Schlaich - Toward A Consistent Design of Structural Concrete

 

Speci Spe cial al R Repor eportt

Toward a

C o n s i s t en en t D Dee s i g n o f S tr t r u c t u r aall C o n c r e ttee

J o r g S c h l aaii c h Dr.-Ing.

Professor at the Ins titute titute of Reinforced Concrete University of Stuttgart West Germany

K u r t S c h a f eerr Dr.-Irtg.

Professor at the Institute of Reinforced Concrete University of Stuttgart West Germany

M a tt t t i aass J e n n e w e iinn Dipl.-Ing.

Research Asso ciate ciate University of Stuttgart West Germany

This report (which is being being considered by C om ite Euro-Inter Euro-International national du Bt ton in connection w ith the revision revision of the Model Cod e) represents the latest and m ost authoritati authoritative ve 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 B- and D-Regions D-Regions .......... ............... .......... ....... 77 77 2 . The Structure s B-

3 . General Design Procedure and Modelling .............. 84 3.1 Scope 3.2 Comm ents on the O verall Analysis Analysis 3.3 Modelling of IIndividual ndividual B- and D-R egions

4 . Dimensioning the Struts, Ties and Nodes ............... 97 4.1 Definitions Definitions and G enera l Rule 4.2 Singular Nodes 4.3 Sm eared Nodes 4.4 Concrete Com pression Struts — Stress Fields Fields C. 4.5 Con crete Tens ile ile Ties --- Stress Fields T, 4.6 Reinforced Ties T, 4.7 Se rviceabil rviceabilit ity: y: Cracks a n d Deformations 4.8 Con cludi cluding ng Re marks 5 . Ex Examp ampes es of Appic icat atio ions ............................. 1 1 0 5.1 The B-Regions 5.2 Some D-Regions 5.3 Prestressed Con crete Acknowledgme men nt..................... ......................................147 .................147

References...........................................146 Appendix— dix— Notation ... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... .....150 ..150

PCI JOURNA JOURNAL+May-June L+May-June 1987

5

 

1 . INTRODUCTION -THE STRUT-AND-TIE-MODEL The truss model is today considered by researchers an d practitioners practitioners to be the rational and appropriate basis for the design of cracked reinforced concrete

beam s loaded in bending, shear and tor-

tie-models condense all stresses in compression and tension mem bers and join them by nodes. This paper describes how strut-andtie-models can be develop developed ed by fol-

sion. However, a design based on the

lowing the path of the forces forces throughou t

standard truss model can cover only

a structure. A consistent design ap-

ities such as point loads or frame corners, corbels, recesses, recesses, holes and other openings, the theory is not applicable.

(including their nodes) are designed

certain parts of a structure. At statical or geometrical discontinu-

Therefore, in practice, procedures which are based on test result results, s, rules of

proach for a structure is attained attained w hen its tension and compression members

with regard to safety an d serviceab ility ility using un ifor iform m design criteri criteria. a. The concep con ceptt also incorporates the

those mentioned above are of similar

m ajor el elem em ents of what is today called   detailing, detailing, and replaces em pirical procedures, rules rules of thumb and guess work by a rational design m ethod. Strut-andtie-mo tiemo dels could lead to a clearer under-

importance, an acceptable design con-

standing of the behavior of structural

cep cept t must validstructure. and consistent for every part be part of any Further-

concrete, and codeslead based such a n approach would to on improved

thum b and past experience are usuall usuallyy applied to cover such cases. Since all parts of a structure including

m ore, since the function of the experiment in design should be restricted to verifyy or dispute a theory but n ot to deverif rive it, it, such a concept m ust be based on

structures, The authors are aware of the en-

couraging fact that, although although they published pape rs on this topic earlier,1.2. earlier,1.2.3 3

physical models which can be easily

they are neither the first nor the only

understood an d therefore are unlikely to be misinterpreted. For the d esign of structural concrete* it is, therefore, proposed to generalize the truss an alogy in order to app ly iitt in the form of strut-and-tie-models to every part of any structure.

ones thinking and work ing al along ong these lines. It was actually at the turn of the last century, when Ritter*' Ritter*' and M cirsc cirsch h   introduced the truss analogy. This metho d

This proposal is justified by the fact

that reinforced concrete structures carry loads through a set of comp ressive str stress ess fields fiel ds wh ich are distri distributed buted a nd interconn ected by tensile tties. ies. The ties ties m ay be reinforcing bars, prestressing ten-

dons, or concrete tensile stress fields. For ana lyti lytical cal 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, forced concreprestressed te, if the latter lattand er is also part plain of a re(unreininforced concrete structure.

76

was later refined and exp expanded anded by Leonhardt, ° Rusch, 7 Kupfer, 8  and others

until Thurlimann's Zurich school, a with Marti l u   and Mu ell eller, er, created its its scientific basis for a rational app application lication 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 Thurliman n had shown that strut-and-tie-models strut-and-tie-models could b e usefully applied applied to deep beam s and corbels. From that point, the present authors

began their eff efforts orts to system atical atically ly expand su ch m odels to enti entire re structures

 

Synopsis Certain parts of structures are de-

ing structural concrete, which includes

signed with al almost most exagg erated accuracy while other parts are designed

both reinforced and prestressed con-

using rules of thumb or judgment based bas ed on p past ast experience. However, 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 generalization of the well known truss an analog alogy y method for beams, are proposed as the appropriate approach for design-

crete structures. This report shows how suitable models are developed and proposes

criteria crit eria according to which the mode l's l's elements can be dimensioned uni-

formly for all 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 initi initially ally prepared for discussion within CEB (Comitd Euro-International du Beton) 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 the prediction of ultimate load and the satisfaction of serviceability requirem ents. From a practical vi viewpoint, ewpoint, true

geom etry of tthe he strut-and-ti strut-and-tie-mo e-mo del at

simplicity can only be achieved if solutions are accepted with sufficient

prestressed concrete follow the same

the elastic stress fields and designing

the mod el structure foll following owing the theory of plasticity. The proposed p rocedure als alsoo perm it itss the dem onstrati onstration on that reinforced and principles although their behavior under w orking loads is qu ite dist distinct. inct.

(hat not p erfect) accuracy. T herefore, it is proposed h ere to treat iin n gene ral the ultimate limit state and serviceability in the cracked state by using one and the sam e m odel ffor or both. As will be shown

It should be m entioned entioned that only the essential steps steps of the proposed m ethod are given here. Further support of the

later, this is done by orienting the

found in Ref. 3.

theory and other information may be

2. THE STRUCTURE S B- AND D-REGIONS Those regions of a structure, in which the Bernou ll llii hypothesis of plane strain distribution distri bution is assum ed valid, are usually 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 (bend ing and torsional mom ents, shear and axial fforces). orces). PCI JOU JOUR RNAL Ma May-June y-June 1987

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

sectional areas and mom ents of iinertia. nertia. If the the tens ile stresses exce ed the ten sile strength of the concrete, the truss model or its variations apply. The B-regions are designed desi gned on the

basis of truss models as d iscussed later on in Section 5.1. 7

 

 

  ^       a   h  

.—h-4

1 1^—h2

h-4

t

^

 

b

IIFH   2. h—L

I

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

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

The above standard methods are not applicable app licable to all the other regions and details of a structure where the strain distribution distrib ution is significantly nonline nonlinear, ar, e.g., near concentrated concentrate d loads, co corners, rners,

discontinuity, disturbance or detaiI).

bends, openings and other dis disconcon-

cracked, they can be readily analyzed by the linear elastic stress method, i.e., applying Hooke s Law. However, if the sections are cracked, cracke d, accepted design

tinuities (see Fig. 1). Such regions are called D-regions (where D stands for

approaches exist for only a few cases such as beam supports, frame corners,

78

As long as these these re regio gions ns are u unn-

 

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

corbels and splitting splitting tension tensi on at pre-

design procedu re for the B-regions are

stressed concrete anchorages. And even these approaches usually only lead to

then readily available and only the

the design of the required amount of

reinforcement; they do not involve

strut-and-tie-m strut-and-tiem odels for the D-regions remain to be developed and added. Stresses and stress trajectories are

stresses. The inadequate (and inconsistent)

treatment of D-regions using so-called

quite smooth in B-regions as compared to their turbulent pattern pattern n ear discontinuities see Fig. Fig. 2). Stress intensities decrease rapidly with the distance from

  detailing, detail ing,

the origin of the stress concentration.

a clear clear check of the concret concrete e

past experien experience ce or good

practice has been one of the main re practice reaasons for the poor performan performan ce and even

This behavior allows the identification

failures of structures. It is apparent,

In order to find roughly the division lines between B- and D-regions, the

then, that a consistent design philosophy phil osophy m ust comprise both B- and D-regions without contradiction. Considering the fact that several decades after MOrsch, the B-region de-

sign is still being disputed, it is only reasonable to expect that the more com plex

D-region design will need to be simplified with some loss of accuracy.

H owever, even a simplified simplified methodical concept of D-region d esign will be preferable to today's practice. The preferred concept is to use the strut-and-tie-model

approach. This method includes the

B-regions with the truss mod el as a special case ofa strut-and-tie model. In using the strutstrut-and-tieand-tie-m m odel ap-

proac h, it is helpful and inform ative to first fir st subd ivi ivide de the structure into its Band D-regions. The truss truss m odel and the PCI JOURNALMay-June 1987

of B- and D-regi D-regions ons in a structure.

following proced ure is proposed, wh ich following is graphically 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 fictitious such a way that it complies complies with the Bernoulli hypothesis and satisfies equilibrium with the sectional forces.

