e)

'►C 1 a^_ I

t F I 71

II
u I I I.

B
I I
I I I
k T I

1 1
1 l

t ^ B
1 1

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

PCI JOURNAL/May-June 1987 133

 a}

b) 2V; T2 =Tp • ltrn, 2 v.

L1 T±J
[)

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

space of the D-region. (b) Prestress plus other loads

Then, T, = P tan a2 = 0.18 P (see Fig. Tensile forces are assumed in the B,-
45b). regions, which are considerably larger
The reinforcement for Ti should be than the prestressing force, as is the case
chosen with due consideration of ac- in an overloaded or partially prestressed
ceptable crack widths in this service- structure. In case there is no additional
ability state of stress, i.e., the tensile reinforcement provided, and the tendon
strength of the reinforcement should not has to take all tension forces the excess
be fully utilized and its slipfree anchor- force (T h0 ,( according to Section 5.3.6)
age should be accounted for. was computed to be:

134
 107 °
a) _
5
Zc
1375kN

m 03 CC 04
C, B' 0Z

5;0
11 d
F 6.15 ^a2 +

moments M

y z
shear forces V

lvi O

b}

C^ C^ C4 C6
C^^ DZ C3 C3
^ B^ - --^ ^ a3 C7
'^ T3
T, TZ Tz T

EF---T:::;4 i:j:
C)

d)
f
stirrupsfl
iz

T5 05Fd
T5=2.1F,

n IDF
T8
T,

_I
__ __
—stondird design B-regon

e)

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

PCI JOURNAL/May-June 1987 135

 T,=T-P=0.33P loads of the other tendons. Thereby,
The load path for T R and the resulting transverse tension T, develops.
model (Fig. 45c) is now quite different The position of T, may have to be
from that for prestress alone, because shifted toward the center of the blister,
the T L on both sides do not meet, but if the bond of the tendon is very poor. It
rather try to equilibrate on the shortest is the tension from these bond forces
possible way with the nearby anchor which now calls for transverse rein-

0
0)
g2 ^a1
I ^-

P- P

curvature forces P aj Z
n, T nag

c)

TzzT-P

crock
T1
Tz

- iii.
INNNI!!!h oil
r
7m:

Fig. 45. Overlap of prestressing tendons: (a) layout, B- and D-regions; (b) model for
prestress only; (c) model for prestress and additional load T d ; (d) layout of the transverse
reinforcement.

136
forceinent T, = 0.53 T2 , not the splitting open several millimeters wide, even if
action as for prestress alone. the B 1-region is reinforced for crack dis-
The overlap of the prestressing ten- tribution in the usual way. Therefore, in
dons results in twice the prestressing this case not very large tendons with
force within the blister, which therefore good bond properties have to be applied
remains basically uncracked and rela- or additional parallel wall reinforcement
tively stiff. The strains of the tendon in must be provided, which takes over
excess of the initial prestress will much of the tendon forces, before the
therefore accumulate within the bond tendons enter the blister.
length of the tendon in the blister and
5.2.7 Beam With Dapped End
cause a crack at the jump of the wall
thickness. It is common practice to suspend the
For large bond lengths this crack may reaction F of the beam in Fig. 46 besides

tII
F
II

11r-11—
s^
f

L
-T3 =F
-'ice

^^

13= zcot*—+
144414 1444111 lUrlill
+]

ILL!1
rz^ t3=1

Fig. 46. Beam with dapped end.

al
^c
Tff] a^

T
bl

______ fll+ifr
Fig. 47. Girder with bent top flange.

PCI JOUJRNALMay -June 1987 137

 rig. 48. 1 he stepped beam.

