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Laurent Labonte M.Eng.
PRESTRESSED CONCRETE CON~INMENT ANCHORAGE ZONES
AN INVESTIGATION OF
ANCHORAGE ZONE BEHAVIOR IN
PRESTRESSED CONCRETE CONTAINMENTS
by
Laurent R.S. Labonté
A thesis submitted ta the Faculty of Graduate Studies and Research in partial
fulfilment of the requirements for the degree of Master of Engineering
Department of Civil Engineering and Applied Mechanics, McGill University Montreal, P.Q. Canada.
1972
December 1971
ABSTRACT
An Investigation' of Anchorage Zone Behavior in Prestressed Concre.te Containments
by
Laurent R.S. Labonté, B.Eng.
Department of Civil Engineering and Applied Mechanics, McGill University.
M. Eng. Thesis December 1971
This report presents the results of a study of the contri-
bution of the lateral reinforcing steel to the bearing capacity of the
tendon anchorage zone in prestressed con crete post-tensioned circular
containment structures.
Experiments were performed on laboratory specimens of two
types: (1) eleven prismatic bloOks containing a single concentrically-
placed bearing plate which were tested to destruction under incremental
loadingi and (2) a model incorporating two anchorage buttresses with a
total of sixteen anchorage zones, in which the tendons were loaded up
to twice the specified design prestressing force. Various anchorage
zone reinforcement details were investigated covering a fairly wide
range of steel percentages.
Models of scale 1/6 and 3/8 compared favorably with the
prototypes, and exhibited a good structural similitude of modes of
failure, ultimate bearing strengths, and strains measured on the
lateral reinforcing bars.
The low level of rein forcing bar strains and the absence of
structural damage observed in the experimental buttresses, indicated
that stress conditions were less severe than could be inferred from Ir
design methods applicable to beam end blocks.
RESUME
Une étude du comportement du béton sous les plaques d'appui dans les cuves de réacteur en béton précontraint
par
Laurent R.S. Labonté
Département de génie civil et de mécanique appliquée Université McGill
Thèse de Maîtrise Décembr.e 1971
Ce ~apport présente les résultats d'une étude de la cont~ibution
de l'armature latérale à la résistance du béton sous les plé:lques dt appui,
dans les cuves de réacteur en béton précontraint.
Des essais en laboratoire on été faits sur les modèles suivants:
(1) onze blocs prismatiques, chacun muni d'une seule plaque d'appui dis-
posée en son centre, et chargé progressivement jusqu'à destruction, et
(2) un modèle comportant deux contreforts, et dont les seize tendons
furent soumis à des efforts allant jusqu'à doubler l'effort de précontrainte
requis en pratique. Plusieurs configurations d'armature latérale furent
étudiées, représentant un champ assez large de pourcentages d'acier.
L'étude de modèles réduits d'échelle 1/6 et 3/8 s'est avérée
rentable, une bonne similarité de comportement ayant été observée entre
ces modèles et des prototypes de pleine grandeur, quant aux modes de
rupture, aux capacités ultimes" et aux déformations unitaires enregistrées
sur les barres d'armature.
L'observation d'un bas niveau de déformations unitaires et
d'une absence de dommage dans les contreforts ~is à l'essai, a permis
de conclure que les contraintes dans un contrefort sont moins élevées
que ne le laissent prévoir les méthodes de calcul applicables aux zones
d'extrémité des poutres précontraintes.
TO MY PARENTS
ACKNOWLEDGEMENTS
The author wishes to express his thanks to the following
individuals and organizations, whose help led to the successful
realisation of this research program:
Dr. M.S. Mirza, Associate Professor, Department of Civil
Engineering and Applied Mechanics, McGi11 University, for his useful
suggestions and constant encouragement throughout the author's
period of study;
Dr. G.M. Sabnis, Senior Engineer, Bechtel Corporation,
for suggesting the research topic and devoting considerable attention
to its progress;
Professor J.O. McCutcheon, Chairman, Department of civil
Engineering and Applied Mechanics, McGi11 University, for his
guidance and support of the structural concrete research group;
Mr. A.J. Bingaman, Chief Civil Engineer, Bechtel Corporation,
for his effort dedicated to the project, and Bechtel Corporation, for
the major financial support of the project;
Mr •. Claudio Zanolin, Director of Planning and Control,
Francon Limited, through whom the Company contributed supplies for
the projecti
Mr. T. Brown, Manager of Nuclear Structures System, Inland
Ryersons Corporation, who was instrumental in obtaining the post
tensioning system through their associates BBR Canada; and to Messrs.
H. van Bodegom, G. Earl, and L. Limperis of BBR Canada, for their helpi
Mr. B. Maiorano, Mr. B. Cockayne and the laboratory
staff, for their technical assistance and moral support;
Miss L. Robinson and Miss Joyce Richards for typing the
preliminary and final manuscripts, respectively.