Thu s, (b) cons ists entirel entirelyy of one or several B-regions. It usually violates the

actual boundary conditi conditions. ons. 2 . Select a self-equilibrating state of stress (c) which, if superimposed on (b) (b),, satisfies satisf ies the real b oundary conditi conditions ons of (a). 3 . App ly the principle of Saint-Wnan t

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

are negligible at a distance a from the equilibrating forces, which is approxi9

 

{ a1

            h

d)

c)

b)

F ^ h 1 ^

F

h

d

h ^

+

d= h

Fig. 3.1. Column with point loads.

{c

hl

lt

M 3h2

V

   

^ ^

M

M1

M

V

c b

 

i U

c)

-dr r+–d2 =h2

{ d1 i

Fig. 3.3. Beam with a recess.

80

B

 

a]

 

ILl

t

r

  m

4 3 T T T T T

h

t

C)

d=h_,

t

+ 

t f :I I I I I I l e

:

Fig. 3.2. Beam with direct supp supports. orts.

( a]

^`.

^

D

^ I ^

d=b

       

Fig. 3.4. T-beam.

( a ) real structure ( b ) loads and reactions applied in

accordance with Bernoulli hypothesis

c) self-equilibrating state of stress

d) real structure with B- and -regions

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

PCI JOURNAL/May-Jun JOURNAL/May-June e 1987

1

 

 

 

Q

F a

6 = 0 i0

 ^d = h

b

F

y ______

  _ iT h

2 y _ 2

 

d=h

d

            x

_"

1. 0

h

10

x h

dd y

1 ,0

h

1 ,0

h

Fig. 4. The principle of Saint-Venant: (a) zone zone of a b ody affected by self-equilibrating forces at th e surface; (b) application applicat ion to a prismatic bar (beam ) loaded at one face.

m atel atelyy equal to tthe he m aximum dist distance ance between the equilibr equilibrating ating fforces orces them selves. This distance distance defi defines nes the range of the D-regions (d). It should be m entioned that cracked concrete members have different stiff-

nesse s in diff different erent directions. Th is sit situu82

ation may influence the extent of the

D-regions but needs no further discussion 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 qualitative aid in developing the strut-and-

 

 

p

h

 

^

c

 

a2h

^ a

B ^B

B

B

B

f^4h

  1 O k ol U I F ^

hI   h

  B

I >Zh^•4h

w

r

f

r

Fig. 5. The identification of their their B- and D-regions (acco rding to Fig. 3) is 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 su bdivis bdivision ion of a structure into B-

short/long/high ong/high beams, deep beams, short/l

standing of the internal forces in the

corbels and other special cases are misleading. For p roper classificati classification, on, bo th geometry and loads m ust be considered (see Figs. Figs. 3, 5 and 6 ).

structure. It also demonstrates, that simple fh rules used today to classify

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

and D-regions is, however, already of

considerable value for the under-

PCI JOURN ALMay-June ALMay-June 1987

83

 

divided into its individual planes, which are treated sep arately. Similarly, Similarly, threedimen sional stress patterns in plane or rectangular rect angular ele elem m ents may be looked at in different diff erent orthogonal planes. There-

fore, in general, only two-dimensional models need to he considered. However, the interaction of models in

different planes m ust be taken into acdifferent count by app ropri ropriate ate boundary con di-

tions.

Slabs m ay also be divi divided ded into B-regions, where the internal forces are easily derived from the sectional forces, and D-regions which need further explanation, If the state of stress is not

predom inantly pl plane, ane, as for examp le iin n the case with punch ing or concentrated loads, three-dimen sional strut-and-tiemod els sshould hould be developed.

3. GENERAL DESIGN PROCEDURE AND MODELLING 3.1 Scope

For the m ajorit ajorityy 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 (and com mon practice) to first carr y  out a general structural analysis. H owever, prior to starti starting ng this analysis, it is is advan tageous to subd ivide the given structure into its its B- a nd D-regions. 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 stati cal system (see Fig. 6 ). The general analysis of linear structures (e.g., beams, frames and a rches) result resultss in the support reactions and sectional effects, the bending moments (M), normal forces

 N), shear force forces s V) and torsional moments(M r ) (see Table 1). The B -regi -regions ons of these structures can then be easily dimensioned by a pplyi pplying ng 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 sub stantial part of B-regions, its its statical syst system em and its bending moments. 84

 

Fig. 7, Prismatic stress fields fields according to the theory of plasticity (neglecting the transverse ten sile stresses due to the spreading of forces in the concrete) are unsafe for plain concrete.

analysis and B -region design provide

sectional effects by a statical system may

re

be omitted and the inner forces or stress-

also the boundary forces for the

es can be determined directly from the

gions of the same structure.

Slabs and shells consist predom-

applied loads following the principles

inantly of B-regions (plane strain dis-

outlined for D-regions in Section 3.3. 3.3. However, for structures with redundant

tribution)   Starting from the sectional effects of the structural analysis, imaginary strips of the structure can be mod-

supports, supp orts, the t he support support reactions rea ctions have hav e to

elled like linear members.

before strut-and-tie-models can be

be determined by an overall analysis properly developed. In exceptional cases, a nonlinear fi-

If a structure consists of one D-region

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

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

Structure

Analysis

Analysis of inner f o r c e s

or stresses in individual region

State I (u n c r a c k e d )

 

B- and D-regions e.g., linear structures, slabs and shells B-regions

Overall structural analysis (Table 2) gives:

Structure consisting of:

A,Js,Jr

D-regions

f3-regions

Boundary forces:

Sectional effects M N V. Via sectional values

D-regions only e.g., deep beams

Sectional effects

 

Supp ort reactions reactions

Linear elastic analysis* (with redistributed redistributed stress pe aks)

Strut-and-tie-models and/or nonlinear stress analysis *

State II (cracked)

Usuall y truss May Le cuuwbiuct[ with overall uiak his.

PCI JOURN AL/May-June AL/May-June 1987

5

 

      p)

    

simple span with cantilever

f or e

in the bolt on

hor

f

single truss multiple truss (steps)

beam

—multiple truss

Mlz

Mx

I

xo

-

—

oc

E

a VIx al

russ

                                  tt

C)

O-region

reg an

 

 

h t e

T

M

xI

ll

8

C w I x }_ 

Twlx- l = V (x} T 

model

Cclx-a1

1 1 1

Cclx

vix a}^^ Tw

S x b

C c IIxx }

^

 

——

_  

w

Tslx x  Ts(x-a) o 8 ,h—a-z cot 8 —

V 2 x Ct   o

––

V(x)

cwlxl b z sn cwlxl

V xJ t w o  ) = z

B

rat 9

Ismeo ed d ogc ogcrd rd stress

(pEr

a nil length of beam

(x= x • V ot 8

V Ix may nclude shear force forcess from torq torque ue

according accord ing to fig fi g 28

Fig   8. Truss model of a beam with cantilever: cantileve r: (a) mod model; el; (b) d distribution istribution of inner fforces; orces;  c) magnitude of inner forces derived from equilibrium of a beam element. 86

 

Table Tab le 2. Overall structural behavior and method of overall structural analysis of statically

indeterminate structures.  mt state

Overall

structural behavior

Essentially uncracked Service

ability

Ultimate capacity

of

Corresponding method of analysis sectional effects and support reactions

Most adequate

Acceptable

Linear elastic Linear elastic (or plastic

Considerably cracked, with steel stresses below yield

Nonlinear

if design is oriented at elastic behavior)

Widely cracked, tbrming plastic hinges

Plastic with limited rotation capacity or elastic with redistribution

Linear elastic or nonlinear or perfectl perfectlyy plastic with structural restrictions

nite element method analysis may be

applied. A follow-up follow-up check with a strutand-tie-model is recommended, espe-

tures for the sectional effects using a

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

cially if the major reinforcement is not

effects derived from p plastic lastic methods

modelled realistically in the FEM analysis.

m ay for simplifi simplification cation purposes also be

3.2 Comments on t he Overall Overall Analysis

In order to be cons istent, tthe he overa ll analysis of statically indeterminate structures should reflect the realistic overall behavior of the structure. The

used for serviceability checks, if the structural design (layout of reinforce-

m ent) is ori oriented ented at the theory of elasticity.

3 .3 . 3 M o d e lll l i n g o f I n d i v i d u a l B - a n d D-Regions 3.3.1 Principles and General Design

guidance for the design o f statically statically iinn-

Procedure After the sectiona l effects effects of the B-regions and the bou ndary forces of tthe he D-

determinate structures. Some of this

regions have been determined by the

intent of the foll following owing paragra ph (sum marized in Table Table 2) is to give some

discussion can a lso be app li lied ed to staticallyy determ inate structures esp eciall icall eciallyy with regard to determining deforma-

tions.