the dapped end (T, = F). But the com- overlapping the reinforcement coming
plete strut-and-tie-model clearly re- from both sides with an elegant loop.
veals that it is not sufficient to simply The strut-and-tie-model supplies facts
add T, to the regular "shear reinforce- for a rational reinforcement layout.
ment" which amounts to the vertical tie
5.2.10 Frame Corner
forces t 3 = F/1 3. In fact, there are addi-
tional vertical tensile forces T 2 = F be- The frame corner with opening mo-
cause the horizontal tie force T at the ment, more often discussed by re-
recess needs to be anchored. The tie searchers than actually occurring in
force T 2 is distributed over a length 1 2 < practice, can be modelled quite differ-
l 3 and therefore t 2 is clearly considerably ently (Fig. 49). Obviously, the design
larger than t 3. If, as usual, an additional engineer has to choose between a rela-
horizontal force H acts at the recess, the tively simple reinforcement combined
necessary amount of vertical stirrups with a reduced moment capacity (Fig.
further increases. 49a,b) or a more sophisticated solution
(Fig. 49c,d,e). The consequent applica-
5.2.8 Tapered Beam With Bent Top tion of strut-and-tie-models makes the
Flange designer aware of what is occurring
The girder in Fig. 47a obviously pro- while offering a rational choice.
duces a vertical tension force T at the
bend of the compression chord. But
5.3 Prestressed Concrete
where does it go? The straight horizon-
tal tension chord cannot equilibrate it. As a last example, it will be shown
The model shows that stirrups in the that looking at prestressed concrete
web are necessary throughout this web beams through strut-and-tie-models
even in regions without shear forces. helps to understand their behavior
Looking at Fig. 47b, it is apparent that which today gets hidden behind so
the compression chord is narrowed by many black box rules. There is a common
the stirrups, resulting in a concentration denominator of all types of prestress:
of compression stresses over the web. post-tensioning, pretensioning and un-
Furthermore, unfavorable tensile bonded prestress can be understood as
stresses in the transverse direction of reinforced concrete which is loaded by
the flange appear. an artificial loading case, i.e., prestress.
As any other Ioading case, it simply
5.2.9 Stepped Beam
has to be introduced into the analysis of
The stepped beam in Fig. 48 is fre- the structure according to the actual
quently used and is usually detailed by history, e.g., for post-tensioning: con-

138
 >H
___ ri $fti
ILJP
b1

d? ^

J l

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

PCI JOURNALJMay-June 1987 139

 7
creting and hardening of the reinforced referred to as prestress losses) are in
concrete, applying the prestress reality simply stress redistributions
(thereby activating dead load), provid- as in any reinforced compressed
ing bond and imposing external loads. member and can be treated accordingly.
After bond is activated, the prestressing If prestressing is introduced in this
steel acts as reinforcement, like regular way into the design of reinforced con-
reinforcement does; only its preloading, crete, all types of prestressed structures
its different surface with respect to the (linear members, plates, deep beams,
strength of bond and its sensitivity shells) can he designed, analyzed and
needs to be taken into account. dimensioned like reinforced concrete
structures: The sectional forces of a pre-
5.3.1 Prestress as a Load stressed beam are determined and com-
By prestressing, forces are artificially bined for the load cases prestress, dead
created with the help of hydraulic jacks; load, live loads, etc., and the resistance
on the one hand, these forces act as of a cross section is derived as for re-
loads on the prestressing steel, on the inforced concrete with prestressing
other as loads on the reinforced concrete steel as additional (passive) reinforce-
structure (Fig. 50). The loads acting on ment.
the prestressing steel and on the rein- Thereby, the same method of analysis
forced concrete are inversely equal. The and dimensioning can be applied to all
design engineer chooses the tendon load combinations in serviceability and
profile, the type and magnitude of the ultimate Iimit states. Consequently, in
prestressing force in such a manner that the ultimate limit state the reinforced
these artificial loads influence the load concrete sections (with reinforcement
paths and sectional effects or stresses A,andA,) of a prestressed beam have to
due to the actual loads (dead and live be dimensioned for the following (ac-
loads and other loads) favorably and as tive) sectional effects:
efficiently as possible.
Normal forces: N = –P + Nf
It is proposed to treat these pre- Shear forces: V = V,, + VL
stressing loads like permanent loads Moments: M = M„ + ML
which never change after the prestress-
ing jack has been removed. All the where
changes of stress in the prestressing
steel, which occur after removal of the P = prestressing force immediately
jack, ought to be attributed to those load after prestressing
cases which cause them. In those load A1, Vp = corresponding moment and
cases the prestressing steel adds to the shear force due to pre-
resistance of the section or member like stressing
nonprestressed steel. N,,, VL , M^, = sectional effects from
The view sometimes expressed that other loads
the prestressing itself increases under The sectional effects given here are
live loads because the stress in pre- meant to include the (partial) safety
stressing steel increases or decreases as factors according to the chosen safety
a result of the bond with the concrete, is concept.
misleading. The stress also changes in The proposed treatment of prestress-
normal reinforcing steel due to these ing leads to the same results as does the
effects and one would never regard this usual method with all stresses in the
as a change of prestressing. tendons regarded as being passive in
Accordingly, the changes of stress in the ultimate limit state. However,
the prestressing steel due to creep, the method proposed here is more gen-
shrinkage and steel relaxation (often eral.

