within the kern of

the cross section, the standard truss

model no longer applies. Instead, all

internal forces, including the shear force

V, may be represented by a single in-clined com pression strut as in Fig. 36b.

Looking closer at the transition be-

tween such sp ecial B-regions and those

represented by the truss model, it is

found that the com pression stresses dis-

tributed over the w hole section have to

converge into the narrow compression

chord of the adjacent B-region, thereby

creating transverse tensile stresses.

These stresses can be assessed by the

more refined mod el in Fig. 36c. S tirrups

may be used to cover them; however,the m odel shows that tensile forces are

inherently quite different from the shear

forces in other B-regions b ut similar to

those in the bottle shaped comp ression

strut.

Speaking of B-region design, there

remains the issue of beams without




a) oment

shear force

compression force

a

^

resultant within

kern-zoneresultant within lever arm z

mod el compatible with the standard truss model

C)

±

kernzoneLi [JE

forces of stirrups

d)

forces of stirrups

Fig. 36. Beam with compressive axial load: (a) sectional forces; (b) simplified model with

a compression strut between the trusses: (c) refined model for a rectangular cross

section with vertical tensile forces; (d) refined model for a I-beam.

1 1 8



shear reinforcement. Sub dividing such

beam s into B- and D-regions first of all

reveals that an arch-an d-tie explanation

only applies if the two D -regions of the

opposite supports touch, leaving no sig-

nificant B-region in between (see alsoFig. 38a). Unbonded tensile chord re-

inforcement also stimulates arch-and-tie

action, in other words it increases the

length of the D-regions. The web of a

real B-region, how ever, can carry shear

loads 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

Aga in, as above, aggregate interlock is

involved and, therefore, an analytical

solution is rather com plicated. On the

other hand, the design engineer must

know and understand what actually

happen s. Thus, it wou ld realistically re-

flect the present situation to replace the

perm issible shear stresses of the current

codes by the equivalent permissible

concrete tensile stresses. Unfortunately,the stresses in the web are not un iformly

distributed. Therefore, calibrated aver-

age adm issible tensile stresses mu st be

given in codes.

For comp leteness, it should be men -

tioned that a B-region contains several

micro-D-regions, which again can be

understood and designed with strut-

and-tie-models. Fig. 35b shows why astirrup must he closed on the top and

bottom in a rectangular beam in order to

take the cross tension from the strut

support.

5.2 Some D Regions

5.2.1 Deep Beam With a Large Hole

(Numerical Example)

Given:Dim ensions (see Fig. 37a)

Factored loadF, F = 3 MN

Concrete compression design strength

= 17 MPa

Reinforcement design yield strength

f„d = 434 MPa

Required:

Strut-and-tie-model

Forces in the struts and ties

Dim ensioning of t ies: reinforcement

Ch eck o f stresses of critical struts and

nodes

Reinforcement layoutSolution:

(1 ) External equilibrium: reactions

A=3x --= 1.07 MN7.0

B=3x 4== 1.93 M N7.0

F=A +B– 3.00 MN

(2 ) Elastic stress analysisThis is a rather comp licated structure.

If the design engineer is not yet suffi-

ciently experienced w ith m odell ing, he

will first employ an elastic finite ele-

ment program an d plot the elastic stress-

es (see Fig. 37b) for orientation of the

strut-and-tie-model.

(3 ) Modelling

The w hole structure is essentially oneD -region. Two sh ort B-regions are dis-

covered in the linear parts to the left and

below the hole (see Fig. 37c).

The load p ath connecting the reaction

B an d its counterpart within F is readily

plotted (see Fig. 37c). The positions of

Nodes 1 a nd 2 are typical (see Fig. 15)

and the forces C and T balancing these

nodes in the horizontal direction arethus also given.

Before continuing modelling on the

left side, the right side m ay be finished:

The strut between Nodes 1 and 2 will

spread and cause transverse tensile

forces as sketched in F ig. 37d ; option-

ally, it can be treated as a bottle shaped

stress field as sketched in F ig. 371i and

described later on.Now the left side's boundary forces

are clearly defined (Fig. 37e) and it can,

therefore, be modelled independently

from the right side. (In pas sing, it m ay

be m entioned that the fictitious separa-

tion line between the two sides is where

the overall shear force of the deep beam

is zero and the bending moment is

PCI JOURNA UMay-June 1987 19



• o 7

FF-3MN

Oi 415t 5 25 X 5

7

a) dimensions [m] and load

i i I \i •

tiJI__

compression stressestens on stresses

b) elastic stresses

120



B

c) load path, right side

I1 \

^ d) complete model, right side

Fig. 37. Deep beam with a large hole.




A

C

e) boundary forces, left side

Bi

I

BZ-rN on

A

f) model 1, left side

) .5A

105E

0 . 5 T

A2= 05 A

CZ=05C

` tI I

I jl

I T4 ^

4 5 0 T=05T

ArO5A

g) model 2 left side

122



node 2

ar p

- '(b.net

bottle

h) complete strut-and-tie-model

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

L LT L L 11

i) reinforcement

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

PCI JOURNA L/May-June 1987 23



moaei

e ned ump€ohed

n 1

T t -

s fig 74 i A h

B i

tl

c l

♦- td

et

r ^^-

stirrup forces

main reinforcement

C - - 1

rlHH

N

Fig. 38. Load near the supp ort; transition from eam to beam.



ma ximum . Therefore, only the horizon-

tal forces 177 =1 CI connect the two

sides.)

Forces C and A meet at Node 3 and

their resultant is given. From the bottom

the reactionA enters the structure verti-

cally and it is assumed that it remains in

this direction until it has passed tho

hole. The B,-region is thus a centrically

loaded column.

In fact, part of the reaction A could

also he transferred via the -region b y

bending moments and shear forces.

Com paring the axial stiffness of B, w ith

the ben ding stiffness of B , this part is

obviously negligible. This fact is also

confirmed b y the elastic stresses, which

are very small there (Fig. 37b). Of

course, some nominal reinforcement

will be provided in the member below

the hole for distribution of cracks due to

imposed deformations.