Plastic methods of analysis analysis usually the static method ) are suitable primarily for a realistic realistic determina tion of ultim ultim ate load capacity, while elastic elastic m ethods are

overall structural analysis, dimensioning

follows, for which the internal flow of

forces has to be searched and qu antifi antified: ed: For uncracked B- and D-regions, D -regions,

standard m ethods are available ffor or the analysis of the concrete and steel stresses (see Table 1). In the case of high

more appropriate under serviceability conditions. According to the theory of

com pressive stresses, the li linear near stress distribution distri bution ma y have to b e m odifi odified ed by replacing replaci ng H ooke's Law with a nonlinear

plasticity, a safe solution for the ultimate load is also obtaine d, if a plastic analysis

materials law (e.g., parabolic stress-

strain relation or stress block ).

is replaced by a linear or nonlinear

If the tensile stresses in individual Bor D-regions exceed the tensile strength of the concrete, the inner forces of those

analysis. Experience Experience further shows that the design of cracked concrete strucPCI JOURNALIMay   J u n e  1987

7

 

regions are determined and are designed 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 straight lines and concentrate their curvature in nodes. 2 . Calculate the strut an d tie forces, which satisfy equilibrium. These are the inner forces. 3 . Dimension the struts, ties and nodes for the inner forces with with due consideration of crack w idth limitati limitations ons (see Section 5). This m ethod implies that the structure is designed according to the lower boun d theorem of plast plastici icity. ty. Since concrete permits only limited plastic de-

advantage that the the sam e m odel can be used for both the ultim ultim ate load and the serviceability serviceabili ty check . If for some reason the purpose of the analysis is to find the actual ultimate load, the model can easily be adapted to this stage of loading by shifting shifti ng its struts and ties in order to increase the resistance of the structure. In this case, how ever, the inelastic rotati rotation on 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 fr fracaction tion of the com pressive str strength. ength. Cases li like ke those given in Fig. 7 w ould be unsafe even if both requirements of the lower bound theorem of the theory of

formations, the internal structural system (the strut-and-tiestrut-and-tie-m m odel) has to b e chosen in a way that the deform ation li lim m it

plasticity are fulf fulfill illed, ed, na m ely, equ ilibilibri rium um and F IA ---f,. -f,. Compatibility evokes

(capacity of rotation) rotation) is not ex ceeded a t any point point before the t he assumed state of

tensile forces, usually transverse to the direction directi on of the lloads oads wh ich may cau se

stress is reached in the rest of the structure. In highly stressed regions th is ductility req req uirem ent is fulf fulfill illed ed by ad apting the struts and ties of the model to the direction directi on a nd size of the internal forces as they would appear from the theory of elasticity. In norma lly or li lightly ghtly stressed regions the direction of the struts and ties in the model may deviate considerably from the elastic pattern without exceeding the structure's ductility. The ties and hence the reinforcement may be arranged according to practical consid-

erations. The struc ture ada pts itself to the assumed internal structural system. Of course, in every case an analysis and safety check mu st be made u sing the fi fi-nallyy chosen model. nall This method of orienting the strut-

and-tie-mod el al along ong the force paths indicated by the theory o f elasti elasticity city obviously neglects some ultimate load capacity which could be utilized by a pure ap plication of the theory of plastic-

ity.. On the other hand, it has the m ajor ity

 

premature cracking and failure. The  bottle-shaped compressive stress field, which is introduced in Section 4.1, fu further rther eliminates such hidden dangers when occasionally the model

chosen is too simple. For cracked B-regions, the proposed procedure obviously leads to a truss

model as shown in Fig. 8, with the inclination of the diagonal struts oriented at the inclination inclination of the diagonal cracks from elastic tensile stresses at the neutral axis. of the strut angle by A reduction 10 to 15 degrees and the c hoice of vertivertical stirrups, i.e., a deviation from the

principal princi pal tensile stresses by 45 degrees, usually (i.e., for normal strength con-

crete and norm al percentage of sti stirrup rrup reinforcement) causes no d istress. Since prestress decreases the inclination of the cracks and hence of the diagonal

struts, prestress permits savings of stirrup reinforcem reinforcem ent, whereas a dditional dditional tensile forces increase the inclination. The distance z between the chords

should usually be determined determined from the plane strain distribution at the points of

 

 

 

 1 0 la L a l h

a)

 

k T II H

   

L / /  t/ .

Cj

I

l

1^`

        i l

--

T

a ^ -

b) 0,5

_ _ _ _

1t5

QG Z

70° 0,3

^

strut tie

fr

d

a/1-01

r2F

/PI a rt/p t

1

Q   0.5 0,5 0.7 9.8 09

1 . 0 1 . 1 11 1 , ?1 , 31 , 6 1 , 5l 5lb d /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 zero sh ear ear and for simplicity simplicity be kept con stant between two adjacent points of zero moments.

Refinemen ts of B-region design will be

PCI JOURNALIMay-June 1987

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

develop a strut strut-and-tie-and-tie-m m odel for each case individually. After some training, 9

 

  i     r   A

p

oodpoth

– —

T

 

 

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

 

1

1

 

P F

F

Fig. 11. Load paths (incl (including uding a U-turn ) and strut-and-tie-model.

this can be done quite simply. De-

veloping a strutveloping strut-and-tie-model and-tie-model is com parable to choosing an overall statical system. Both procedures require require som e design experience an d are of similar rel rel-evance for the structure. Developing the m odel ofa D-region is

much simplified if the elastic stresses

and principal stress directions are available as in the case of the example show n in Fig. 9. Such an elastic analysis is

readily facilitat facilitated ed by the w ide variety of computer programs available today. The direction of struts can then b e taken in direction accordance with the mean direction of 90

principal compressive stresses or the

more im portant struts and ti ties es can be located at the center of gravit gravityy of the corresponding stress diagram diagram s, C and T in Fig. 9a, using the y diagram given there. H owever, even if no elast elastic ic analysis iiss available and there is no time to prepare one, it is easy to learn to develop s trutand-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 Method 3.3.2 First, it must be ensu red that the outer

 

 

a         e

1 `^

r

i

T -I+II I ^  

I

r

I I

I I I I I I

ILL Y

I

   

 

  

I

I

D

i

-

Q

x ^

drl

7

L i J

III I III I

    

LV

C

l

t1 C 7

Vc, :c.

Y

^

 

 

}

-

I

 

_P

p

c_

L ^ ^

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

a

 

a

IF  

 

d I4

dr

I

,

T

jCr t

t

8

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 ili ilibrium brium of the D-region is s atisfied atisfied by determining all the loads and reactions (supp ort forces) acti acting ng on it. In a boundary adjacent to a B-region the

loads on the D-region are taken taken from the B-region design, assuming for example that a linear distribution of stresses (p) PCI JOURN AL/May-June AL/May-June 1987

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

such a w ay, that tthe he loads on on e 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 1

 

 

    B

a}

B

 

   

Ti

v

Al

Tc

shear force' A

B

c

moment

b)  B   m P

         

H

f/

c

T

C

strut tie

_^lo a dp a th

-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 gravit gravityy of the corresponding stress diagram diagram s and have there the directi direction on of the applied Ioads or reactions. They tend to take the

stress concentrations (support reactions or singular loads).

shortest possible streamlined way in

pletely used up with the load paths depletely scribed; there there rem ain resultants ((equa equa l

between. Curvatures concentrate near 92

Obviously, there will be some cases where the stress diagram is not com-

 

      

a good

b bad

U I IiL n T T T Y _

r

P

I

I

 

 

L 1ru . L U 1 1 1 1 1 P

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 w hirl a ass ill illusustrated by forces B or Figs. 11 and 13 a. trated Un til now , equili equilibrium brium on ly in tthe he direction of the applied loads has been

con sidered. After plotting all lload oad paths with smoo th curves and replacing tthem hem by polygons, further struts struts and ti ties es m ust be added for transverse equilibrium acting between the nodes, including including

those of the U-turn. While doing so, the ties must be ar-

ranged with proper consideration of

bined approach is applied in Fig. 12 and the num erical exam ple iin n Sec ti tion on 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 div divided ided into a B-region and a D-region. The bottom of the D -region is acted on by the stresses (p) as

derived for the the adjace nt B -region. These stresses are then resolved into into four compone nt s: T The he t wo com pr essi essive ve forces Cs + C. = F, which leaves two The forces C, equal forces T Z and C Y .   and C, are the components, respec-

practicality of the reinforcem ent layout practicality (generally parallel parallel to the the c onc rete sur-

tively, on the left hand an d right hand

face) of crack distribution requireme ntsand

terrnined by the load F. By laterally shifting the load components into the

The resu lting lting m o dels are quite o ft ften en

given positions, transve transve rse stresses are generated. The corresponding horizontal struts

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 appropriat appropriatee m odel has to be adapted to the load c ase an d is iin n m o st cases determined by e quili quilibri brium um conditions after only a few struts or ties have been chosen. A very po werful m eans o f develo pi ping ng new strut-and-tie-models for complicated cases is the combination of an

elastic finite element method analysis with the load path method. This comPCI JOURNAL/May-June 1987

side of the vertical plane which is de-

and ties are located at the center of

gravity o f stress diagra gravity diagram m s in typical sections which are de rived from from an elastic analysis (Fig. (Fig. 1 2.1b). Their no des with the vertical struts also determine the

position o f the diagonal struts (see Fig. position 12.lc). T h e e x a m p l e i n F i g. g. 1 2 . 2 s h o w s t h a t of Fig. 1 2.Ic disappears, if the 3 the loadtie F Tmoves toward the corner of the D-region. 3

 

a

p

 

1i

  AI

b

Q

e

+ 2l

f-;   ` r 1

 

f

     

high

tl

^

Y

 

I

 

medium

1

r   `

long

low

_ ,I

  

r +

L 4

I

 

i r

+

4 i

r

I

7Tt

f

'-, 4

 

r^ l y ^^  ^^

Fig. 15. The two m ost frequen frequen t and most useful strut-and-tie strut-and-tie mo dels: (a) through (b), (b), and som e of the ir variations variations (c) throug throug h (e). 94

 

3.3.3 Model Optimizati Optimization on 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 et hod and how t hey ar e connect ed ttoo the ove rall sectional effects of the D -regions. gio ns. The ties T   and T x in Fig. 13a would require inclined reinforcement, which is undesirable from a practical

viewpoint. Therefore, a tie tie arrangem arrangem ent has been c h o s e n i n F i g . 1 3 b w h i c h c a n b e s a t iiss fied by an orthogonal reinforcing net with the bars parallel para llel to the edges.