140
 e} E' i :i
Fig. 50. Loads due to prestressing (anchor forces, friction forces,
deviation forces due to the curvature of the tendon) acting (a) on the
prestressing steel; (b) on the reinforced concrete member.

Also, the different degrees of pre- changes in temperature or settlements;

stressing (full prestressing, partial pre- these diminish or even disappear in the
stressing, no prestressing) and different whole structure if the stiffness decreases
applications Iike pretensioning, post- due to cracking, those from prestressing
tensioning and unbonded cables can all are only redistributed.
be treated alike with these principles: If it can be assumed that the stiff-
Forces due to prestressing as permanent nesses are not altered by loading, mo-
active loads which never change and ments resulting from different loading
prestressing steel contributing to the re- cases may be superimposed. However,
sistance after the cable is anchored. because part of the reinforced concrete
girder passes from the untracked to the
5.3.2 Prestressing of Statically
cracked state and because plastic de-
indeterminate Structures
formation of the concrete and the steel
In structures with statically indeter- in the reinforced concrete girder is pos-
minate supports the properties of the sible, the local stiffnesses change with
materials and the geometry of the variations of the load and, therefore, the
structure have to he considered when distribution of moments also changes.
determining the sectional effects. Those When this occurs the individual mo-
from prestressing can be split up in: ments can no longer be superimposed.
V„= P sin 6+V, Nevertheless, if the theory of linear
M,= –Ve +M,12 elasticity is taken as a basis for calculat-
where S denotes the inclination of the ing the forces, the resulting overall mo-
prestressing cable and e denotes the ca- ments (including M1 ) can be adapted
ble's eccentricity. V., and M me , the stat- to the real loadbearing behavior by re-
ical indeterminate portion of the pre- distribution.
stressing effects resulting from support
reactions due to prestressing, are of the 5.3.3 The Prestressed Concrete Beam
same kind as the statically indetermi- With Rectangular Cross Section
nate moments which result from dead In Fig. 52 the strut-and-tie or the truss
loads or live loads. M, is not the result model of a simple prestressed concrete
of restraints such as those due to beam with a straight eccentric tendon is