Figs. 37f and 37g give two different

strut-and-tie-models for the Ieft side of

the deep beam. It is seen that there issome similarity with a beam having a

dapped end. The model in Fig. 37f is

based on 45 degree struts and on tiesalso at 45 degrees from the struts. This

gives a reinforcem ent layout, which for

practical reasons is orthogonal.

The m odel in Fig. 37g assumes a 45

degree tie at the corner of the hole

which is know n to b e effective in similarcases like opening frame corners or

dapped beams. Each model in itself

would be sufficient but - looking at the

elastic stresses of Fig. 37b - a combi-

nation of both appears to be better thaneither of them. Therefore, it is assumed,

that each m odel carries half the load.

Finally, Fig. 37h sh ows the superpo-

sition of both mo dels of the left side in-

cluding the right side as described be-fore. Wh en com paring model and elastic

stresses one finds a satisfactory coinci-

dence. The geom etry of the mod el is in-

deed o riented at the elastic stress fields.

(4) Design o f the ties

The reinforcement requirement for

the respective tie forces is given in the

table below.

Tie

Force

[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__-

T 1.07 24 . 6 2x7 5 0.08_-

'I'1 7

24 . 6 2x7 5 0.08_ -

1 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, ° 0.4 02 9.2 or bottle check

(see below)„ 0.402 9.2

Note: cm .155 in.

(5) Check of the concrete stresses

Stresses under the bearing plates:

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

(biaxial compression)

Cr _1 07-=.).4MPa<0.8fd

0.5 x 0.4

1.930 = ---=9.7MPa<0.8ff4

0.5 x 0.4(compression with transverse tension

from tie)

The most heavily loaded node is the

support B. Following Fig. 22a and the

detail of Node 2 in Fig. 37h:

9 7Pa = v =

[ 1 + 0= tan 29° l cos t 29°

l

0.5

1= 8.1MPa<0.8f4

Bottle shaped stress field at right side:

pa=8.1MPa

pa= 8 1=0.48

fed 17

From F ig. 23a for a = 0.68 m and b

2.0 m and hla = 3.4, the permissible

PCI JOU RNAL/May-June 1987 125



p. ffa for uncracked plain concrete is

0.62 > 0.48.

The stress field is safe without trans-

verse reinforcem ent. If no dependen ce

is placed on the concrete's tensile

strength, the reinforcem ent as given by

Fig. 23a or, equivalent, as calculated

from T,,, and T„ may be p rovided.

Chord C u nder the loaded plate:

C–Ir1-1.07 MN

Required depth of comp ression zone:

d 1.07

0.4x1.2f,,,

1 7

0.4 x 1.2 x 17= 0.13 m < 0.4 m

Nodes I and 3 were taken 0.2 m below

the upper surface and are the cen troids

of the rectangular compression zone

which h as therefore an available depth

of0.4 m.

(6 ) Check of the anchorage lengths of

the reinforcing bars

The anchorage length l e f of the tie Tin Node 2 must com ply with the anchor-

age length given in the code, consider-ing the favorable transverse compres-

sion from Reaction B. Anchorage begins

at the left end ofthe bearing and ex tendsover the whole length of the bearing.

Accordingly, T is anch ored with cross

pressure over the column type B1-region

to the left of the hole, Less favorablestress conditions with transverse tension

exist in the node w here T, and T, meet.

Therefore, the T,, bars should be loop

anchored whereas the T, bars pass

through tho se loops and find su fficient

anchor length in the B,-region below the

hole. Anchorage in the other nodes is

less critical.

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

me nt resulting from the abov e design is

shown. The design engineer should pro-

vide further reinforcement, such as a

mesh on either surface of the wall, nomi-

nal colum n reinforcemen t at the left of

the hole and stirrups below the hole.

5.2.2 Transition from B eam to D eep

Beam — Variation of Span-to-DepthRatio

If a single load is app lied at a distance

a C h near the support (Fig. 38a), the

load is carried directly to the supp ort bya compression stress field; therefore, no

shear reinforcement is required be-

tween the load and the support. How-

ever, the transv erse tensile stresses in

the com pression strut may ca ll for diag-

onal reinforcement or vertical rein-

forcement instead.With increasing a, this single strut

model gradually blends into the trussmod el as shown in Fig. .38b, c. Follow-

ing the principle of minimum strain

energy, the com pression mem ber C, and

the tensile member T, (Fig. 38a) of the

chord combine (Fig. 38b) and thereby

cancel the compress ion force C, and an

equivalent part of the tension force T1

Simu ltaneously, the transverse tensile

forces of the strut (Fig. 38a) blend into

the vertical ties of the truss m odel (Fig.38b), which are now also needed for

hanging up the shea r forces.

In the typical truss models on the

right sides of Fig. 38c and d and of Fig. 8

the inclination B, of the support strut is

shown to be m uch steeper than 8 of the

interior struts. This can be proven b y the

principle of m inim um potential energy

if models with different strut inclina-tions are compared. On ly the geom etri-

cal relation:

cot H , =a

+c2z 2

provides constant vertical web tension

t , or equal distances e betw een all verti-

cal ties and thereby distributed stirrupforces which are proportional to the

shear forces (Fig. 39). The force in thetensile chord would have to increase

near the support, if 9 were constant

throughout, or in other words, the

supplem entary chord force from a c on-

stant shear force is constant along the

6



i,i T w = V w i z•cof

0) T w = V

rcote ----

TryW =V I z^ w

T

111111 l l I w = V

b

^ T w = V

cot 9t

cot ^= z

Fig. 39. Detailed geom etry of D 2 -region from F ig. 3 8:

(a) d irect support; (b) indirect su pport.

chord only if B, satisfies the above given

relation.

5.2.3 Corbels

Some of the so-called problems of

corbel design result simply from the

fact that the individually different

boundary conditions are not con-

sequ ently considered. Each case leadsto a differen t strut-and-tie -model (Figs.