Thereby, special m ethods such as those given in Ref. 17, which con sider deviations of reinforcement directions from

the principal stress directions, are not n e e de d. Doub t s could ar ise is e as t o w het her t he correct model has been chosen out of several possible ones. In selecting the

m odel, it iiss h elpful to realize that loads try to use the path w it ith h the least fforces orces and deformations. Since re inforced ti ties es are much more deformable than con-

crete struts, the the m odel with the least and shortest ties is the best. This simple

criterion criteri on for optimizing a model m ay be

b)  

 

ON

c)

0 5 5

 

1 011

^ ` --

i

I

B

1511

r

d ^

    a -o

 

rr r

r r

I II

5 rr

(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 gi ven . PCI JOURNAL/May-June 1987

  5

 

 

t

     

 

column

I

b anchor pla late tendon

^ioad^

}

a )

_

T

    H

1

Fig. 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 foll follows: ows:  .F ; l ;  e„,; = M ini inimu mu m

wher e

F, = force in striitortiei 1, = length of memberi e m s  = me an stra strain in of me mbe r i This equation is derived from the

principle of minimu m s trai train n ene rgy for linear elastic behavior of the struts and ties after after cracking. The c ontribution of the concrete struts can generally he

om itt itted ed bec ause the strains of the struts ar e usually usually m uch smaller tthan han t hose of the steel ties. This criterion is also helpful in eliminating less desirable

m odels ((see see F ig. 14).

Of course, it should be understood

that there are no unique or absolute optimum solutions, Replacing a continuous set of smooth curves by indi-

vidual polygonal line line s is an approxim ation ti on in itself and leaves am ple room for subjective decisio decisio ns, Fu rthermo re, individual divi dual input such as the size of the region or reinforcem reinforcem ent layout are always different. But an engineer with some experience in strut-and-tie-modelling

will alway alway s find a satisfactory solution. 96

 

3.3.4 The Pedago gical Va Value lue o f Mo delling

A n y o n e w h o s p e n d s t iim m e d e v e l o p iin ng strut-and-tie-models strut-and-tie-m odels will observe that some types of D-regions appear over

and over again even in apparently very different dif ferent str structures. uctures. The two m o st fr freequent D-regions, which are even related to one another because they have the same characteristic stress distribution along their centerline, are given in Fig. 15 with som e of their variations. Fig. 1 6 show s appli applications cations of the fi first rst

type of m odel (Fig. 15a) to four di different fferent structures: The distribution of cable

forces in a bridge dec k (Fig. 16a); a wall with big open ings (Fig. 16b); a box gir girdder with anchor loads from prestressing tendon s (Fig. 16c); and a detail of iinternternal forces in a rectangular beam wh ich sho ws that st stirr irrups ups n eed to he clo sed. (Fig. 16d). In In all of these c ases the pattern of intern al forces is basically identical. To recognize recognize such com mon fea features tures of structures is of considerable pedagogihe lpful ttoo the design cal value value and v e r y

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

fusing if the same facts are given dif

ferent designations only because they appear under different circumstances. Fo r exam ple, the deep beam in Fig. 17 "does not care" whether its identical

loads result from from supports or from prestressing tendons. It, therefore, therefore, does not

react with "bending" in the one case

and with spli splitti tting ng tension in the other case, but sim sim ply with ttension ension and com pression. Num erous other strut-andstrut-and-ti tie-models e-models are found in Ref. 3.

4. DIMENSIONING DIMENSIONING THE STRUTS TIES A ND NODES 4 .1 .1 D e f i n i t iioo n s a n d G e n e r a l R u l e Fig . 18 shows som e t y pi pical cal examples of strut-and-tie strut-and-tie mode ls, tthe he co rresponding stress stress trajectori trajectories es and reinforcem ent layout. Node regions are indicated by shading. Looking closely at these ex-

amples as w ell as tthose hose in the previous and following sections, the following

conclusions may be drawn: Dim ensio ning no t o nly m eans siz iing ng and reinforcing the ind ividual ividual struts and

TTT-node The principle remains the same if

mo re than three struts and tties ies mee t.

S t r u t s a n d T i es es Wh ereas the T, ar aree es sentially sentially li linear near or one-dimensional elements between two nodes, the C . and T,, are two- (or three-) dimen sional stress fiel fields, ds, tending to spread in between two adjacent nodes. This spreading, indicated indicated by the

ties for the forces they carry, but also

bulging of the struts in Figs. 18 and 19 a, bulging can result in transverse tensile and co mpressive stresses stresses which then m ust

close relation between the detailing of

he considered either by introducing introducing

ensu ring tthe he load tr transfer ansfer betwee n them by che cking the node regions. There is a

the nodes and the strength of the struts bearing on them and of the ties an-

chored in them because the detail of the nod e chose n b y t he d esig n eng ineer affects the flow of forces, The refore, it iiss necessary to check whether the strutand-tie-model and-tie-mo del initially chosen is still

valid after detail detailing ing or nee ds correc tion. Thus, m odell odelling ing and dim ensioning is in principle an iterat principle iterative ive process. There are basically three types of struts and ties to be dimensioned: C,: Conc rete struts iin n com pression TT : Concrete ties in tension without r einfor cement Ties in tension with reinforce-

me nt (mild (mil d st eel r einf einfor or cem ent or prestressing steel) There are essentially e ssentially four types of

nodes depen ding on the combination of struts C and ties T (see (see F ig. 19 ): CCC-nod e

C C T-no de CTT-node PC I JOURNAL/May-June 1987

these stresse s into the fail failure ure criterion of the C struts and the Tc ti ties es 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 th e model are resul-

tants of the stress fields. Since by de finifiniti tion on the curvatures o r deviati deviations ons o f the forces are conce ntrated 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 ari aring ng (or spreadi spreading) ng) the node over a substantial length of the strut (see for example F igs. 18a2 and 18b2). To cover cover all cases of compression

fields fields including those of the B -regions, three typical configurations are suffi-

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

  7

 

      2)

all

I_

3)

1_{ .) .) IGIl lG Gla]{ ]{)1III II1iTI[J [JIIJ IJi] i]11111

 

C

 

   

bll

2)

3)

s

          C

C

C

C 1)

2) not reinforced

3) reinforced

 C

EE

`

Fig. 18. Some typical examples of strut-and-tie-models, strut-and-tie-models, their stress fields, nodes and corresponding correspon ding reinforcement (if any is i s 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 no des o f tthe he m o del are a si sim m pli pli-fied idealization idealization of reality. The They y are formally derived as the intersection inte rsection

points of three or selves m ore straight struts or ties, ti es, which them represent either either straight strai ght or c urved stress fields or reinforcing bars or tendon tendon s. A node as introduced into the m odel implies implies an abrupt chan ge of dir direction ection of forces. In the actual reinfo reinfo rced co n crete structure this deviation deviati on u sually occu rs over a certain

d1)

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 str stress ess fields jjoining oining each other or with tensile ties, which consist of many closely distributed reinforcing bars, the deviation of forces may be

sm eared (oin spread) o vercase so me Therefore, irn spr theead) former thelength. 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 of nodes.

2) struts not reinforced

3) struts str uts 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

a

i

 

 

anchor pl pla4e a4e

t

 

I^-he12^1

   

anch rOge length

b3

b4

 

withart with crosspressure

inchomge length length ith loop

nnc hort^ge 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 loads from an anch or plate (a,) (a,) or bearing plate (a 2 ) into int o (bottle sh shaped) aped) compression fields; (b) CCT-nodes. CCT-nodes. A diagonal diagonal compression

strut an d th e verti cal sup p ort reacti reaction on are bal an an ced by rei n forcem en t wh i ch i s an ch ored by an an ch or p l ate beh i n d th e n ode ( b,), b,), bon d w i th th i n th e n ode ( b  ), bond within and behind the n ode (b,), (b,), bond and radial pressu re (b,). (b,).

 

 

S

Failure Criteria for Concrete

The strength o f the the co nc rete iin n co m pression fields fields or within nodes depe nds to a large ex tent on it itss m ulti ultiaxial axial st state ate of stress and on disturbances from cracks and reinforcem reinforcem ent. favor(a ) Transverse com pression is favorable especially if it acts in both transverse directions, directions, as for exam ple in confi fined ned regions. Confi Confinem nem ent m ay be provided by transverse reinforcem reinforcem ent or by bulk conc rete surroun ding a relat relatively ively small com pression fi field eld (s (see ee F ig. 21). (b ) Transverse tensile stresses and the cracks caused by them are detri detrime me ntal ntal.rs .rs The concrete may fail considerably

below its cy linder linder strength if tthe he trans-

the prisms between those cracks are ragged and narrow. The reduction of

com pressive str strength ength is small or nom inal if the ten sile forces are carried by the reinforcem ent and the cracks are wide enough apart. particular, cracks which are not (c ) In particular, parallel to the com pressive stresses are detrimental. In 19 82, an em piri pirical cal formula for calculating the strength of parallel concrete com pression fi fields elds with transver transverse se tension was pu bli blished shed by Colli Collins ns et al., al.,2a 2a and 2 years later a simil similar ar 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

verse tension causes closely spaced cracks approximately parallel to the

width, crack distance distance an d crack direc-

principal pri ncipal compression stresses such that

tion by the transverse tensile strain

c 

C2)

dz

d 

t

 

I 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,), (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 10 1

b         

a}

`

       

1;

 

I I

r

^tllfft/f

J

 

j r

I

        J

I

1

G

1411111J4

i

r

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 avai available lable in the analysis. For practical purposes, the following simplified strength values of f re proposed for dimensioning all types of struts and nodes: , : or an undisturbed and f * = 1.0 f

uniaxial state of compressive stress as shown in Fig. 20c; fir

0.8 f: f tensile strains in the cross direction or transverse tensile reinforcement may cause

cracking parallel to the strut with

norm al crack width; tthis his applies

also to node regions where tension steel bars are anchored or

crossing (see (see F ig. 19b); f , 0.6 f : as above for skew cracking or skew reinforcement; = 0 4 d   or skew cracks with extraordinary crack width. Such

cr acks must b e expect ed , if mod elling of the struts de parts significantly from the theory of elasticity'ss flow of internal F orces (e.g., ity'

102

 