TABLE OF CONTENTS
Page
LIST OF FIGURES v
LIST OF TABLES vii
NOTATION viii
UNITS OF ME~SUREMENT x
CHAPTER
1 INTRODUCTION 1
2 SURVEY OF PREVIOUS WORK 9
2.1 NATURE OF ANCHORAGE ZONE STRESSES 9
2.2 THEORETICAL INVESTIGATIONS 13
2.2.1 Morsch's solution 13
2.2.2 Magne1's solution 15
2.2.3 Bortsch's solution 17
2.2.4 Guyon's solution 17
2.2.5 Sievers' solution 19
2.2.6 Iyengar's solution' 19
2.2.7 Som and Ghosh's solution 21
2.2.8 Ramaswamy and Goel's solution 23
2.2.9 Gergely and Sozen's solution 23
2.2.10 Lensdhow and Sozen's solution 24
2.2.11 Yettram and Robbins' solution 24
2.3 EXPERIMENTAL INVESTIGATIONS 26
2.4 DESIGN OF BEAM ANCHORAGE ZONES 32
2.5 DESIGN OF BUTTRESS ANCHORAGE ZONES 34
i
CHAPTER
3
4
DESIGN AND FABRICATION OF TEST SPECIMENS
3.1 BUTTRESS DESIGN
3.1.1 Design loads
3.1.2 Bearing stresses
3.1.3 Anchorage zone reinforcement
3.2 BLOCK MODEL DESIGN
3.2.1 Madel detai1s
3.2.2 Choice of sca1es
3.2.3 Specimen designations
3.3 BUTTRESS MODEL DESIGN
3.3.1 General features
3.3.2 Reinforcement
3.3.3 Post-tensioning system
3.4 MODEL FABRICATION
3.4.1 Mate ri aIs
3.4.2 Instrumentation
3.4.3 B10ck mode1 fabrication
3.4.4 Buttress mode1 fabrication
BLOCK MODEL TESTS
4.1 TEST PROCEDURE
4.2 EXPERIMENTAL DATA
4.3 GENERAL BEHAVIOR
4.4 DISCUSSION OF ULTIMATE LOADS
4.4.1 U1timate loads of unreinforced b10cks
4.4.2 U1timate loads of reinforced b10cks
ii
Page
36
36
36
37
41
49
49
52
54
56
56
58
64
67
67
72
73
75
76
76
76
85
88
88
89
CHAPTER Page
4 (continued)
5
4.5 DISCUSSION OF STRAIN DATA 92
4.5.1 General observations on steel strain data 92
4.5.2 Strains under maximum initial prestressing 93 force
4.5.3 Attainment of maximum a11owab1e stress or 95 yie1d point in one bar
4.5.4 Total force in transverse rein forcement 100
4.6 DISCUSSION OF' END B~OCK DESIGN 113
4.7 DISCUSSION OF SIMILITUDE 115
BUTTRESS MODEL TEST 117
5.1 TEST PROCEDURE, INSTRUMENTATION AND TEST DATA 117
5.2 GENERAL BEHAVIOR 122
5.3 DISCUSSION OF STRAIN DATA 124
5.3.1 General observations 1~4
5.3.2 Effect of middle plane tendons 124
5.3.3 Effect of sustained loading and thermal 126 gradient
• 5.3.4 Effect of direct loading on anchorage zone
127
5.3.5 Effect of loading on adjacent anchorage 131 zones
5.3.6 Effect of loading on opposite side of buttress
5.4 CORRELATION BETWEEN BLOCK MODEL AND BUTTRESS MODEL TESTS
5.5 STRESS DISTRIBUTION IN BUTTRESS
5.6 SUGGESTIONS FOR FUTURE RESEARCH
133
135
136
138
5.6.1 Study of vertical anchorage rein forcement 138
5.6.2 Further study of horizontal anchorage 141 re in forcement
5.6.3 Study of unreinforced anchorage zones 142
5.6.4 Finite e1ement ana1ysi~ 143
iii
CHAPTER Page
6 CONCLUSIONS 146
LIST OF REFEREijCES 149
iv
FIGURE
1.1
1.2
1.3
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
LIST OF FIGURES
Prestressed concrete containment
Buttress
Experimental specimens
Stresses under concentrated bearing'load
Factors influencing the stress distribution under bearing load
Morsch's solution
Magnel's solution
Bortsch's solution
Guyon's solution
Sievers' solution
Som and Ghosh's solution
Lenschow and Sozen's solution
Tensile stress distribution and'total tensile force
Buttress design details
Anchorage zone details
Block specimen details
Block models - comparative dimensions
Buttress model - isometric view
Buttress model - thermal system
Buttress model - plan
Buttress model - elevation
Post-tensioning system
Micro-concrete load-deformation characteristics
Reinforcement load-deformation characteristics
v
Page
2 ,
3 , i, 1
i
7
10
12
14
16
18
18
20
22
25
33
38
42,43
50
55
57
59
60
61
65,66
70
71
'FIGURE
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
B10ck mode1 fai1ure mechanism
U1timate strength graph
Strain distribution (a) Wide face , (b) Narrow face
Total force in transverse steel B10cks 1-160, I-160N, 11-160,
1-375, I-375N, 11-375
Fai1ure mechanism
Total force in transverse steel, a11 mode1s (a) Wide face (b) Narrow face
Strain gage locations in buttress mode1
Superficia1 damage in buttress mode1
vi
Page
86
91
98,99 '
101-106
109
111,112
119
123
TABLE
3.1
3.2
3.3
3.4
3.5
3.6
3.7
4.1
4.2
4.3
4.4
4.5
4.6
4.7
5.1
5.2
5.3
5.4
5.5
LIST OF TABLES
Bearing stress calculations
Amount of reinforcing steel per tendon anchorage zone for various end details
End details in order of increasing tensile capacity
End details in order of increasing steel weight
Required lateral reinforcement capacity
Buttress model dimensions and rein forcement
Micro-concrete properties
Block model ultimate strengths
Transverse'steel strains Blocks 1-160, I-160N, 1I-160~