40 and 41). Furthermore, from Fig. 40d

it becomes evident that the comp ression

strengths of the struts, which a ccording

to Fig. 23 a re a function of alb, depend

via a, , , on the detailing and via a, s and a R

on the depth x of the bending compres-

sion zone of the column.

The modelling of a deep corbel isshow n in detail in Fig. 41: The stresses

in the column of Fig, 41a may be

evaluated according to standard

m ethods or the the ory of plasticity and

are then applied to the D-regions to-

gether with the corb el load F 2 . The fact

that the vertical struts at both sides of

the B 2 -region in Fig. 41b do not coincide

(the stress diagrams, of course, do) is

due to the different grouping of stresses

for the struts with respect to mod elling

of the D-regions D 2 and Dg.

Balancing the applied forces between

the two boundaries of each D-region

(Fig. 41b) shows that two tension

chords develop, the first one as us-

ual, right below the upper edge of the

corbel, and a second one in the lower

part of the corbel. Their forces are in this

case T, = 0.23 F 2 and TZ = 0.32 F 2 , e-

spectively.The compression struts of the same

corbel are not critical becau se the sing-

ular nodes in the D, and D , regions with

maximum comp ressive failure strengthsof 1.2f according to Fig. 23 keep the

com pressive stresses in the rest of the

structure relatively low.

5.2.4 Details of a Box Girder Bridge

The critical D-regions of a box girder

bridge are described by strut-and-tie-

PCI JOURNAL+'May-June 1987 27



0

I ;\ I,

1

^

b

r

i

^ t r

ss ry

f /

1

• I

I

it/ _J 1

t

I

m

128



RI4i

♦ i r

r

Fig. 40. D ifferent supp ort conditions lead to different strut-and-tie-m odels and different

reinforcement arrangements of corbels.




mod els as shown in fig. 42.

In Fig. 43 the diaphragm is pre-

stressed: The shea r forces V, carried by

the diagonal struts C, in the webs of the

box girder (Fig. 43a) are in this case

transferred to the central supp ort via a

prestressed web reinforcement de-signed for T, = V,/sin/ 3 = 1.01 V , and

curved tendons designed for:

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

5.2.5 Beam With Opening

Fig. 44a shows the sectional forces

and the sub division of the beam into B-

and D -regions. Following the p roposed

procedure, the boundary forces of the

B-regions have to be a pplied as loads to

the D -regions.

The B -region will be represen ted by

the standard truss, wh ich yields for 9 =

31 degrees:

CM ot (z _C ^ sin B

T, M 2 1 cot 9z

_ 3 55F ,1.15 + --1.66

= 3.92 F4

850KN

F =425 KN Fz LJy

110

t ° 3 Tz

0 4 ct

Fig. 41. Deep corbel projecting from a column: (a) B- and D-regions; (b) boundary forces

of the D-regions and their models; (c) reinforcement layout.

1 3 0



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

T Z 1 15 ^ – 5.35 Fa

The lever arm z = 1.15 m is deter-

mined from the assumption of planestrain in the cross section in the middle

of the opening by standard methods.

Therefore, B., has to carry the axial com-

pressive force C = T,, (eccentrically with

respect to the axis of the B -region) plus

the differential mom ents AM =M – M

plus the total shear force V.

Under the combined action of these

forces, the B, region show s the transi-

t ion from the column type B-region on

its left end (resultant C ) to the truss

type B-region at the other en d (C g , C5

T 3 ) (see Fig. 36). For simplicity, the

model of the B,-region is extended

somew hat into the D 2 and D 3-regions,leaving over for modelling only the D,

and D 3 regions in Fig. 44b.

With known boundary conditions

from B , B, and k, the mod el for the D,-region can be developed (Fig. 44c):

Looking for the counterparts of C„

and T, at the opposite side of the D,-

region it helps to split up C 3 into three

forces C , C' and C'.,' in order to estab-

tablish the load paths, which balance

their horizontal com ponen ts: (C 5 = C1;,_

Vertical equilibrium in the D 2-regionis established by a v ertical tension tie TK

and b y a vertical com pressive strut CTheir forces depend on the choice of

their position. Kno wing that they repre-

sent transverse stresses which are insidethe D -region and that these stresses

tend to fill the available space, the re-

sultant tension T S is chosen in the mid-

dle of the D2-region and the resultant

compression C. at the right end of D,,

(which is inside D 2 ). Then:

Ts = (T –T 1 ) tan B

= (5.35 F – 3.92 F,,) tan 46°

= 1.48F4The tie force T 5 may be interpreted as

the transverse tension n ecessary to an-

cho r the differential force T, – T l of the

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

other end of the opening may be treated.

The transverse tension forces are:

T =V =Fdand

Tg =(T,–T.) tan H2= T,–T –V cot 0)tan H2

= (8.79 F 4 – 5.35 F, – 1.66 F4 tan 50'

= 2.12F4

A striking result of this example arethe stirrup forces which in som e places

considerably exceed the norm al shear

reinforcement for a beam w ithout open-

ings (Fig. 44d).

This example also shows, however,

that it sometimes takes some effort to

develop a realistic strut-and-tie-model.

But if it is considered how many tests

have been carried out and how many

papers have been written on beams w ith

openings w ithou t finishing the case, it

was w orth this effort.

5.2.6 O verlap of Prestressing Tendons

In Fig. 45a the flow of forces in the

given cross sec tion is investigated w ith

its B- an d D -regions. In a longitudinal

section through the blister, the B-regions

of the cross section m ay also appear as

D-regions, because the cu rvature forces

of the tendons are line loads.

(a) Prestress only

Loa ds acting on the D-regions are theanchor forces of the tendons P. the

known concrete stresses along the

boundaries hetween B- and D-regions

and the curvature forces Pa l from the

tendon s (Fig. 45b). The horizontal com-

ponents of all forces follow two load

paths, wh ose position and d irection is

given at all D-region boundaries.