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

co de o f practice. Acco rding to the C EB Code,f^ d is determ ined by: 0.85fc ye

whe re y, = 1 .5 is the partial safety fa factor ctor for the the con crete in compression and the coefficient coeffici ent 0.85 acco unts for sustained lo ading, ading, In the C EB C o de, 2  0 = 1.0 in all cases and the load factors for dead and live live loads are 1.35 and 1.5, respectively. The increase in strength due to two or three-dimensional states states of com pressive stresses stresses may be taken into account if the the simultaneou sly acting transverse com pressiv pressivee stresses are considered reliable. Skew cracks are no t expected, if the

due to redistribution of internal

theory of elasticity is followed sufficiently closely during modelling. modelling. This

forces in order to exploit a maxim um ulti ultimate mate c apacit apacity). y).

me ans that the angle between struts and ties entering a singu singu lar node sh ould not

   

a^

--

7

-- - ,con ,confining reinforcement

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

be too small. How ever, skew cracks may also be left over from a previou s loading case w ith a diff different erent stress situation. Before deciding on one of the given

strength values, both transverse directions tions m ust always he considered. General rule

Since singu lar nodes are bottl bottlene ene cks of the stresses, it can be assume d that an entire D-region is safe, if the pressure under the m o st heavil heavilyy lo aded bearing plate or ancho r plate is lless ess than 0.6 ff (or in unusual cases 0.4 f,.) and if all

significan t tensile forces are resisted by significan reinforcement and further if sufficient development lengths are provided for the reinforcement. Only refinements

will be discussed in the following Sections 4.2 through 4.5. 4.2 Singu lar Nodes

Singu lar node s equili equilibrate brate the forces of the ties and struts acting acting on the m relatively abruptly as compared to the smeared nodes. The deviation deviation of the forces occurs over a short length or small

area around the theoretical nodal poi point. nt. Suc h nodes originat originatee m ainl ainlyy from single loads or support reactions reactions and from co n-

therrnore, geometrical discontinuities (e.g., (e.g ., reentrant corners) can c ause stress concentrations which are represented by a singular node. Although numerous possibilities exist for detailing detailing node s (and despite the fact that they all behave somewhat differ-

ently), in most c ases their forces balance each 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 essentia lly a

lo ad transfer tr ansfer via co ncrete co m 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. 19 b,) b,).. Even in these cases the flow of forces

can be visualized by strut-and-tiem odels with singular nodes at the ribs of the bar. If the the se mode ls requi require re 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 of

practice shou ld be used. practice In sum m ary, then, dimensioning singular nodes means: (a) Tuning the geometry of the node

centrated forces introduced by the

with the applied forces.

reinforcement through anchor plates, bond, or radial pressure inside bent

For CCC-nodes it is helpful, though not at all all mandatory, to assume the bor-

PCI JOURNAL/May-June 1987

 

reinforcing bars such as loops. Fur-

03

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 parti particular cular case the u nequ ivo caI geo m etri etrical cal rel relaation a,: a 2 : a   - C,: C2 :   Fig. 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, a arr-

edges of a node are satisfactory. Departing even more from hydrostatic

stress and still disregarding the nonuniform stress distribution in the

node m ay lead to com pati patibil bility ity stresses, which are not covered by the strength values val ues given above.

rangements of forces which lead to

When designing a singular CCTnode, the design engineer must be aware that the c urvature of the load path

stress ratios down to 0.5 on adjacent

and the corresponding compression

 

al

 

a

        B

  1' f 

iC

...-lo_--__._ p tanplcns2 ^

2

C 01 •

—

         

    a3

Q2

_C

o3,o1 tcnp

a a Z c c s W Q1=ap

op f*

±t + o

ci

Hp

1h,net1

i b,net

aZ

b

0

ao.2[0

1^

Q }

cd

 f ^

y

`b,net

——

C)

1_.

 

Fig, 22. A propo sal for the dimensioning of typical singular CC CCT T-nodes with different reinforcement layouts: (a) multil multilayered ayered tie a 3  relatively large, o < a, ; (b) single layer tie ( a 3  small a-z>v,); (c) (c) special case in betwe en     = o o ); (d) same type as (b) with pressure from a com pression field. 104

 

field is largest at the origin of the c onfield centrated load, i.e., i.e., next to the bearing plate or anchor plate (see Figs. 9 and

10). The ties and plates should be arranged ac cordingl cordingly. y. (b ) Checking whether the concrete pressures within within the node are withi within n the li limits mits giv given en in S ection 4.1 . This con dit dition ion is au tomatically satisfied 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 saf safe. e. If tthe he reinforcem ent is anchored in the n ode region, cracks ar aree po ssible ssible and c o rrespo ndingl ndinglyy the co ncrete strength stren gth for cracked concrete

applies. For CCT-nodes with bonded rein-

transverse compression stress trajec-

tories of the struts m eet the bar and are deviated;; the bar deviated b ar must extend to the

other end of the n ode region in order to catch the outermost fibers of the deviated compression stress field (Fig.

1 8b 3 , c3).

4 ..3 3 Smeared Nod es Since D-regions usually contain both sm eared and singular nodes, the latt latter er will be critical critical and a check of con crete stresses str esses in smeared no des is unnec essary. Ho we ver, iiff a smeared C C T-no de

forcement arranged according to Fig. 22, a check of conc rete st stresses resses a respec-

is assumed to remain uncracked, the tensile stresses of the corresponding

struts is sufficient. Since in m ost cases it

concrete stress fiex eldam need to this be che cked (Section 4.5 ). Anfield ple of case is Node 0 in F ig. 18c1 and the stress ffield ield in Fig. 18c2. Safe anchorage of reinforcem reinforcem ent bars in smear ed nod es m ust b e ensur ed folfol -

tively, o in the adjacent compression

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

struts controls, only one of them needs struts to be analyzed. An analysis on the basis of Fig. 22 rewards reasonably the arrangement of multilayered reinforce-

m ent distr distributed ibuted 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 in the nodes (exce pt for CCC -nodes). In the case of anc hor plates, tthis his invo lves a check o f 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 ti tiee wh ere it crosses the node is better than good bond quality because strain compatibility compatibility

with the bo nded bar will tend to crack the node's concrete. In the case of directly directly anch ored reinforcing bars, hook or loop anchorages (with cross-pressure in CCT-node s) ar aree preferred. Generally, the minimum

radius allow allow ed by the applicable code is selected. For straight bar anchorages, the

PCI JOURNALJMay-June 1987

 

length o f the anch o rage is selected fo ll-lowing the designated code. The design engine er mu st ensure that it iiss located within and behind the node (Fig. 19 b2 and b 3 ). Anchorage begins where the

lowing the rules for singular nodes

(Section (Secti on 4.2).

4 ..44 C o n c r e t e Co Co m p r e s s i o n S t r u t s —Stress —S tress Fields C The fan-shaped and prism atic st stress ress stresses fieldsaccordingly do not evelop and the transverse uniaxial concrete

applies. appli es. If transverse stresses, strenth cracks or tension bars c ross the strut, the strength may be based ba sed on the values given giv en in Sec ti tion on 4.1 .

The bottl bottle-shaped e-shaped com pressi pression on stress

field (Fig. 20b) applies to the frequent

case of com pressive forces being introduced into concrete which is unreinforced in the transverse direction.

Spreading of the forces c auses biaxial or triaxial tria xial compression unde r the load and transverse tensions farther away. The transverse tension (com (com bined with llonongitudinal com pressive stresses) can re05

— — biaxial biaxial com pression failure failure in bottle neck

 

m

crocked, but w it ith h tr transverse ansverse reinforceme reinforceme nt  de gree w) in the belly region uncrocked, plain conc rete

nfinement

b

 

o

pa

  Y

IF po

a t

la

 — b

21

  II ate

^

^ 1 , 3 1 a

Fig. 23. Dimensioning plane bottle shaped stress fields: (a) diagrams giving safe pressure values P   with regard to cracking and crushing of plain unreinforced unrei nforced concrete concrete

stress fields, yielding of transve t ransverse rse reinforcement reinforcement and biaxial compression compression failure in the t he 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 crit eria fr from om Sec tion tion 4 .1 apply. The general rule given at the end of

Sect ion 4.1 usuall usuallyy m akes t he calculation of stresses within the stress field unnecessar y . How ever, 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 pla plane ne bottle shaped stress fields in D-regions.   ' 3 • " This stress field 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 distance 1 of the anch or plate plate from the section where the stress trajectories are

For compression struts with transverse reinforcement the failure loads

analyzed with the mode l iin n F ig. 23c are also given given in F ig. 23a. It can be see n that a reinforcemen t rati ratio: o: w = 

y

tfd

0 . 0 6

(where a, is the cross section of reinforcement per unit length) le ngth) approximately compensates for the tensile

strength of the the c oncrete. If it it is desired not to rely so h eavily on the concrete tensile strength, stre ngth, lower

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

reinfo rcement rati reinfo ratioo s m ay be used w ith ith reduce d val values ues of g a ff f ,  as shown in Fig.

20b). The diagram for compression

23a.

fields fields without transverse reinf reinforcem orcem ent (bold line) is based on an elastic analysis, a conc rete tensile strength off = f115 and a biaxial com pressive tensile failure fai lure criteri criterion on as given in F ig. 26b. It can be see n, that for cert certain ain geome trical relati relations ons a p ressure at the an chor platee as lo w as 06f co u ld cause crackplat 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 rall rallyy appears to be considerably consid erably on the safe side and further research in this area is required.