1-375, 1-375N, 11-375, 1-1, 11-1
Estimated ultimate bearing capacities of end blocks
Transverse steel strains at maximum initial prestressing force
AttaiIlment of critical strain level in any bar
Total force in transverse steel B10cks 1-160, I-160N, 11-160,
1-375, 1-375N, 11-375
Total tensi1e force, theoretica1 predictions
Strain readings, buttress mode1 (a) Gages in anchorage zones lB and le (b) Gages in anchorage zones 2A and 2e
Effect of midd1e plane tendons, sustained loading, and thermal gradient
Effect of prestressing force app1ied direct1y on anchorage zone
Effect of prestressing force on adjacent anchorage zones
Effect of prestressing forces on opposite side of buttress
vii
Page
40
45
47
47
48
62
68
78
79-84
91
94
97
101-106
109
120 121
125
128
128
134
NOTATION
Symbols used in this report were chosen to conform as
closely as possible'to the proposed ACI standard: "Preparation of
Notation for concrete~(46) Stresses'were dasignated f or f c s
according to whether they occurred in concrete or steel, and
assigned subscripts x or z to indicate coordinate directions, x
being in the direction of the prestressing force and z in a trans-
verse orthogonal direction. The total splitting force on a trans-
verse plane was denoted Z, and that component ascribed to steel was
subscripted s. The total force on the bearing plate was expressed
as P by analogy with prestress-ing force. Use of the subscript p
for the bearing plate was found unambiguous and in agreement with
its intended sense of indicating prestressing in general.
A c
A cp
A p
b
b c
b cp
b p
= area of con crete under uniform compressive stress
= net area of the portion of the concrete anchorage zone that is geometrically similar to and concentric with the area of the anchorage plate
= net area of anchorage plate
= transverse width of con crete anchorage zone and bearing plate
= width of concrete anchorage zone
= width of con crete anchorage zone associated with area A cp
= width of anchorage plate
viii
) f. = compressive strength of concrete at time of initial prestress (*)
C1
f = P/b b cm c = mean bearing pressure on net area of concrete
anchorage zone
f = PIA cp . P = allowable bearing pressure on net area of
bearing·plate
= splitting tensile strength of concrete (*)
f = cu specified compressive strength of concrete or strength of concrete at time of testing
(*)
f = shear stress in a~chorage zone con crete cv
f = compressive stress in concrete along prestressing force direction cx
f = . transverse stress in anchorage zone concrete cz
f = calculated stress in prestressing steel at design load (*) sp
f = ultimate strength of prestressing steel (*) su
f = stress in lateral reinforcing bars . sz
f. = yield strength of non-prestressed reinforcement (*) y
h = length of anchorage zone
P - load on anchorage plate
= prestressing force at design working stress f . = 0.6 f sp su
= prestressing force in initial jacking operations
P su
(stress 0.8 f assumed) su
ultimate capacity of prestressing steel =
P = ultimate capacity of anchorage zone u
x = coordinate along prestressing force
A sp
z = coordinate paraI leI to face of anchorage plate
z = total splitting force
z = total force in lateral rein forcement s
f su
(*) Denotes symbols and definitions extracted from Ref. (46).
ix
UNITS OF MEASUREMENT
In keeping with the present trend for standardization
of units of measure through the use of the modernized metric system,
all important -dimensional quantities were expressed in terms of the
International System of units (SI, Système Inte~ational), as well
as the customary British units currently in use in the United States
and Canada.
The system and its usage are well documented in the ASTM
publication: "Metric Practice Guide (A Guide to the Use of the
International System of units)~(4~) The basic quantities length,
mass and time are assigned the units metre (m), kilogram (kg) and
second (s), respectively. Angles are expressed in radians (rad),
while tempe ratures may be written either in Celsius (C) units,
identical to the current "centigrade degrees", or Kelvin (K) units
for thermodynamic measurements. All other quantities are expressed
in terms of units derived from the above. The unit of force is the
newton (N), defined through Newton's law as the force required to
2 give the unit mass a unit acceleration, viz.: 1.0 N = 1.0 kg.m/s •
The unit of pressure or stress is the pascal (pa), equal to the unit
2 force per unit area, 1.0 Pa = 1.0 N/m • Other units ?sed herein
3 are self-explanatory, e.g. kg/m, unit of density.