The relatively small shift of the load

path in between m ust be accomplishedby deviative forces within the D-

regions. This is a rather strict condition,

which limits the freedom of choice for

the bends of the load paths. A further

help in finding appropriate nodal points

of the model is again the tenden cy of the

deviative forces to fill the available

PCI JOURNA LIMay-June 1987 3 1



compress ionTrz3tensron

W

I I I I I I I

b) ompressionrTr p

I

, SI I •ft

^ I 1

I

^

d

. v^ 5r

T ^r

132



► C 1a_

t 71

IIu I.e

BI II I I

k T I

1 l

t Fig. 42. Strut-and-tie-models of typical D-regions of a box girder bridge: (a) tensile

flange with opening; (b) compression flange with opening; (c) web supported by

diaphragm; (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 2 V ; T 2 = T p • l t r n , v .

L T±J

IC

Fig. 43. Diaphra gm of a box girder bridge: (a ) D-regions and

model of the web nea r the dia phragm: (b) dia phragm a nd

model: (c) prestressing of the web a nd the dia phragm .

spa ce of the D-region.

Then, T, = P tan a 2 = 0.18 P see Fig.

45b).The reinforcement for T i should be

chosen with due consideration of ac-

ceptable crack widths in this service-

ability state of stress, i.e., the tensile

strength of the reinforcement shou ld not

be fully utilized a nd its slipfree a nchor-

a ge should be a ccounted for.

(b) Prestress plus other loa ds

Tensile forces a re a ssumed in the B,-

regions, which are considerably largerthan the prestressing force, a s is the case

in an overloa ded or partia lly prestressed

structure. In ca se there is no additiona l

reinforcement provided, and the tendon

ha s to take a ll tension forces the excess

force (T h 0 ,( according to Section 5.3.6)

wa s computed to be:

134



a) _ 107 °Z c 1375kN

m

C B 0Z 3C 0 4 0 . 1 5 a2 +

moments M

lv

y zshear forces V

O

b

C ^ 4 C 6

C^Z C 3 C 3 B ^ 3 C 7

3T , Z z TC

E F T : : : ; i jizT

T8

d s up5 = 2 1 F

5 05F n IDF

—stondird design B-regon

Fig. 44. Beam with opening: (a) B- and D-regions, sectional forces: (b) reduced

D-regions at both ends of the opening with bounda ry loads from the B -regions:

(c) strut-a nd-tie-models of the 0 2 -a nd D ,-regions; (d) distributed forces for the

design of the stirrups; (e) reinforcement layou t.

e

PCI JO UR NAL /M ay-June 198 7

1 35



T,=T-P=0.33P

The loa d pa th for T R a nd the resulting

model (Fig. 45c) is now quite different

from tha t for prestress a lone, beca usethe T on both sides do not meet, but

ra ther try to equilibra te on the shortest

possible way with the nearby anchor

loads of the other tendons. Thereby,

transverse tension T, develops.

The position of T, may have to be

shifted towa rd the center of the blister,

if the bond of the tendon is very poor. It

is the tension from these bond forces

which now calls for transverse rein-

0)

g 2 a^-

P- curvature forces a Z

n, T agc)

TzzT-P

crock

T

T z mN N N h

oi i iFig. 45. O verlap of prestressing tendons: (a) layou t, B- a nd D-regions; (b) model for

prestress only; (c) model for prestress and a dditiona l loa d T d ; (d) lay out of the transverse

reinforcement.

1 36



forceinent T, = 0.53 T 2 , not the splitting

a ction a s for prestress a lone.

The overlap of the prestressing ten-

dons results in twice the prestressing

force within the blister, which therefore

rema ins ba sica lly uncracked a nd rela-tively stiff. The stra ins of the tendon in

excess of the initial prestress will

therefore accumulate within the bond

length of the tendon in the blister a nd

cause a crack at the jump of the wall

thickness.

For la rge bond lengths this crack ma y

open several millimeters wide, even if

the -region is reinforced for crack dis-

tribution in the usua l wa y. Therefore, in

this case not very large tendons with

good bond properties have to be a pplied

or additiona l pa ra llel wa ll reinforcement

must be provided, which takes over

much of the tendon forces, before the

tendons enter the blister.

5.2.7 Beam W ith Dapped End

It is common pra ctice to suspend the

rea ction F of the bea m in Fig. 46 besides

tII

T =F ]

II

3 ice

F s ^ r — =zcot —

144414 444 lUrlill

r t3=1z^

I L LFig. 46. Beam with dapped end.

al

f f T

_ _ _ _ _ _ l l i r^c bl

Fig. 47. Girder with bent top flange.

PCI JOUJRNALMay -June 1987 37



rig. 48 1hestepped beam.

the dapped end (T, = F). But the com-plete strut-and-tie-model clearly re-

veals that it is not sufficient to simply

add T, to the regular shear reinforce-

ment which a mou nts to the vertical tie

forces t 3 = F/1 . In fa ct, there are a ddi-

tional vertical tensile forces T 2 = F be

cause the horizontal tie force T at the

recess needs to be anchored. The tie

force T 2 is distributed over a length 1 2 <

l 3 and therefore t 2 is clearly considera blylarger than t . If, a s usua l, a n additional

horizontal force H a cts a t the recess, the

necessary amount of vertical stirrups

further increa ses.

5.2.8 Ta pered Beam W ith Bent TopFlange

The girder in Fig. 47a obviously pro-

duces a vertical tension force T at the

bend of the compression chord. But

where does it go? The straight horizon-

ta l tension chord ca nnot equilibra te it.

The model shows that stirrups in the

web a re necessary throughout this web

even in regions without shear forces.

Looking a t Fig. 47b, it is a ppa rent tha t

the compression chord is na rrowed by

the stirrups, resulting in a concentra tion

of compression stresses over the web.Furthermore, unfavorable tensile

stresses in the transverse direction of

the flange appear.

5.2.9 S tepped Bea m

The stepped beam in Fig. 48 is fre-

quently used a nd is usu a lly detailed by

overlapping the reinforcement comingfrom both sides with an elegant loop.

The strut-a nd-tie-model supplies fa ctsfor a ra tiona l reinforcement la yout.