The com pressive st strength rength of compression reinforcemen reinforcemen t may be added to the concre te strength if the reinforcem ent is prevented from buc kli kling. ng. 4.5 Concrete Tensile Ties — Stress

Fi e l d s T ^ In the case of unc racked tensile stress fields, fields, the tensile strength o f concre te is used. Although it is dif diffi ficult cult to deve lop design c rit riteria eria for this case, it wou ld be even w o rse to m ainta aintain in the fo rmalist rmalistic ic view that the tensile strength of con-

Better knowledge knowledge is also needed on substantially nonsymmetrical stress fields which originate from singular

crete cannot and therefore m ust not be

nodes w ith ttension ension ties crossing or anchored there. Com pari parisons sons with test results suggest that checking the singular

and-tie-mod els will inevitably show that

node (Section 4.2) and applying the

diagrams of Fig. 23 is al also so safe for those cases. The bottle shaped stress field provides a safe lower bound for unrein-

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

equilibrium can frequently only be

satisfied if ti satisfied ties es or ten sile forces can be accepted in places where, for practical reaso ns, reinfo reinfo rceme nt canno t be pro vided, i.e., if the tensile strength of concrete is u tilized. tilized. ,  It shou ld be apparent that no anchorage, no lap, no frame

forced compre ssion struts, wh ereas an indiscriminate application of the theory of plasticity plasticity to cases such as those shown in Fig. 7 (mainly Fig. 7a) wo uld perm it prismatic stress fields fields between two op-

com er, no slab w it ithout hout stirrups stirrups an d (as

posite anchor plates with 1.0 ff  as a failure stress and cou ld lead to a premature

not recognize this fact and, therefore,

PCI JOURNALiMay-June 1987

 

failure.

shown) no unreinforced strut or compression member can work without

using the tensile strength of concrete. Unfortunately, most code s of practice practice do surrogates surrog ates such as bond, shear 07

and o ther misno m ers have been intro duced. As a result, co des have beco m e unduly imprecise and complicated. Un til further rese arch w ork is avai availlable in this field, the following simple

guidelines are proposed, which appear to yield safe results when com pared to tests: The tensile strength stren gth of co concrete ncrete

shou ld only be u tilized tilized for equili equilibrium brium

forces where no progressive failure is

transverse direction and in greater

depth. If the tensile stress field is crossed by a com pression field, the redu ced biaxial strength mu st be considered. The graph (see Fig. 26c) provides a safe assumption. 4.6 Reinfo rc ed Ties T.

microcracks have to be taken into account, even in "uncracked" and unloaded concrete. As shown in Fig. 24,

Usually, reinforcing steel should be provided provid ed to resist tensile te nsile forces. The axis of the steel reinforcement must coincide with the axis of the tie in the

redistribution of stresses w hich avoids redistribution progressive cracking cracking may be assum ed to be possible if at any part of the stress fi field eld a cracked fail failure ure zon e w ith an area

from the cross section A. (reinforcing steel) o r A, (prestressing (prestressing steel) and the

expe cted. Thereby, restrai restraint nt forces and

JA, can be assumed, without the in-

creased ten sil silee stresses in the rem aining section exceeding the tensile strength

As a preliminary proposal, it is sug-

gested that:

and ?A,,110 where A, = area of tensile tensile zone and  o  diameter o f lar largest gest 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 in the ben ding tensil tensilee zone of beam s (Fig. 25). The tensile stresses stresses m ay be analyzed with a linear elastic materials law. law. Stre ss peaks in the outer fibers or at failure

zones m ay be distri distributed buted over a width of 5 cm (2 in.) but not more than 3 d a rule wh ich finds its justifi justification cation in fracture me chanics of concrete. 13 The design engineer will have to decide case by c ase wh ich fracti fraction on of the tensile strength strength can he use d for carryi carrying ng loadss and w hich fract load fraction ion has been u sed up by restraint stresses. The latter

stresses are u sually la large rge in the longit longituudinal directi directioo n o f a structural m em ber 1 08

mo del. The dimensioning of these ties is is quite straightforward; it follows direc tly

yield stren gth f,,, and steels:  

Jet .

 

and at its surface, but are sm all aller er in the

f

of the respective

_-AJ _-A Jr,+A ,A ,Aff ,,

Since it iiss proposed here (see S ection 5.3) to introduce introduce prestr prestress ess as an external load into into the analysis and dime nsioning, the acting tie force T   is the resu lt of al alll external loads (including prestress). Then, however, part of the strength of

the pre stressed steel is already u ti tili lized zed by prestressing and on ly the rest, Af,, is available to resistT,.

4 ..7 7 Serviceability: Cracks and Deflections If the forces in the reinforced ties

under w orki orking ng loads are used and their effective concrete area A , ,ef as defined in t he CEB M od el Cod e or in in R ef. 19 is attributed to them, the known relations for crack control can be appli applied ed directis propose d that ly . o n In principle, it is the same m odel be used at the ulti ultim m ate limit 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

ilure zonF failure failur e zone AA c

Fig. 24 . Assum ption of a fail failure ure zon e for the chec k of the the ten sile stren gth of a con crete ten si on ti tiee

   

concrete stresses

fe

falu rez o n eAA

with torture zcne without

c a c k

Fi g, 25 . Progressi ve fail fail ure of a beam because of a l ocal fai fai lure lure zon e, wh i ch increases maximum tensile stresses a., to U., f^.

detailing (provision of minimum rein-

forceme nt, adequate selection of bar diam eters and bar spacing) is usually better than sophisticated crack calculations. Having determined the forces of the

m odel, tthe he analysis for tthe he de formations PCI JOURNAL;May-June 1987

 

is straightf straightforward. orward. Sinc e the co ntrihu-

tion of the concrete struts is usually

sm all all,, it is suffi sufficient cient to use a m ean value o f their cro ss sectio sectio n even tho ugh this varies over their length. For the ties, tension stiffening above crack analysis. follows from the 09

a)

cr f fct

^ c f c

c)

b)

 

ct

tc

 

1 0.5fc

fct

fC f

com pressive-tensil pressive-tensilee strength of concrete and two simplified assumptions for analytical application, Fig. 2 6. The biaxiia al

4.8 Concluding Remarks

Des pite the fact that the m ajor pri princinci-

even if used use d only to find out if, and where, reinforcement is needed. ne eded. In a

pleschhave been work set forth in in this mu research remains to chapter, be done

which is not over-reinforced, the con-

with respect to the more accurate dimensioning of the concrete ties and

struts. This, however, should not preclude application of the proposed procedure as a whole. Current design

m ethods of D-regions are, in fact, worse because they simply ignore such un-

solved questions, Fo ll llow ow ing the flow flow of forces by strutand-tie-m and-ti e-m odels is of con siderabl siderablee value

structure with reasonable dimensions

crete co m pressive str stresses esses are u sually not the main conc ern. Furtherm ore, it it iiss much more important to determine where the tensile strength of the con-

crete is utilized, utilized, and the n to reac t with reinforcem reinfor cem ent if possibl possible, e, than to qu antify the strength of the concrete ties. In the ne xt chapter the applicati application on of the foregoing principles is elaborated

upon with many desi design gn exam ples.

5 EXAMPLES OF APPLICATION With an unlim unlim ited it ed num ber o f exam ples it might be shown, that tracking

do wn the internal ffoo rces by strutstrut-andandtie-models results in safe structures and quite often provides simple solutions for problems which appear to be rather

com pli plicated. cated. It should, how ever, also he admitted that it som etimes takes som e effort to find the appropriate model.

How ever, the strut-and-ti strut-and-tie-m e-m odel is always worthwhile because it can often am

 

reveal weak points in a structure w hich otherwise could remain hidden m the

d esig n eng ineer if he appr oache s t hem by standard procedures. The m ajo ajorr advanta advantage ge of the m ethod is to improve the d esign of the criti critical cal Dregio ns. Ho we ver, tthe he au tho rs also believe 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-

L LI H Ui I

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

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

Vf

treated (see (see Refs. 1 an d 3 for addi additi tional onal design exam ples) ples).. Finally, the basic approach to pre-

stressed con crete is discussed and ill illusustrated trat ed with several exam ples.

5 .1 .1 T Th he B Regio ns The plane rectangular webs consid-

ered here apply not only to beam s with r ect ang ular cr oss sect ion ion b u t may also be part of a T, 1, double tee or box be am (see Fig. 27). The she ar lloading oading may result from shear forces or from torsion (see Fig. 28), the axial forces from external loads or prestress (see Section

 

CT

v w +

_ 0 5M1 bm V w

V hm

–bm bm— -

Fig. 28. Shear forces as a result of torsion.

5.3).

As discussed above, the web of a Bregio n m o dell delled ed w ith ith the same crit criteria eria as proposed for the strut-and-tie 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 pressivee st r esses accor d ing t o t he t heor y of siv elasticity. The design of beams for bending,

shear and torsion is is then nothing m ore PCI JOURNAUMay-June 1987

 

than the we ll known analysis of the truss forces and the chec k of the the com pressi pressive ve stresses of the the c oncrete an d the tensile stresses of the reinforcem reinforcem ent. Since this analysis of the truss includes the cho rds, possible problems like the staggering

effect eff ect or t he quest ion why b ot h chor d s are simultaneously in tension at the

points of infl inflection ection (mom ents – 0, shear forces 4 0) ar aree solved au tomaticall tomatically. y. 111

fs

O  

 

eqao refeld, Thurston resler, Scorde/ s

    

 

®

•

Rujagopalan, Ferguson Petterson eorhardt, eorhard t, W alther alther ,Y addadrn addadrn et a .

•

•

Taylor

a

Braestnup raestnup et al

e

Rodriguez et al Guralnick

a

Lyngberg, Orden, Sorensen

 

•

of

  • L_ i —                   

  2

0,

   

 

a 

Q

V E

fut

^__

^

g y m ^ 8 ^  `

5

 

 

ZL

r

^1o^ s^^^

•

 

•

 

 

R

V

•

0

bZ

9

005

010

0,15

d ZO

OZ5

030

tc

Q35

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

 

It should be mentioned also, that in contrast to the bending theory, truss models are capable of dealing with stresses perpendicular perpendicular to the beam's

axis and variable shear forces along the axis:

(Note that that these sh ear forces are not com patible with Bernou lli's lli's assum ption of plane plane strains as defined for the B-regions.)