Multiples and submultiples of units are formed by pre-
fixing one of the following letters onto the symbol (only those used
hereafter are listed):
x
giga (G) (10)9
mega (M) (10)6
kilo (k) (10) 3
centi (c) (10)-2
mi11i (m) (10) -3
micro (11) (10)-6
Thus, MN represents a meganewton, or one million newtons, and mm
is the fami1iar mi11imeter. Conversion constants to 3 significant
digits for the principal Imperial units used in the foregoing are as
fo11ows:
1 inch = 25.4 mm = 2.54 cm, 1ength
1 foot
1 . 2 1n
= 0.305 m, 1ength
2 2 = 645 mm = 6.45 cm , area
1 pound = 0.454 kg, mass
1 pound = 4.45 N, force
1 kip = 1000 pounds = 4.45 kN, force
1000 kips = 4.45 MN, force
1 psi = 1 pound/in2 = 0.690 kPa, stress or pressure
1 ksi = 1 kip/in2 = 1000 psi = 0.690 MPa, stress
1000 ksi = 0.690 GPa, s~ress or e1astic modu1us
1 degree F = 0.555 C = 0.555 K, temperature difference
32 F = 0 C = 273.15 K, standard freezing point tempereture of water
1 degree = 0.01745 rad., pl anar angle
1 circumference = 360 degrees = 2 1T rad, p1anar angle
xi
AlI measurements and calculations were perforrned in the
British system, and conversion to SI units was done subsequently.
In tables or calculations, SI conversions are not included when re-
sults are of an intermediate nature. Conversions were conducted
as suggested by ASTM, preserving the original degree of precision
of the original measurement. Thus dimensions precise to the
nearest 1/16-inch were converted to the nearest integer mm, while
those to the nearest kip were converted to the closest multiple of
5 kN. Non-dimensional ratios were used wherever possible.
Strains were expressed as pure numbers accompanied by the symbol ~,
a non-dimensional unit introduced to replace the (10)-6 representation.
AlI dimensions on structural drawings are in inches,
except where explicitly stated otherwise, ànd are not converted to
SI units to avoid conflicts in the precisio~ of duplicate measurements.
xii
CHAPTER 1
. INTRODUCTION
The containment structure of a nuclear réactor is the
last barrier that shields the environment from radioactive matter.
Unlike the reactor itself, where the nuclear reactions take place,
the containment is, under normal conditions, subjected to only small
intensities of radiation. Exceptional accident conditions may,
however, arise, for example through the rupture of a pipe in the
power generatin~ system, where the stru~ture might be partially
flooded or filled with high-temperature steam. Under such circum-
stances a crack-free containment structure would be most desirable.
In con crete construction, a viable means of preventing
tensile cracking 1s the introduction of prestressing forces. A
circular vessel must be prestressed in the longitudinal and circum
fe~ential directions; its domed roof might be prestressed in the
pattern illustrated in Figure 1.1. The prestressing force is ob-
tained by post-tensioning large cables with effective working
strengths of the order of 1000 kips (4.45 MN) or larger. These
cables, once tensioned, are anchored against bearing plates resting
on, or embedded into, the ring beam and the protruding buttresses
shaped in the slip-forming operation. (Fig. 1.2).
In the region behind a bearing plate, locally concentrated
pressures diffuse, according to Saint-Venant's principle, into a more
uniform distribution of pressures. The transition from a discon-
tinuous stress distribution to a more uniform one induces shearing
1
2
PRESTRESSED CONCRETE CONTAINlVlENT
2 4
.... 3 .. -- ,A -- , \...... -- ... _-
......... ........... ------ ----_.......... ,'"
...... ------ ... _---...... -... -..... ------
...-" ___ ----... ---.... ,..tII ...------_ .........
Figùre 1.1
1 RING GIRDER 2 BUTTRESS 3 HOOP TE'NDONS 4· LONGlïUDINAL TENDONS 5 DOME TENDONS
1
1
1
1
1
3 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
4
and transverse tensile stresses in the transition region, known as
the lead-in zone or the anchorage zone •
. The determination of the behaviour of anchorage zone
concrete, and the requirements for satisfactory detailing of lateral
reinforcement to resist the tendency for cracking and spalling, are
subjects whicn are often left to engineering intuition and judgment.
The Building Codes of the Prestressed Concrete Institute(l) and the
American Concrete Institute (Proposed ACI 318-71 (45) and the current
ACI 318-63(2» recommend the rein forcing of anchorage zones to re-
sist bursting and spalling forces under the maximum jacking force,
but offer no rules for detailing the amount and the location of the
steel. As of 1960, according to Zielinski and Rowe(3), some codes
, in Europe either made no mention of anchorage zone design, or followed
a line similar to that of the ACI; others, such as the German code,
were based on old experimental data or formulae of questionable
relevance. The Canadian Standards Association Code CSA A135-l962(8)
specifies the following methods to determine the required reinforcement:
"3.8.3 Reinforcement of Anchorage Zones.
3.8.3.1 Anchorage zones shall be reinforced to resist tensile bursting and spalling forces introduced by the concentrated loads due to the action of prestressing.
3.8.3.2 The rein forcement of anchorage zones shall be determined by either of the following methods: (a) Computation based on recognized elastic or ultimate theories, in which case the working load shall be 'the total effective prestressing force, and the ultimate design load shall be twice this value; or (b) Testing, in which case the anchorage zone shall be capable of resisting the total ultimate strength of the tendons."
5
While recognizing the val~e of the experimental method, the Code
states no precise standard of performance in the way of service
ability or load factor to qualify the phrase "capable of resisting
the total ultimate strength of the tendons".