5.2.10 Frame Corner

The frame corner with opening mo-

ment, more often discussed by re-

searchers than actually occurring in

practice, can be modelled quite differ-

ently (Fig. 49). Obviously, the designengineer ha s to choose between a rela-

tively simple reinforcement combined

with a reduced moment capacity (Fig.

49a ,b) or a more sophistica ted solution

(Fig. 49c,d,e). The consequent a pplica -

tion of strut-a nd-tie-models ma kes thedesigner aware of what is occurring

while offering a ra tiona l choice.

5 3 Pres t ressed Conc re te

As a last example, it will be shown

that looking at prestressed concrete

beams through strut-and-tie-models

helps to understand their behavior

which today gets hidden behind so

ma ny black box rules. There is a common

denominator of all types of prestress:

post-tensioning, pretensioning and un-bonded prestress can be u nderstood a s

reinforced concrete wh ich is loaded by

a n a rtificia l loading ca se, i.e., prestress.

As any other Ioading case, it simply

ha s to be introduced into the ana lysis of

the structure according to the actual

history, e.g., for post-tensioning: con-

138



_ H f t i

__ riL P

d? J l

Fig. 49. Different strut-and-tie-models and the corresponding reinforcement for a frame

corner with positive moment.




creting and ha rdening of the reinforced

concrete, applying the prestress

(thereby activating dead load), provid-

ing bond and imposing externa l loa ds.

A fter bond is a ctiva ted, the prestressing

steel a cts as reinforcement, like regula r

reinforcement does; only its preloa ding,its different surfa ce with respect to the

strength of bond and its sensitivity

needs to be taken into a ccount.

5.3.1 Prestress as a Loa d

By prestressing, forces a re a rtificiallycreated with the help of hydraulic jacks;

on the one hand, these forces act as

loa ds on the prestressing steel, on the

other a s loads on the reinforced concrete

structure (Fig. 50). The loads a cting on

the prestressing steel and on the rein-

forced concrete are inversely equa l. The

design engineer chooses the tendon

profile, the type and magnitude of theprestressing force in such a ma nner that

these a rtificia l loa ds influence the loa d

pa ths a nd sectiona l effects or stresses

due to the actual loads (dead and liveloa ds and other loa ds) favora bly a nd a s

efficiently a s possible.

It is proposed to treat these pre-

stressing loads like permanent loads

which never cha nge a fter the prestress-

ing jack has been removed. All the

changes of stress in the prestressing

steel, which occur a fter remova l of the

jack, ought to be attributed to those loadca ses which ca use them. In those loa d

ca ses the prestressing steel adds to the

resistance of the section or member likenonprestressed steel.

The view sometimes expressed that

the prestressing itself increa ses under

live loads because the stress in pre-

stressing steel increa ses or decreases a s

a result of the bond with the concrete, is

misleading. The stress a lso changes in

normal reinforcing steel due to these

effects and one would never rega rd thisa s a change of prestressing.

A ccordingly, the cha nges of stress in

the prestressing steel due to creep,

shrinkage and steel relaxation (often

referred to as prestress losses) are in

reality simply stress redistributions

as in any reinforced compressed

member and can be trea ted accordingly.

If prestressing is introduced in this

way into the design of reinforced con-

crete, all types of prestressed structu res(linear members, plates, deep beams,

shells) can he designed, analyzed and

dimensioned like reinforced concrete

structures: The sectiona l forces of a pre-

stressed bea m a re determined a nd com-

bined for the load ca ses prestress, dead

load, live loads, etc., a nd the resista nce

of a cross section is derived as for re-

inforced concrete with prestressingsteel as additional (passive) reinforce-

ment.

Thereby, the sa me method of ana lysis

a nd dimensioning ca n be a pplied to a ll

load combinations in servicea bility a nd

ultima te Iimit sta tes. Consequ ently, in

the ultimate limit state the reinforced

concrete sections (with reinforcement

A ,andA ,) of a prestressed beam ha ve to

be dimensioned for the following (ac-

tive) sectiona l effects:

N orma l forces: N = –P + Nf

Shea r forces: V = V,, VL

Moments: M = M„ ML

where

P = prestressing force immediately

a fter prestressing

A1 V = corresponding mom ent and

shear force due to pre-stressing

N V , M^ sectional effects from

other loa ds

The sectional effects given here are

meant to include the (partial) safety

factors according to the chosen safetyconcept.

The proposed trea tment of prestress-

ing lea ds to the sa me results a s does the

usual method with all stresses in the

tendons regarded as being passive in

the ultimate limit state. However,

the method proposed here is more gen-eral.

140



E i : iFig. 50. Loads due to prestressing (anchor forces, friction forces,deviation forces due to the curvature of the tendon) acting (a) on the

prestressing steel; (b) on the reinforced concrete member.

Also, the different degrees of pre-

stressing (full prestressing, pa rtial pre-

stressing, no prestressing) and different

applications Iike pretensioning, post-

tensioning a nd unbonded cables can a llbe treated alike with these principles:

Forces due to prestressing as permanent

active loads which never change and

prestressing steel contributing to the re-

sistance a fter the cable is anchored.

5.3.2 Prestressing of Statically

indeterminate Structures

In structures with sta tica lly indeter-

minate supports the properties of thematerials and the geometry of the

structure have to he considered when

determining the sectional effects. Those

from prestressing can be split up in:

V„= P sin 6+V,

M = –Ve M 12

where S denotes the inclination of the

prestressing ca ble and e denotes the ca -

ble's eccentricity. V., and Mm e the sta t-

ical indeterminate portion of the pre-

stressing effects resulting from support

reactions due to prestressing, are of the

same kind as the statically indetermi-

na te moments which result from dea d

loads or live loa ds. M , is not the resultof restraints such as those due to

cha nges in tempera ture or settlements;

these diminish or even disappea r in the

whole structure if the stiffness decrea ses

due to cracking, those from prestressing

a re only redistributed.If it can be assumed that the stiff-

nesses a re not a ltered by loa ding, mo-

ments resulting from different loading

ca ses ma y be superimposed. However,

beca use pa rt of the reinforced concrete

girder pa sses from the untra cked to the

cracked state and because plastic de-

forma tion of the concrete a nd the steel

in the reinforced concrete girder is pos-

sible, the local stiffnesses change withva riations of the loa d a nd, therefore, the

distribution of moments a lso cha nges.