W hy then, if everything appea rs to be so simple, is there a shear riddle and wh y has the shear battl battlee been waged for so man y decades . In fact, the endless discussions on shear could be put to bed if design engineers contented

45 degrees, depending depending on the am ount of stirrup sti rrup reinf reinforcement orcement an d the w idt idth h of the web expressed by b lb. If, in addition, axial compressive forces such as prestress act, a is a priori smaller than 45

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

given by th e theory of elasticity.

11

However, the fact remains that the

explanation of th e real stirrup stresses only by the the standard truss wo uld imply the assu mp tion of struts with less than realistic reali stic incli inclinations, nations, in ex trem e cases down to d = 15 deg rees. Other kn own explanations are also not satisfactory: An inclined compression chord would si-

themselves with the sam e level of accu-

mu ltaneously reduce the inner lever arm

racy (and simplicity) in the B-regions as they do for the D -regions. -regions. Since in principle it it ma kes no sense to

z, which is not compatible with the Bregion regi on assu mption of plane strains and

design B- and D-regions, which are

regions wh ich extend fr regions from om the supports; the pure arch or direct struts with tie action can on ly be applied if the tie iiss not bonded w it ith h the surrounding concrete or to short beam s, i.e. i.e.,, beam s without B-regions; the dowel action of the reinforcem ent, tthou hou gh leading iin n the right direction, directi on, can on ly to a sm all degree be responsible for the effect effect und er discus-

parts of the same structure, at different levelss of s ophisti level ophistication, cation, there wou ld be very good reasons for such a compara ble accura cy in every aspect of tthe he ana lysis. How ever, at present this see seems ms n ot to be appropriate mainly b ecause of h ist istorical orical reasons, i.e., since so m any researchers have invested so m uch time investigating the B-regions, they they have foun d that (under certain conditions) savings in

stirrup reinforcem reinforcem ent are possible com pared to simple truss design.

can, therefore, develop only in the D-

s ion.

Wh at does real really ly happen in the web? On the foll following owing pa ges it will be shown , that it is the the con crete's tensile strength

Fig. 29 shows the we ll kno wn plot (i (in n dim ensionless coordinates) of the ultimate shear forces V. versus the the am ount of stirrups a„ (cm 2 /m ) required ttoo carry V. for beams in pure bending and shear. If the straight line line according to the truss mo del with 45 degree struts ((M M Orsch) is

pure truss design of B-regions B-regions m ay now jump ah ead to Section 5.2. T The he authors wrote what follows not with the inten-

compared with the compiled test re-

tion of adding one more paper to the

and aggregate interlock in the web

which really causes the reduction of the stirrup stresses. Read ers who a re satisfied with this explanation or with the

suIts, it is is found that a large discrepancy

shear dispute, but only to show the ef-

results, mainly in the region of low to

fectiveness of the strut-and-tie-model

med ium values of V,,. The discrepancy is reduced if sm aller ang les of iinclination nclination of the struts tha n those taken from the elasti elasticc stresses at

approach even for such cases.

from test results it is observed that the

angle a of the initial crack from pure

shear can be u p to 10 degrees lless ess than

 

have reach ed the tensile str strength ength of the

 

the neutral axis are assumed. In fact,

PCI JOURNAL.iMay-June 1987

After the principal tensile stresses

concrete, the web cracks at angles as discussed above. Consequently, Consequently, following the direction of the load, individual pieces of the web, only con-

trolled in their movement by the

13

bI

al  

i

 

F

 

load path 1

C Q

fps

 t r

 

stirrup tie-canerete strut

stirrup

 

c

food path 2

fl

r

concrete tie - concrete strut

Fig. 30. Internal forces for ces in the web due to shear: s hear: a) kinematics kinematics of load path 1 if acting alone 8 = a); b) through throu gh c) load paths pat hs 1 and and 2 in the web web if if acting acting combin combined ed 0 < a).

fl flanges, anges, try to fall down. T here, they are caught by the stirrups which hang up the load via T into the adjacent piece

evoking C in the struts for verti vertical cal equilibrium (Fig. (Fig. 30 30a). a). Th The e ch chords ords ((or or flanges) provide horizon tal equili equilibrium brium with ad ditional tensile fforces orces F. This is

the principal load path path 1, if the concrete's tensile strength is disregarded

(Fig. 30 b).

for equ ilibri ilibrium um . Both load paths jointly carry the load and therefore their their comb ined comp ressive struts together together assu m e the inclination 0 -_ a, As long as it can be sustained by the concrete, the concrete tensile

force perpend icular to the struts iiss respon sible ffor or the fact that th e stirr stirrups ups need to carry only part of the shear

loads. However, it also also causes the con -

lock in the crack and it appears reason able to assum e, that the resisti resisting ng force R acts in the d irection of ', i. i.e., e., parallel to the crack. The force force R has two com po-

crete of the struts to be biaxially loaded, thus either reducing their comp ressi ressive ve strength or resulting in a second array of cracks w ith iinclinations nclinations less tthan han a , depending on the load case. Only if 0 = a does load path 2 disappear (Fig. 30a). Wh en this occurs the com pressi pressive ve struts are uniaxially loaded and can the refore develop their maxim maximum um strength. Therefore, the maximum capacity of a beam for shear forces iiss achieved if the struts are parallel to to the crack s and if the corresponding large am ount of sti stirrups rrups is provided.

nents, a compressive force C, with an

What can so simply be described in

inclination inclinati on B < a an d a concrete tensile

words mu st also be accessible ttoo a relatively transparen transparen t analysis. This is possible, if if the com patibili patibility ty betwe en load

Looking closer, it is recognized that the kinematics as described evoke an

additional load pa th 2 (Fi additional (Fig. g. 30c) w hich comb ines with the lload oad path 1 (Fig. 30b) but which is usually neglected: The vertical movement v has two compo-

nents, the crack opening w perpendicular to the crack and a slidi sliding ng A parallel to the crack (Fig. 31). The sliding A is obviously resisted by aggregate inter-

force 'I perp perpendicular endicular to it (F (Fig. ig. 32).

Again, the chords are activated 114 11 4

 

paths 1 and 2 is solved by plastic su-

perposition of b oth the stirrup and the perposition interlock inter lock forces, making u se of the fact that aggregate interlock is sufficiently ductile. How ever, since a biaxial fail failure ure criterion crit erion 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 is imp ortant to note that the bea m   rem em bers the iinitial nitial incli inclination nation a of the diagonal cracks. This permits the

logical introduction of the effect of prestress into web design, resulting in savings of stirrup reinforceme nt but also in an ea rlier rlier com pressive strut fail failure. ure. The an alyti alytical cal 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 interse cts the abscissa (where 0 = d2), because a m ini inimu mu m of sti stirrup rrup reinfo reinforcercem ent is necessary iin n order to guarantee that a truss model can develop at all all.. As show n in Fig. 34, it iiss necessary to avoid that the flanges flanges separate from the web after a diagonal crack crack has formed. Of course, the longitudinal and transverse spacing of the stirrups stirrups m ust further be limited to ensure a pa rall rallel el diagonal com pression fiel field. d. If the spacing is too large, the smeared diagonal compressive strength strength m ay be less than that taken

v = v e r tit 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

L

T c R c R

T c  =

r e s u l ta ta n t

of

c

the tension

field

C c = r e s u l t a n t of the compression field

 

R = force of the aggregate interlock

Fig. 32. Agg regate inte interlock rlock force R and corresponding compression C. and tension

for Fig. 33, because the stresses con-

T, in

centrate in the nodes with the stirrups (Fig. 35). To use load path 2 in the practical design of the webs of B-regions — if it is considered desirable to be more soph is-

the concrete.

to replace the curved bran ch of the diagram in Fig. 33 b y a straight line. It iiss only important that the design engineer is aware that reliance is- placed placed on the

ticated there than in the D-regions, where aggregate interlock or the concrete tensile strength is neglected simple diagrams may be derived from

concrete tensile strength in the web if this method is used to reduce the re-

quired am ount of sti stirrups. rrups. Fig. 29 corn-

pares the amount of stirrup reinforcer men t derived in thi thiss way from the strutand-tie-model with that from different

Fig. 33. Even easier (and (and correspon ding to the present present CEB M odel Code), it may he suffici sufficient ent to to reduce in tthe he low shear range the act acting ing shear force for stir stirrup rup design by an am ount wh ich is at attri tributed buted to load path 2. For that purpose, it may be su ffi fficient cient

  codes and tests.

One might also ask whether the full design strength f can be exploited in

the compression chord of a beam, be-

PCI JOURNALJMay-June 1987

15

 

ms s

pv f

c

b  

 

0,40

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

0,35

                  / o            

—•   urnlnik 4

•   oddodm M J , Hong ST . Mattock A H

Krefeld WJ , Thurs ton CW

0   3 0 ---

A  eonhordt F alther R

°  ynberg BS , Ozd en K Sorense Sorensen n HC

Petterson T

^_-

Ram apo lnn KS , Ferguson P M o Regan PE

0   25

0  odrquesJJ,BionchmiAC,Viest1M, Kesler C E

02 0 _

0,15

—

010

—

,OS

/

o  aylor R.

—

^ 0

—

6

off'

df _

a

 

o

250

^

 

0 ^•

o

c  

19 .