Theoretical analyses of the anchorage problem, whether
tiley be closed-form classical solutions of elasticity equations, or
numerical solutions of these equations by finite difference or finite
element methods, have the disadvantage of using a mathematical model
based on a linearly elastic, homogeneous, isotropie material forming
a two-dimensional body. AlI of these assumptions pose serious
limitations on an analysis of the bearing problem in plain concrete
alone. Although some iterative methods can bypass the inelasticity
problem, and the material may be taken to be homogeneous and isotropie
until the beginning of crack formation" the lack of information on
the load-deformation response and failure criteria for concrete under
triaxial stresses reduces the accuracy and makes it difficult to
recognize failure conditions when they arise; furthermore, the three-
d~mensional nature of the problem cannot be ignored. With the
presence of rein forcement and anchoring devices, the transfer of force
between concrete and steel through bond and dowel action introduces a
degree of complication in the problem which rules out aIl exact methods
of solution, save the finite element method, still pending the avail
ability of data on the triaxial behaviour of concrete and steel-concrete
interfacial effects.
Indirect modeling techniques attempt to improve the
accuracy of solution by removing one or more of the constraints inherent
6
\
in the mathematical analysis. Photoelasticity has been used in
the context of anchorage zone stress distributions, usually as a
verification of a proposed theoretical solution. Through the use
of frozen-stress techniques, it was possible to obtain a three-
dimensional analysis of the anchorage zone stresses, although this
method also suffers from several of the limitations of the classical
theory of elasticity.
Direct modeling of the actual structure or of the relevant
compone nt reproduces all material properties and loading conditions
and therefore gives a satisfactory assessment of behavior from
which the engineer can evaluate the safety and economy of a chosen
design. The prohibitive cost of the method can be considerably re-
duced by using small-scale models constructed from materials whièh
faithfully simulate the properties of the materials used in the
prototype structure.
The purpose of this investigation was to examine a parti-
cular anchorage zone detail in a prestressed concrete containment
structure. Two lateral steel designs, evolved from engineering
judgrnent, were analyzed according to available formulae, and subse-
quently tested in the laboratory in simple small-scale models
(Fig. 1.3a), referred to as "block models". Ultimate and cracking
loads, and strains in the steel and concrete at incremental load
levels, were recorded.
In order to appraise the similitude of the models, tests
were performed on models of approximately one-third and one-sixth
scales, and compared with the data available for concrete blocks
comprising full-size anchorage hardware.
7
(a) Block model
(b) Buttr ess model
er1mental 1.3 Exp . specimens
· 8
A more complex.one-sixth scale model was also fabricated
incorporating two buttresses and sixteen prestressing tendons,
(Fig. 1. 3b), referred to as the "buttress model". Many features
of the full-size structure were reproduced in this model, either
inherently from its form, or intentionally through special design,
viz.: continuity of concrete mass; presence of re-entrant corners;
self-weight of structure; continuous and distinct bearing plates;
tensioning sequence; thermal gradient from cernent hydration or from
accident conditions. The buttress model could thus give an indica-
tion of the correlation between the behavior observed in a simple
block test and that in the actual structure.
From the results, observations were made on the proposed
reinforcement details, on available design rules and sources of data,
on the similitude between models of different scales, and on the
effect of simplification of the model, in the design and analysis of
anchorage zones for post-tensioned cylindrical containment construction.
CHAPTER 2
SURVEY OF PREVIOUS WORK
2.1 NATURE OF ANCHORAGE ZONE STRESSES
Compressive stress trajectories in the region under a
locally concentrated bearing pressure follow curved paths, approach
ing a uniforrn distribution at sorne distance from the bearing surface
(Fig. 2.la). The curvature of these trajectories results in a tri-
axial stress state wherein a principal tensile component of stress
acts orthogonally to the paths of principal compression.
Working stress design xequires sufficient reinforcing
steel to develop the full tensile force in the concrete. The arnount
and arrangement of the steel is dependent on the distribution of the
tensile stresses and the magnitude of the resultant tensile force.
The distribution of tensile stresses, as deterrnined from theoretical
and photoelastic investigations to be detailed later, is shown in
Figs. 2.lb to 2.ld. Surface measurements of transverse strains,
on a prismatic test specimen under concentric loading, corroborate
these distributions (Fig. 2.le).
As the magnitude of the tensile stresses increases with
the curvature of the compression trajectories, a reduction of the
ratio of the bearing plate width to the underlying con crete width,
results in a steeper gradient of compressive stresses, and hence in
higher tensile stresses. Inclination or eccentricity of the line
of action of the bearing force, the presence of simultaneously applied
9
10
(a) ApproximatC' compression trajectories and associatcd transverse forces
(27)' -[From Leonhardt .. J
(c) Lines of equal tranverse stress (isobars); compression zone cross-hatched
[From Leonhardt(27) after Tesar(22)
and Guyon (18) J
(b) Principal stress trajectories from photoelastic investigation
[From Guyon (~8) after Tesar (22) J
(d) Transverse stress distributiol along central axis
[From Leonhard(27) J
(e) Transverse strains in prismnti beam end-blocks
[From Zielinski & Rowe(3) 1
Figure 2.1
Stresses under conccntratcd bearing load.