When this occurs the individual mo-

ments ca n no longer be superimposed.

Nevertheless, if the theory of linear

elasticity is taken as a ba sis for ca lculat-

ing the forces, the resulting overa ll mo-

ments (including M 1 ) can be adapted

to the rea l loadbea ring beha vior by re-distribution.

5.3.3 The Prestressed Concrete Bea m

With Rectangula r Cross Section

In Fig. 52 the strut-and-tie or the truss

model of a simple prestressed concretebea m w ith a straight eccentric tendon is

PCI JOURNAL/May June 1987 41



shown a nd in the preceding Fig, 51 , for

comparison dra wn in the sam e ma nner,

of a reinforced concrete bea m, The de-

sign engineer selects the prestress P and

combines it with the su pport force A = Fto a resultant, R , entering the bea m a t

the support.If the resulta nt meets the a ction line

of the load F within the kern of the sec-

tion, the condition is full prestress, if Fis the working load F. Then the modelha s no tensile chord (Fig. 53). If the re-

sulta nt meets the compression chord of

the beam before it meets F. such that

vertical ties (stirrups) and, therefore,

also a truss with a tension chord is

necessary to transport F. into the in-

clined resulta nt, the condition is pa rtia lprestress (Fig, 52).

It is appa rent tha t the vertica l tie and

inclined strut forces of this truss a re (a l-

most, see below) the same as for the

reinforced concrete beam , becau se the

shear forces a re the sa me. The span of

the truss is only shorter a nd, therefore,

the forces in the tensile chord a re less,whereas the compression chord a lso in-

cludes the a dditiona l prestressing force.

If the load is increased from F,,, to the

ultimate F,,, of course, the inclination of

the resulta nt R increa ses. The fully pre-

stressed bea m now becomes pa rtia lly

prestressed, the initially partially pre-

stressed beam gradua lly a pproa ches the

reinforced concrete bea ms. (This, inci-

dentally, shows that ma nipulating safety

completely via a factored load is mis-

leading in contrast to pa rtia l safety fac-

tors being put on the ma teria l and theload.)

Full prestressing transforms the gird-

er under service loads into a horizon-

ta l colum n (Fig. 36 ); its eccentric nor-

mal force is generated artificially and

the external load w hich a ctua lly ha s tobe carried by it is relatively small.

W ha tever amount of prestress P is cho-

sen, it shortens tha t pa rt of the girder,

where a truss must form to ca rry the load

and replaces it by a direct and shorter

load pa th. It is, thus, directly a ppa rent

that prestressing improves the load-

bearing behavior as compared to the

nonprestressed reinforced concrete

girder.

Since prestress does not fully utilize

the strength of the high tensile steel

used for tendons, it can be used asT.I ,w Y i to cover a pa rt of the tensile force

of the chord T,.,. If the prestressing

steel is not bonded w ith the concrete, it

is unsu itable to serve as reinforcementfor T o r . It only affects the sectional

forces in the reinforced concrete via a d-

ditional anchor and deviation forces,

a nd a ll tensile forces of the chord hav e

to he ta ken by regu la r reinforcing steel

to sa tisfy equilibrium.

The resultant entering the bea m from

the supported end, as discussed, has the

tendency to spread in the web of the

beam a s in the bottle sha ped compres-

sion field (Fig. 52b ). Transverse tension

forces develop a s well a s a force whirl

in the corner (Fig. 53) which ca uses high

tensile stresses nea r the anchor plate.22

These tension forces have to bechecked. If they cannot be covered by

the tensile strength of the concrete (see

Section 4.5), stirrups have to be pro-

vided (see Section 4.6).

Tensile edge forces, splitting tensile

forces, tensile end forces, etc., in the

zone of introduction of anchoring forces

are thus simply part of the strut-and-

tie-model. In fa ct, they require no spe-cial names suggesting that they are

something special or specific to pre-

stressed concrete. The problem of

superposition of the reinforcement for

the shea r force a nd the splitting tensile

force is resolved by the model.

5.3.4 The Prestressed Concrete I-Girder

If a beam is not plane (rectangular)

but ha s a distinct profile like the T, I orbox girders, with relatively large cross-

sectiona l a reas of the flanges, the resul-

tant entering the beam as discussed

above, will spread on a path different

from that in the rectangular beam;

though of course for the same applied

142



-- -- compression

tension

Fig. 51 . Strut-and-tie-model of a reinforced concrete bea m loa ded

with two single loads.

a) esult t R dge cut

:Ja —T chord

moment shear force

normal force

b1

Detail l-1 Fig. 52 . (a) S trut-and-tie-model of a pa rtia lly prestressed beam with rectangula r

cross section: (b) detailed strut-and-tie-model of the bea m a rea, w here the

resultant is within the beam section.

forces P and A the simplified model is

the same for any type of cross section

(compare Fig. 54a with Fig. 53a ).The detailed strut-and-tie-model of

the prestressed I-girder (Fig. 54 ) shows

that a truss a lready develops in the a rea

where the tota l resulta nt force rema ins

within 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, only

a pa rt of the prestressing force can join

PCI JOURNAL , May-June 1987 43



L 1 i t 4 t _J.:k4

b) force whirl 1F

t Qq^^^

A

Fig. 53. (a) Simplified strut-model of a b ea m w ith its recta ngula r cross section

fu lly prestressed ; (b) deta iled strut-a nd-tie-model.

01 A

blweb

c) top f[ange

.-.

centroid of the forces coming from the web

d) bottom flange

Fig. 54. S trut-and-tie-models of a n I-girder with full prestress: (a ) simplified model;

(b) through (d) deta iled models of the web, top fla nge a nd bottom fla nge, respectively.