PCI JOURNAL/May-June 1987 141

 shown and in the preceding Fig, 51, for that prestressing improves the load-
comparison drawn in the same manner, bearing behavior as compared to the
of a reinforced concrete beam, The de- nonprestressed reinforced concrete
sign engineer selects the prestress P and girder.
combines it with the support force A = F Since prestress does not fully utilize
to a resultant, R, entering the beam at the strength of the high tensile steel
the support. used for tendons, it can be used as
If the resultant meets the action line T.I,wY i to cover a part of the tensile force
of the load F within the kern of the sec- of the chord T,.,. If the prestressing
tion, the condition is full prestress, if F steel is not bonded with the concrete, it
is the working load F. Then the model is unsuitable to serve as reinforcement
has no tensile chord (Fig. 53). If the re- for T orn . It only affects the sectional
sultant meets the compression chord of forces in the reinforced concrete via ad-
the beam before it meets F. such that ditional anchor and deviation forces,
vertical ties (stirrups) and, therefore, and all tensile forces of the chord have
also a truss with a tension chord is to he taken by regular reinforcing steel
necessary to transport F. into the in- to satisfy equilibrium.
clined resultant, the condition is partial The resultant entering the beam from
prestress (Fig, 52). the supported end, as discussed, has the
It is apparent that the vertical tie and tendency to spread in the web of the
inclined strut forces of this truss are (al- beam as in the bottle shaped compres-
most, see below) the same as for the sion field (Fig. 52b). Transverse tension
reinforced concrete beam, because the forces develop as well as a "force whirl"
shear forces are the same. The span of in the corner (Fig. 53) which causes high
the truss is only shorter and, therefore, tensile stresses near the anchor plate.22
the forces in the tensile chord are less, These tension forces have to be
whereas the compression chord also in- checked. If they cannot be covered by
cludes the additional prestressing force. the tensile strength of the concrete (see
If the load is increased from F,,, to the Section 4.5), stirrups have to be pro-
ultimate F,,, of course, the inclination of vided (see Section 4.6).
the resultant R increases. The fully pre- Tensile edge forces, splitting tensile
stressed beam now becomes "partially" forces, tensile end forces, etc., in the
prestressed, the initially partially pre- zone of introduction of anchoring forces
stressed beam gradually approaches the are thus simply part of the strut-and-
reinforced concrete beams. (This, inci- tie-model. In fact, they require no spe-
dentally, shows that manipulating safety cial names suggesting that they are
completely via a factored load is mis- something special or specific to pre-
leading in contrast to partial safety fac- stressed concrete. The "problem" of
tors being put on the material and the superposition of the reinforcement for
load.) the shear force and the splitting tensile
Full prestressing transforms the gird- force is resolved by the model.
er under service loads into a "horizon-
tal column" (Fig. 36); its eccentric nor- 5.3.4 The Prestressed Concrete I-Girder
mal force is generated artificially and If a beam is not plane (rectangular)
the external load which actually has to but has a distinct profile like the T, I or
be carried by it is relatively small. box girders, with relatively large cross-
Whatever amount of prestress P is cho- sectional areas of the flanges, the resul-
sen, it shortens that part of the girder, tant entering the beam as discussed
where a truss must form to carry the load above, will spread on a path different
and replaces it by a direct and shorter from that in the rectangular beam;
load path. It is, thus, directly apparent though of course for the same applied

142
 -- -- compression
tension

Fig. 51. Strut-and-tie-model of a reinforced concrete beam loaded

with two single loads.

edge cut
a) result t R F

L :J r:
a —T chord

moment -

shear force

normal force

b1

Detail l-1 ::::

Fig. 52. (a) Strut-and-tie-model of a partially prestressed beam with rectangular

cross section: (b) detailed strut-and-tie-model of the beam area, where the
resultant is within the beam section.

forces P and A the simplified model is where the total resultant force remains
the same for any type of cross section within the kern zone of the girder sec-
(compare Fig. 54a with Fig. 53a). tion (Fig. 54b). This is because the lon-
The detailed strut-and-tie-model of gitudinal forces here are mainly con-
the prestressed I-girder (Fig. 54) shows centrated in the flanges. Therefore, only
that a truss already develops in the area a part of the prestressing force can join

PCI JOURNAL, May-June 1987 143

 L.1it4t _J.:k4 H
b) force whirl
1F
it pQq^^^

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

01 F

blweb

c) top f[ange

.-.