11
adjacent bearing forces, or a reduction of the area over which
compressive stresses can distribute can have a beneficial effect
on the reduction of transverse tensile stresses (Fig. 2.2).
Thé early studies of anchorage zone problems were devoted
to establishing the stress distribution in concrete. Basic equations
of statics were used to obtain a first approximation. Simplifying
assumptions were made to account for the general form of the stress
distribution. The mathematical theory of elasticity was also
employed to derive the stress field. Experiments using indirect
photoelastic models, were conducted to confirm and supplement the
theoretical background.
Due to the simplifying assumptions introduced in both the
statics and the elasticity approaches, discrepancies were noted
among the several solutions in their respective predictions of the
location and magnitude of principal tensile stresses. However, the
resultant tensile force values calculated using different theories
tended to agree. From these consideratione, the practical rule of
ignoring the tensile strength of concrete, and reinforcing for the
total lateral tensile force, was found to be appropriate and not
excessively conservative.
The recent investigations of anchorage zone behaviour have
been aimed at evaluating experimentally the overall behavior and
bearing capacity of the end zones of post-tensioned prestressed con-
crete beams. These tests emphasized practical design considerations,
such as the detailing of the lateral reinforcing steel.
12
.... ~ =·,n".fH • ~ cl =/,',.,/.,
a '" cS . ~
c:I
. (a) Effect of bearing plate width [From Leonhardt(27)]
c
(b) Effect of adjacent bearing forces [From Leonhardt(27)]
(c), Effect of inclination
, (27) [From Leonhardt after
Sargious (23) ]
Figl\re 2.2 Factors influcncing the stress distribution under bearing load
13
The design of buttress detai1s in prestressed concrete
nuc1ear containments has,up to the present, been conducted fo11owing
the same lines as the design of beam end zone detai1s. On1y a
1imited amount of research has been done to assess the re1evance of
the availab1e design methods for containment structures.
A brief review of the principal inves'tigations of the
anchorage problem in post-tensioned construction is undertaken in the
fo11owing sections. A survey of the literature on anchorage zone
stresses published between 1923 and 1964 is available from the
Prestressed Concrete Institute(lO). A summary of theoretical re-
search prior to 1960 can be found in the papers by Zie1inski and
(3) (9) Rowe and Iyengar • A review of papers in foreign languages is
contained in the bibliography on prestressed nuclear vessels pub
li shed by the Oak Ridge National Laboratory(ll) •
2.2 THEORETICAL INVESTIGATIONS
2.2.1 Marsch's solution(12) (1924) (Fig. 2.3)
Compressive stress trajectories were assumed to follow a
parabolic law, based on measurements of tensi1e strains at three
positions on a test block. Assuming a uniform pressure distribution
over the bearing plate, and assuming furthermore that,in accordance
with Saint-Venant's princip1e, the pressure was uniform over the plane
at a distance h = b beyond the loaded face, equilibrium between the c
14
t:l 0 'r! 01..1 :s r-I 0 li)
li)
,d N • tJ
li)
J.I :~
ct')
N
15
resultant tensile force Z and the bearing force was expressed in
the equation:
Z. (h/2) = (P/2) (b - b )/4 P c
The ratio of lateral force to bearing load was found to be:
Z/P = 0.25(1-b /b ) . p c
and the maximum tensile stress in a rectangular prism of width b was
given by f = 1.5 Z/b b. cz c
2.2.2 Magnel's solution(13,14) (1949) (Fig.2.'Ù
The tensile stress distribution due to the bending action
of the prestressing force on planes parallel to the central axis, was
assumed to be a cubic function,
f cz
in which the coefficients were derived from boundary conditions.
Tensile stresses and shear stresses were then expressed, respectively,
as
f = K M/b 2b cz z c and f = K V/b b cv v c
where M and V are the bending moment and shear force on the plane
considered, and the parameters K and K are third and fourth degree z v
functions of x. Principal stresses could be calculated knowing f cz
and f , assuming that the normal stress f disperses at an angle of cv cx
45 degrees (TI/4 radians). Points of zero and maximum tensile stresses
occurred at x = 0.25 b and x = 0.5 b respectively. p p
16
z
A -------------
h
(a) ·Eccentric prestressing force on beam end-zone
1 1 1 1 1 1 1 1 1 1 1
fil Il ~
X 811. Il
(b) Stress distribution on transverse section
X ... h 2'
(c) Stress and shear coefficients:
z
g o
i f = K M/b2b cz z c
Figure 2.4 Magnel's solution
a
17
A modified form of the theory uti1ized a second degree
expansion for f , and rep1aced the previous assumption on the cz
distribution of f by a fourth degree polynomial in x. cx
2.2.3 . (15 16) Bortsch's solut~on ' (1935) (Fig. 2.5)
C1assica1 e1asticity equations were solved for the case
of a cosine distribution of bearing pressure on ~ semi-infinite sur-
face. Infinite series expansions were derived to express the
longitudinal, transverse and shearing stresses. Because of the
semi-infinite boundary condition, resu1ts are applicable on1y in the
range b lb = 0 to 0.2. P c
Maximum tensi1e stresses varied from 0.45 f to 0.38 f
in the range b lb = 0.1 to 0.2, and occurred at x p c
(f Pib b, mean bearing pressure in end block). cm c
cm cm
0.2 b to 0.3 b P P
2.2.4 Guyon's solution(17,18) (1951) (Fig. 2.6)
Guyon's approach was simi1ar to that of Bor.tsch, and made
use of Fourier series to eva1uate fcx' fcz' fcv due to knife-edge
loading on a semi-infinite strip. The stresses were tabu1ated(18)
for various locations of interest (nine y coordinates from -b to +b , P P
eight x coordinates from 0 to 2b) and severa1 different points of p
load application.