1 44



with the support force to flow into the

web. The effective resulta nt force in the

web is, therefore, smaller and at a

greater angle than in the prestressed

girder with a recta ngula r cross section.

It is, however, still at a smaller anglethan in the nonprestressed reinforced

concrete girder.

The chords in the two flanges are

linked to one a nother via the web struts

a nd ties. In this wa y, compression forces

a re introduced into the fla nges (Fig. 54c,

d). The strut-a nd-tie-model shows tha t

the spreading of the forces from the

width of the web to the width of the

fla nge generates tra nsverse forces in the

flange. The transverse reinforcement

must be distributed in accordance with

the length and intensity of the introduc-

tion of forces.

5.3.5 The Loa dbearing Behavior of the

Web

The strut-and-tie-model of the pre-

stressed girder with rectangular crosssection shows, tha t the stirrup forces in

the pa rt of the girder where the tota l re-

sultant force rema ins in the kern zone of

the section result from the sprea ding of

the compression forces (Fig. 52b, 5 3b).

The shea r reinforcement in tha t a rea

is in reality a tensile splitting rein-

forcement. A s soon a s the truss model

develops in the web (which as shownha ppens for a rectangu lar bea m further

a wa y from the support than for a beam

with a profiled section), the dimen-

sioning of its struts a nd ties follows a s

discussed for the B-regions of reinforced

concrete.

W hether the web cra ck develops from

the bending crack or begins in the web

itself (as specified for example by the

German Code DIN 4227 as zones a a nd

b ) ha s no effect on the loa dhea ring be-

havior of the members in the cracked

web. After the web ha s started to cra ck

the prestressing normal force is only

present in the compression chord, like

any normal compression force. In the

truss (or B-region) the web itself is in-

iluenced by the axial force due to pre-

stressing only via the inclina tion a of the

web cra ck, which is sha llower than for a

nonprestressed beam, corresponding to

the sha llower inclina tion of the princi-

pa l tensile stresses in the concrete whencracking begins.

This effect of a can be considered in

the web design as discussed in Section

5.1 (B-regions). From there it is shown

tha t the grea test possible inclina tion of

the dia gona l strut is para llel to the crack

in the web. The ultima te loa d ca pa city

of the diagona l struts in a web of a pre-

stressed concrete girder is therefore

somewha t sma ller tha n that of the web

of a girder without prestressing, how-

ever, it requires less stirrup reinforce-

ment for a simila r beam a nd loa d.

5.3.6 Dimensioning the Prestressing

Steel for the Different Types of

Prestressing

As a lrea dy mentioned a bove, the pre-

stressing steel can a nd will serve a s reg-ular reinforcement, if it is bonded with

the concrete, in other words it acts as the

tensile chord of the truss, developed for

the structure loa ded with prestress and

other loads. If the capacity of the pre-

stressing steel still a va ila ble a fter pre-

stressing Tp r r i ca nnot a lone cover the

force of the chord, reinforcement must

be supplemented in such a wa y that thetotal chord force T rd can be taken by

the prestressing steel (p) a nd the rein-

forcing steel (s):

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

In this equation To r , represents the

total chord force from the various loads

(including prestress). The loads are to

be mu ltiplied by the pa rtia l sa fety fa c-

tors y which a re, of course, different for

various kinds of load. The right hand

side of the equation stands for the re-

sisting chord forces, divided by the ap-

propriate partial safety factors.

The force T f the prestressing

steel which is still available for the

chord a fter prestressing is equ a l to its

PCI JOURNALMay - J u n e 1987 45



permissible tota l force T,, 0 , minus the

prestressing force yAP

T .. , = Tn,tat – Y P

As a rule, under ultima te load the pre-

stressing steel is strained beyond its

yield point a nd, therefore, T.,,, = A fnu.Then, the following simple equation

applies:

Tp, iwr r = Yr

If this is not the ca se, f n the a bove

equa tion ha s to be replaced by the stress

of the prestressing steel correspond-

ing to its tota l strain

€ p rot = €: Ep. ttorri

where

= stra in from prestressing (a ctive

strain)

e , , ,wr= strain of chord a fter

bonding (pa ssive stra in)

__ T .cpw _.z

A,+A8)Es

Some codes restrict the stra in of thechord after decompression of the con-

crete to 0.005 (DIN 4227 ) or 0.01 0 (CEB

M odel Code). Considering the a rbitra ri-

ness of these quantities, the small de-

compression stra in including the stra ins

from creep and shrinkage can be ne-

glected, which means €v ch r s limited

from 0.005 to 0,010, respectively.

II n„ hand is provided after pre-

stressing, the prestressing steel cannotbe considered as reinforcement in this

wa y. Ra ther, it a cts a s a tic member. Its

stress increment can only be deter-

mined from an internally statically in-

determina te ana lysis. If the truss model

conta ins a tension chord, supplementa ry

bonded reinforcing steel must be pro-

vided.If pretensioning is used a nd the ten-

don profile is straight, the prestressingforce a cts as a n externa l norma l force on

the reinforced concrete girder. The pre-

stressing steel is then pa rt of the rein-

forcement.

5.3.7 Result

Considering prestressing forces as

externa l loa ds is not only a n a dva ntage

with rega rd to service loa d design, but

also for checking ultimate load designa nd a ll other checks, beca use the loa d-

bearing behavior of the entire pre-

stressed concrete girder can then be

simply explained in terms of a strut-

and-tie-model.

Prestressing steel is used for two dis-

tinct purposes. O n the other ha nd, its

prestress applies fa vorable loa ds to the

reinforced concrete girder, while on the

other hand, it works as passive rein-

forcement when it is bonded with the

concrete. In this la tter respect it is not

different from reinforcing steel. O ther-

wise, if it is not bonded it acts as a tie

member.

As a result of this a pproach, the task of

designing a ny type of prestressed con-

crete girder becomes the task of de-

signing a reinforced concrete girderwith regard to bending, shear forces,

a nd norma l forces, which am ong others

include the additional loading case of

prestress.