centroid of the forces coming from the web

d) bottom flange

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

144
with the support force to flow into the iluenced by the axial force due to pre-
web. The effective resultant force in the stressing only via the inclination a of the
web is, therefore, smaller and at a web crack, which is shallower than for a
greater angle than in the prestressed nonprestressed beam, corresponding to
girder with a rectangular cross section. the shallower inclination of the princi-
It is, however, still at a smaller angle pal tensile stresses in the concrete when
than in the nonprestressed reinforced cracking begins.
concrete girder. This effect of a can be considered in
The chords in the two flanges are the web design as discussed in Section
linked to one another via the web struts 5.1 (B-regions). From there it is shown
and ties. In this way, compression forces that the greatest possible inclination of
are introduced into the flanges (Fig. 54c, the diagonal strut is parallel to the crack
d). The strut-and-tie-model shows that in the web. The ultimate load capacity
the spreading of the forces from the of the diagonal struts in a web of a pre-
width of the web to the width of the stressed concrete girder is therefore
flange generates transverse forces in the somewhat smaller than that of the web
flange. The transverse reinforcement of a girder without prestressing, how-
must be distributed in accordance with ever, it requires less stirrup reinforce-
the length and intensity of the introduc- ment for a similar beam and load.
tion of forces.
5.3.6 Dimensioning the Prestressing
5.3.5 The Loadbearing Behavior of the Steel for the Different Types of
Web Prestressing
The strut-and-tie-model of the pre- As already mentioned above, the pre-
stressed girder with rectangular cross stressing steel can and will serve as reg-
section shows, that the stirrup forces in ular reinforcement, if it is bonded with
the part of the girder where the total re- the concrete, in other words it acts as the
sultant force remains in the kern zone of tensile chord of the truss, developed for
the section result from the spreading of the structure loaded with prestress and
the compression forces (Fig. 52b, 53b). other loads. If the capacity of the pre-
The "shear reinforcement" in that area stressing steel still available after pre-
is in reality a "tensile splitting rein- stressing Tp,r , r, i cannot alone cover the
forcement." As soon as the truss model force of the chord, reinforcement must
develops in the web (which as shown be supplemented in such a way that the
happens for a rectangular beam further total chord force T rd can be taken by
away from the support than for a beam the prestressing steel (p) and the rein-
with a profiled section), the dimen- forcing steel (s):
sioning of its struts and ties follows as
discussed for the B-regions of reinforced Tchord = TA,ehorrt + T..char,i

concrete. In this equation To"r, represents the

Whether the web crack develops from total chord force from the various loads
the bending crack or begins in the web (including prestress). The loads are to
itself (as specified for example by the be multiplied by the partial safety fac-
German Code DIN 4227 as "zones a and tors y which are, of course, different for
b") has no effect on the loadhearing be- various kinds of load. The right hand
havior of the members in the cracked side of the equation stands for the re-
web. After the web has started to crack sisting chord forces, divided by the ap-
the prestressing normal force is only propriate partial safety factors.
present in the compression chord, like The force T 5 of the prestressing
any normal compression force. In the steel which is still available for the
truss (or B-region) the web itself is in- chord after prestressing is equal to its

PCI JOURNALMay -June 1987 145

permissible total force T,, , , 0 , minus the determinate analysis. If the truss model
prestressing force yAP, contains a tension chord, supplementary
T.., rd = Tn,tat – Y"P bonded reinforcing steel must be pro-
As a rule, under ultimate load the pre- vided.
stressing steel is strained beyond its If pretensioning is used and the ten-
yield point and, therefore, T.,,, = A ,, fnu. don profile is straight, the prestressing
Then, the following simple equation force acts as an external normal force on
applies: the reinforced concrete girder. The pre-
stressing steel is then part of the rein-
Tp, iwr,r = yPP forcement.
Yr
If this is not the case, fp ,, in the above 5.3.7 Result
equation has to be replaced by the stress Considering prestressing forces as
Q ,, of the prestressing steel correspond- external loads is not only an advantage
ing to its total strain with regard to service load design, but
€ p,rot = €:, p + Ep.(ttorri also for checking ultimate load design
where and all other checks, because the load-
= strain from prestressing (active bearing behavior of the entire pre-
stressed concrete girder can then be
strain)
simply explained in terms of a strut-
e,, , , wrr = strain of chord after
and-tie-model.
bonding (passive strain)
Prestressing steel is used for two dis-
__ T,,.cpw,_.z tinct purposes. On the other hand, its
(A,+A8)Es prestress applies favorable loads to the
Some codes restrict the strain of the reinforced concrete girder, while on the
chord after decompression of the con- other hand, it works as passive rein-
crete to 0.005 (DIN 4227) or 0.010 (CEB forcement when it is bonded with the
Model Code). Considering the arbitrari- concrete. In this latter respect it is not
ness of these quantities, the small de- different from reinforcing steel. Other-
compression strain including the strains wise, if it is not bonded it acts as a tie
from creep and shrinkage can be ne- member.
glected, which means € v c ho rd is limited As a result of this approach, the task of
from 0.005 to 0,010, respectively. designing any type of prestressed con-
II n„ hand is provided after pre- crete girder becomes the task of de-
stressing, the prestressing steel cannot signing a reinforced concrete girder
be considered as reinforcement in this with regard to bending, shear forces,
way. Rather, it acts as a tic member. Its and normal forces, which among others
stress increment can only be deter- include the additional loading case of
mined from an internally statically in- prestress.