°

,05

10

15

20

0,25

 ,0

,3 5

,4 0

e baccording to the strut- and-tie- m od el Fig. 33. Req u ired ired amount o f vertical stirrups in a w (for crack inclination a = 38 degrees), compared with simple truss ana analo logy gy and tests.

cause the chord is subject also to tensile

the forces in the compression chord as

strains in both transverse directions. In

much as it increases the tension chord

the horizontal direction such stresses

forces. However, if the chord forces are

are quite obvious in the flange of Tbeams with transverse bending and

correctly derived from the truss model,

transverse tension from the flange con-

sumed to be only 0.8 ff , according to

nection. In addition, there is vertical tension from compatibility compatibility with stirr stirrup up

Section 4.1.

tensile strains. Some cross tension also exists in both directions in rectangular beams (Fig, 35).

very frequent B-region, namely, beam or column with rectangular cross section column

The neglect of this fact in practical de-

shear forces and bending moments. moments. This

sign is to some extent balanced by the

case is typical also for prestressed

usual neglect of the staggering effect in

beams. beam s. If the axial force is large enough

the compression compression chord, which reduces

to keep the resultant of the normal force

 

the compression strength should be as-

Consider, in addition, a special but

loaded by an axial force in addition to

         

 

compression chord

tirrup

^ —

web

—

s

stirrup

T

C

eb

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 T he compr com pression ession strut in the web with stirrups.

and of the moment within the kern of the cross section, the standard truss

chord of the adjacent B-region, thereby

model no longer applies. Instead, all

internal forces, including including the shear force

These stresses can be assessed by the more refined mod el iin n Fig. 36c. S tir tirrups rups

V, may be represented by a single in-

may be used to cover them; however,

clined cli ned com pression strut as in Fig. 36b.

represented by the truss model, it is

the m odel shows that tensil tensilee forces are inherentlyy q uite diff inherentl different erent from the shear forces in other B-regions b ut similar to those in the bottle bottle shaped comp ressi ression on

found that the com pression stresses distributed tri buted over the w hole section have to converge into the narrow compression

strut. Speaking of B-region design, there remains rem ains the issue of beams beams without without

Looking closer at the transition be-

tween such sp ecial B-regions and those

PCI JOURNAL/May-June 1987

 

creating transverse tensile stresses.

  17

 

a)

o me n t

shear force

compression force

a

    Li [JE ^ ^

resultant within kern-zone resultant result ant wit ithin hin leve lever arm z

mod el com patib atible with with the the standard truss truss model

C)

± ^ ^ kernzone

forces of of stirrups 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.

118

 

shear reinf reinforcement. orcement. Sub divi dividing ding such beam s into B- and D-regions first first of al alll reveals that an arch-an d-tie explanation only applies if the the two D -regions of the

Strut-and-tie-model Forces in the struts and ties Dim ensioning of ties: rei reinforcement nforcement Ch eck o f stresses of cri critical tical struts and

opposite supports touch, leaving no sig-

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

nodes Reinforcement layout Solution:

inforcement also stimulates arch-and-tie action, in other words it increases the

(1 ) External equilibrium: reactions

length of the D-regions. The web of a

real B-region, B-region, how ever, can carry shear loads only with the help of the con-

A=3x --= 1.07 MN 7. 7.0 0

4 = 1.93 1.9 3 M N B=3x =

crete s tensile strength. There are several approaches based on rational mod-

7 .0 F=A +B– 3.00 MN

els to exploit this finding.12.16 Aga in, as above, aggregate interlock is

(2 ) Elastic stress analysis

involved and, therefore, an analytical

solution soluti on is rather com plicat plicated. ed. On the other hand, the design engineer must know and understand what actually happen s. Thus, it wou ld realist realistical ically ly reflect the present situation to replace the perm issible shear stresses of the current

This is a rather comp licated structure. If the design engineer is not yet suffi-

cientlyy experienced w it cientl ith h m odelli odelling, ng, he

will first employ an elastic finite ele-

men t program an d plot the elasti elasticc stresses (see Fig. 37b) for orientation of the

strut-and-tie-model.

codes by the equivalent permissible permissible

(3 ) Modelling

concrete tensile stresses. Unfortunately, the stresses in the web are not un iformly distributed. distri buted. Therefore, calibrated average adm issible tensile stresses mu st be given in codes. For comp leteness, iitt should be men tioned that a B-region contains several

The w hole structure is essential essentially ly one D -regi -region. on. Two sh ort B-regions are di disscovered in the linear parts to the left and below the h ole (see Fig Fig.. 37c). The load p ath connecting the reaction B an d its counterpart within F is readily

micro-D-regions, which again can be understood and designed with strut-

Nodes 1 a nd 2 are typical (see Fi Fig. g. 15) and the forces C and T balancing these

and-tie-models. Fig. 35b shows why a

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

nodes in the horizontal direction are

stirrup must he closed on the top and bottom in a rect rectangular angular beam in order to

thus also given.

take the cross tension from the strut

leftt side, the right side lef side m ay be fi finished: nished:

support.

The strut between Nodes 1 and 2 will spread and cause transverse tensile

5.2 Some D Regi Region ons s

forces as sketched in F ig. 37d ; optionally, all y, it can be treated as a bottle shaped stress field field as sketched in F ig. 371i and described later on.

5.2.1 Deep Beam With a Large Hole

(Numerical Example)

Given:

Before continuing modelling on the

Now the left side's boundary forces

Dim ensions ((see see Fig. 37a)

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

Factored loadF, F = 3 MN

therefore, be modelled independently

 

Concrete compression design strength  = 1 7 M P a

Reinforcement design yield Reinforcement y ield strength f„d  = 4 3 4 M P a

from the right side. (In (In pas sing, it m ay be m entioned that the ficti fictitious tious separation line line between the two sides is where the overall shear force of the the deep beam

Required:

is zero and the bending moment is

P C I J O U R N A U M a y - J u ne ne 1 9 8 7

 

19

• o7

 

FF-3M 3M N

O i

      4 1 5

t

^5

7

a ) dimensions [m] and load

25

X 5

 

iiI

\i •

tiJI__

compression stresses tens on stresses b ) elastic stresses

120

 

B

c) load path, right side

I1 \

 

^

d) complete model, right side

 

Fig. 37 . Deep beam wit h a large hole. hole.

PCI JOURNAL/May-June JOURNAL/May-June 1987

21

 

A

C

e) boun b ounda dary ry forces, forces, left left side

Bi

I BZ-rN rN^on

A

).5A

105E

f) model 1, left side

0.5 T

A2=05A

  I I

I

  

 

  CZ=05C

`

t

I

j l

g) model 2 left side

T4

 

^

4 5 0  T = 0 5 T

A r O 5 A

122

 

bottle

node 2

h) complete strut-and-tie-model 

 

ar  p

-

'(b.net I-2 t5 4-J i---4 2x7 5

L^`L 11

L ^ L T

i) reinforcement

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

PC I JOURNA L /May-June /May-June 1987

23

 

N

    C -- 1 moaei

ump€ohed main reinforcement   e n e d n1 T t s fig 74

i^A i^A

h

Bi

tl

 

 

cl

♦-

t

d

e

t^

rlHH

r

    ^^-

stirrup forces

Fig. 38. Load near the supp ort; transition transition from

eam to beam .

 

ma ximum . Theref Therefore, ore, only tthe he horizontal forces 177 =1 CI connect the two

sides.)

the respective tie forces is given in the table below.

Forces C and A meet at Node 3 and

their resultant resultant is given. From the bottom the reactionA enters the structure vertically and it is assumed that it remains in this direction until it has passed tho

hole. The B,-region is thus a centricall centricallyy loaded loade d col column. umn. In fact, part of the reaction A could

also he transferred via the

-region -regi on b y

Tie

Force [MN I

Req'd. A, [cm 2 l

T

1.07

24.6

8#7

T,

0.535

12.3

4 7

T.

0.535

12.3

4#7

T,

0.535

1 2.3

2x7 5

U se

( c ? Fm   I

0.08_

-

bending moments moments and shear forces.

Com paring the axial stiff stiffness ness of B, w ith the ben ding stif stiffness fness of B   , this part is obviously negligible. This fact is also

confirmed b y the elastic str stresses, esses, which are very small there (F (Fig. ig. 37b). Of course, some nominal nominal reinforcem rei nforcement ent will be provided in the member below

the hole for distribution distribution of crack s due to imposed deformatio deformations. ns.

T

0.535

1 2.3

2x5 4

0 .3

T 

1.07

24.6

2x7 5

0.08_

'I'

1

24.6

2x7 5

0.08_

1 

0.535

12.3

2x5 4

0 .3

1 ',

0.535

1 2.3

2x7 5

0.08_

T9

0.663

1 5 .2

4 7

T, °

0 .4 0 2

9 .2

or bottle check

„

0 .40 2

9. 2

(see below)

Figs. 37f and 37g give two different

7

Note: cm

some similarity with a beam having a

(5) Check of the concrete stresses Stresses under the bearing plates:

based on 45 degree struts and on ties also at 45 degrees from the struts. This gives a reinforcem reinforcem ent layout, which for

practical reasons is orthogonal. The m odel iin n Fig . 37 g assumes a 4 5 degree tie at the corner of the hole ho le which is know n to b e effect effective ive in similar cases like opening frame corners or dapped beams. Each model in itself would be suffi sufficient cient but - looking at the elastic stresses of Fig. 37b - a combi-

nation of both appears to be better than either of them. Therefore, it is assumed, that each m odel carri carries es half the load. Finally, Finall y, Fi Fig. g. 37h sh ows the superposition of both mo dels of the left side including the right side as described before. Wh en com paring model and elastic stresses one finds a satisfactory coincidence. The geom etry of the mod el iiss indeed o riented at the elastic stress fiel fields. ds. (4) Design o f the ties The reinforcem rei nforcement ent requirem require ment for

-

__-

 

strut-and-tie-models for the Ieft side of the deep beam. It is seen that there is dapped end. The model in Fig. 37f is

__-

.155 in. in.

3.0 - = 10.7 MPa < 1.2f.4 0.7 x 0.4

(biaxial compression) _1

C r

07

-=.).4MPa