146



ACKNOWLEDGMENT

This pa per is a progress report of the

work in this field at the University of

Stuttga rt. The au thors want to acknowl-

edge the contributions of several former

a nd present members of their Institute,

mainly K. H. Reineck, D. Weischede,

H. G. Reinke a nd P. Ba uma nn.

The authors further received va lua ble

contributions a nd encoura ging support

through ma ny critica l discussions with

colleagues; thanks go mainly to

B. Thiirlima nn (Zurich) a nd M . P. Col-

lins (Toronto), who also promote the

idea of a consistent design of concretestructures and to the members of the

CEB Commissions concerned, in par-

ticula r T. P. Tassios (A thens), G. Ma cchi

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

chat (Paris).

Fina lly, the authors wish to thank the

reviewers of the PCI JOURNAL who

offered us critical but constructive help.

In particular, we wish to thank J. E.

Breen (Austin) and J. G. MacGregor

(Edmonton). It is by no mea ns their fault

if the paper is still a burden to the

reader.

Lastly, the authors hope that this

paper will generate fruitful discussions

in 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 prominent

forum.

NOTE: Discussion of this report is invited. Please submit

your comments to PCI Headquarters by February 1, 1988.

PCI JOURNAt1May-June 1987 47



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praktisches Verfahren zum methodi-

schen Bemessen and Konstruicren im

Stahlbetonhau (A Pra ctica l M ethod for

the Design and Detailing of Structural

Concrete), Bulletin d'Information No.

150, Comite Euro-International du

Beton, Paris, M a rch 1 98 2.

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Consistent Design of Reinforced Con-

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IABSE, Vancouver, British Columbia,

September 1984.

3. Schlaich, J., and Schafer, K., Kon-

struieren im Sta hlbetonbau (Design a nd

Deta iling of Structura l Concrete), Beton-ka lender 19 84, Part II, W . Ernst & S ohn,

Berlin-M unchen, pp. 787 -1005 .

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(The Hennebique System), Schweize-

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5- Mbrsch, E., Der Eisenbetonbau, seine

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7 . Rusch, H., Ober die Grenzen der An-

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82, No. 1 , Ja nua ry-Februa ry 198 5, pp.

46-56 (see also Ref. 25).

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and Torsion Design of Prestressed and

Nonprestressed Concrete Beams, PCI

JOURNAL, V. 25, No. 5, September-

O ctober 198 0, pp. 32-100.

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methodischen Konstruieren im

Stahlbetonbau (Investigations on the

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Stuttgart, 1 983.

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zugfestigkeit bei der Stahlbeton-bemessung (On the A ssessment of the

Concrete Tensile Strength in the Design

of Structu ra l Concrete), Thesis, Institut

for M a ssivbau, Stuttga rt, 19 86.

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spnichung a nd Verformun g der Schu b-

zone des schlanken profilierten

Stahlbetonbalkens (Stresses and De-

forma tions of the Shea r Zone of Slender

Profiled Reinforced Concrete Beams),Forschungsheitrage for dir Baitpraxis

(Kordina -Festschrift), W. E rnst & Sohn,

Berlin, 1 97 9, pp. 225-236.

1 5, J ennewein, M. F. Zum Verstandnis der

Lastahtragung and des Tragverhaltens

von S tahlbetontragwerken m ittels Sta b-

wekm odellen (Explana tion of the Loa d

Bea ring Behav ior of Structural Concreteby Strut-and-Tie-Models). Thesis being

prepared, Institut fur M a ssivba u, Stutt-

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out Shear Reinforcement, Work in

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Stuttgart.

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bei der Stahlbetonbemessung mittels

Sta bwerkm odellen (Concrete Compres-

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Concrete by Strut-and-Tic-Models).

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tigkeit bei den Nachweisen zur Trag-

und Cebrauchsfahigkeit von unbe-

wehrten and bewehrten Betonbau teilen

(On the Assessm ent of Concrete Tensile

Strength for the Ultima te Capa city a nd

the Serviceability of Concrete M embers

1 48



With and Without Reinforcement),

DA fStb.-Heft341, Berlin, 1 98 3.

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forced Concrete and Prestressed Con-

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Delhi, 1 98 6, pp. 259-268.20. Niyogi, S. K., Concrete Bearing

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AS CE, V. 1 00, No. ST8, August 19 74, pp.

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21 . Schober, H., Ein Modell zur Be-

rechnung des Verbundes a nd der R isse

im Sta hl- a nd Spa nnbeton (A M odel for

the Assessment of Bond a nd Cra cks in

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29, No.2, M a rch-April 1 98 4, pp. 28-61 .

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ETH Zurich, July 1 978 .




APPENDIX -- NOTATION

Geometry

a = width of a nchor pla te

1 = width of compression field or

plated = length of D-region

d,, = diameter of largest aggregate

e = eccentricity

h = depth of beam

= spa n, length

t = thickness

V vertical movement

w = crack width

x = depth of bending compression

zone

z = lever arm of interna l forces

A = sliding pa ra llel to cra ck

a = cra ck inclina tion (see Fig. 30)

diagonal compression strut

angle

= a rea of concrete tensile zone

A A , = a rea of assumed failure zone

effective concrete area for ten-

sion stiffeningA, = cross section of reinforcing steel

a, = cross section of reinforcing steel

per unit length

_ mecha nica l degree oft f , einforcement

Forces and Moments

C = compression force, compression

strut

F load

M = bending moment

M torque

M„ = sta tica l indetermina te portion of

moment from support reactions

due to prestressing

P = prestressing force

p = load per unit Iength

p ressure under an a nchor plate

R = resultant force

T = tens ile force, tensile tie

V = shear force

Strength

f, = specified compressive strength

of concrete concrete compressive strength

for design of undisturbed timi-

a xia l stress fields

f = average concrete cylinder

strength

= specified tensile strength of

concrete

f specified yield strength of rein-

forcing steelf specified yield strength of pre-

stressing steel

y = pa rtial sa fety fa ctor

Subscripts

concrete or compressive

d design

p = prestressing, prestressing steel

s = steel

t = tensile, tension

u = ultimate

w = working, web

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