146
 ACKNOWLEDGMENT

This paper is a progress report of the (Pavia), P. Regan (London) and J. Per-
work in this field at the University of chat (Paris).
Stuttgart. The authors want to acknowl- Finally, the authors wish to thank the
edge the contributions of several former reviewers of the PCI JOURNAL who
and present members of their Institute, offered us critical but constructive help.
mainly K. H. Reineck, D. Weischede, In particular, we wish to thank J. E.
H. G. Reinke and P. Baumann. Breen (Austin) and J. G. MacGregor
The authors further received valuable (Edmonton). It is by no means their fault
contributions and encouraging support if the paper is still a burden to the
through many critical discussions with reader.
colleagues; thanks go mainly to Lastly, the authors hope that this
B. Thiirlimann (Zurich) and M. P. Col- paper will generate fruitful discussions
lins (Toronto), who also promote the in the interest of producing quality con-
idea of a consistent design of concrete crete structures. To this end, we wish to
structures and to the members of the thank the Editor of the PCI JOUR-
CEB Commissions concerned, in par- NAL for offering us such a prominent
ticular T. P. Tassios (Athens), G. Macchi forum.

NOTE: Discussion of this report is invited. Please submit

your comments to PCI Headquarters by February 1, 1988.

PCI JOURNAt1May-June 1987 147

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Alumni," Universite de Liege, 1964, out Shear Reinforcement," Work in
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PCI JOURNAL/May-June 1987 149

 APPENDIX -- NOTATION
Geometry M„2 = statical indeterminate portion of
a = width of anchor plate moment from support reactions
1, = width of compression field or due to prestressing
plate P = prestressing force
d = length of D-region p = load per unit Iength
d,, = diameter of largest aggregate p Q = pressure under an anchor plate
e = eccentricity R = resultant force
h = depth of beam T = tensile force, tensile tie
I = span, length V = shear force
t = thickness
V = vertical movement
Strength
w = crack width
x = depth of bending compression f, = specified compressive strength
zone of concrete
z = lever arm of internal forces f = concrete compressive strength
for design of undisturbed timi-
A = sliding parallel to crack
a = crack inclination (see Fig. 30) axial stress fields
H = diagonal compression strut f,, = average concrete cylinder
angle strength
= area of concrete tensile zone = specified tensile strength of
AA, = area of assumed failure zone concrete
= effective concrete area for ten- fa = specified yield strength of rein-
sion stiffening forcing steel
A, = cross section of reinforcing steel fp„ = specified yield strength of pre-
a, = cross section of reinforcing steel stressing steel
per unit length y = partial safety factor
_ mechanical degree of
t f, ,, reinforcement
Subscripts
C = concrete or compressive
Forces and Moments d = design
C = compression force, compression p = prestressing, prestressing steel
strut s = steel
F = load t = tensile, tension
M = bending moment u = ultimate
M T = torque w = working, web

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