For groups of forces or eccentric forces, a simple approxi-
mate solution known as the "symmetric prisms method" was suggested.
For a single eccentric prestressing force, or a group of forces pro-
ducing a linear distribution of stresses, (Fig. 2.6b), subdivision
Figure 2.5
1 1 1 1 1 1 1
----T--.J 1 1 1 1 __ ,-J
1
Figure 2. 6(b)
18
---.1~-_._. __ . __ ._.-._._.-.-
Eortsch's solution -Cosine distribution on semi-infinite plate
Figure 2.6(a)
1--------.., 1 1 1 1 f 1 1
Guyon's solution -Knife-edgeloading on semi-infinite strip
Guyon's solution - Method of symmetric prisms for several loads or one eccentric load producing a linear prestress distribution
---~---,--------._-~ l, ' l, 1 1 1 1
••• ..1 .. --. 1 1 .. 1 1 1 1 ----____ 1 ,
Figure 2.6(c) . Guyon's solution -Method of successive resultants for nonlinear prestress distribution
1 :-----.. 1 1 1 , 1 , , 1 ,
Figures 2.5 and 2.6
19
of the anchorage zone into symmetric bearing areas is emp10yed,
and tensile stresses are eva1uated at the line of action of each
force. When the forces do not result in a uniform distribution of
stresses, the unbalancedforces create additional tensile stresses.
Bursting stresses must then also be eva1uated along the line of
action of the resultants of the various pairs of forces (Fig. 2.6c).
2.2.5 Sievers' solution (19,20) (1952) (Fig. 2.7)
Sievers app1ied Bleich's ana1ysis of deep bearns(21) to the
prob1em of symmetrically placed bearing loads on a finite rectangular
anchorage zone. stresses were first expressed as infinite series
to allow the use of an Airy stress function in the solution of the
biharrnonic elasticity equation, and were later written as approximate
expressions in which f and f were exponentially decaying functions cz cv
of x.
2.2.6 Iyèngar's solution(9) (1960)
The general equations for a serni-infinite strip were
formulated for the four cases of normal and tangential loading, sym-
metrical and antisymmetrical about the central axis of the bearing
block. By superposition of any number of the four basic cases,
solutions for eccentric and inclined tendons could be generated.
The approach to the problem and the general form of results were
similar to those of Guyon.
In comparing his results to those of the previous theories,
Iyengar discarded the methods of Morsch and Magnel as being too
20
Figure 2.7
, l-,
Sievers' solution -Use of deep beam analysis in anchorage zone problem
21
approximate and not satisfying e1asticity. Guyon's theory agreed
c10sely with that of Iyengar in its evaluation of tensile stresses,
h 'l l' h' l' (21) 'abl 'd th '1 w ~ e B e~c s so ut~on apprec~ y overest~mate ese tens~ e
stresses. Saint-Venant's principle was found to be satisfactory in
estimating the distance over which significant stress gradients occur.
2.2.7 Som and Ghosh's solution(36) (1964) (Fig. 2.8)
The anchorage zone prob1em was treated as a two-dimensiona1
boundary value problem. Known stress conditions at the bearing sur-
face and at the terminal section of the anchorage zone were represented by
Fourier series and imposed as boundary conditions in the solution of
the biharmonic equations. The three loading conditions shown in
Fig. 2.8 were investigated.
The length of the anchorage zone was determined analytically
(Fig. 2.8b). In the case of axial symmetric prestressing by multiple
cables (case II), the length of the anchorage zone was found equal to
the width of the block.
The maximum tensile transverse stress in aIl three cases
was observed te be lower than that predicted by Guyon. In case 1
(single concentric prestressing force) the difference was small, whereas
in cases II and III, considerable discrepancies were noted. Magnel's
solution fell between Guyon's and Som and Ghosh's in cases II and III,
but differed largely in case 1.
(36) In a discussion of the paper, Iyengar and Chandrashekhara
questioned the validity of certain boundary conditions, assumed or
implied, in the method of Som and Ghosh. It was also pointed out
22
.----._--
'I TI ID
(a) Loading conditions
.----_.----.----.----.----.---;--
(b) Analytical determination of anchorage zone length
Figure 2.8 Som and Ghosh's solution
1 }
23
that an approximate, conservative solution, rather than Gllyon's
original solution, had been used in the comparisons which showed
Guyon's method to overestimate the lateral stress.
Do~ge's discussion(36) focussed on the application of
statics to study the variation of the anchorage zone length with.
the position of the load, to determine the transverse stresses, and
to verify the validity of a stress distribution .obtained by any
method.
2.2.8 Ramaswamy and Goel's solution(28) (1957)
Ramaswamy and Goel made use of McHenry's "lattice analogy"
method(29) to obtain a numerica1 solution of the stress distribution
in a beam end b10ck of fini te proportions under the action of a
centra11y applied 1ine load. A comparison with Guyon's resu1ts
indicated that the bursting zone was 1arger in extent than predicted
by Guyon, and the tensile stresses in the spal1ing zone were sma11er.
A1so, the maximum tensi1e stress in the bursting zone was found to
have a value 0.60 f compared with the value of 0.50 f in Guyon's cm cm
analysis (f = P/b b, mean pressure in end b1ock) • cm c Use of the
method was suggested for design practice.
2.2.9 Gergely and Sozen's solution(38) (1967)
Because of the many uncertainties invo1ved in the prediction
of crack initiation and propagation, an approach aimed at finding the
effect of lateral rein forcement on the limitation of cracking was pre-
ferred to the e1asticity approach, which had been attempted numerical1y
24
by the finite difference method in an earlier investigation by
Gergely, Sozen and Siess(39) •
A simple design procedure for lateral steel was derived
on the basis of an equilibrium analysis of a cracked anchorage zone.
Experimental results confirmed the prediction of the
crack location and of the force in .the reinforcement, and were used
in the derivation of a force-slip re1ationship from which design
considerations to limit crack widths resulted.
2.2.10 Lenschow and Sozen's solution(37) (1965) (Fig. 2.9)
A physical analog was conceived in which the anchorage zone
had a fictitious discontinuity along the plane of potentia1 cracking,
which was bridged by e1astic elements representing the tensile restraint
of the uncracked concrete. Predicted stress distributions across
the boundary compared favorably with those of the classica1 solutions.
A1ternatively, by considering the 1ateral restraint to be
provided by reinforcement, a similar analog was used to determine the
effect of latera1 rein forcement on crack widths. Results were con-
firmed experimental1y.
Formulas were derived from consideration of the ana1og, and
their practical design was demonstrated in two numerica1 examples.
2.2.11 Yettram and Robbins' solution(54,55,56) (1969-1971)
Anchorage zone stresses were determined by using a finite
e1ement procedure for the e1astic analysis of three-dimensional solids.
. 1 . d mb . f . (54) Ax~a 1y post-tens~one me ers of un~ orm rectangu1ar cross-sect~on ,
CRACK WIDTH
25
(a) Bearn end-zone
(b) Fictitious discontinuity along planes of potential crackin~
CRACK j . __ . __ I._L_EN_~~H __ '_i-
FORCE IN REINFORCEHENT
(c) Modified ana log accounting for contribution of reinforcement to crack restraint
Figure 2.9 Lenschow and Sozen 1 s solution
26
uniform rectangular and I-type sections with eccentric and multiple
anchorages(55), and l sections with end blocks(56) , were investigated.
The three-dimensional analysis of uniform sections under
concentric loading revealed significant transverse variations in the
stress distribution and showed a marked difference between centroidal
and surface stresses. The two-dimensional solutions of Iyengar and
Guyon gave good agreement with the values of the average stresses,
although they did not recognize the transverse variation, and could
occasionally be misleading.
For groups of anchorages, Guyon's equivalent-prism method
was found to be satisfactory. For flanged sections, Leonhardt's
evaluation of flange bursting stresses proved grossly conservative,
while the spalling stresses were considerably reduced due to the
presence of the flanges; end blocks for these sections were given
detailed consideration.
2.3 EXPERIMENTAL INVESTIGATIONS
Morsch supported his analytical findings with tests carried
out mainly on stone blocks. In the few tests on plain and reinforced
concrete specimens, transverse reinforcement was found to have no
visible influence on the cracking and ultimate load of the blocks. The
use of higher concrete strength was considered preferable to lateral
reinforcement.
Hagnel's two block tests showed that his method under
estimated both the magnitude of tensile stresses and the distance from
27
the face of the plate to the point of maximum stress.
Guyon's theory agreed ~ith the results of a photoelastic
study by Tesar(22) for the case b lb = 0.1. P c
Points of maximum
and zero tensile stress were correctly located, and the experimental
maximum f = 0.45 f was noted to be close to Guyon's value of cz cm
0.42 f • cm
Photoelastic test results were used by Bortsch and Sievers
to qualitatively verify their respective solutions.
sargious(23) applied the photoelastic method for the
determination of stresses and tensile forces in the anchorage zone of
prestressed beams, under the action of an eccentric inclined pre-
stressing force, in the vicinity of the support reaction.
Christodoulides(24,25) made use of two-dimensional photo-
elasticity, three-dimensional frozen-stress techniques, and fullscale
concrete blocks with embedded strain gauges, to model a beam end-zone
subjected to two symmetrical prestressing forces. The photoelastic
stress measurements were confirmed by the internaI strain measurements.
Results agreed fairly closely with Iyengar's theoretical solutions,
but differed markedly from those of previous investigations. The
theories of Magnel and Guyon considerably underestimated both the
tensile stresses and the resultant tensile force. The critical stress
values were as follows:
maximum shear stress, f = 2 f • cv cm'
maximum principal compressive stress, fcx = 4 fcmi
maximum principal tensile stress, f = 0.6 f • cz cm
28
Spieth(30) deterrnined the pressure distribution in con-
crete subject to high local pressure, from which a formula for
allowable bearing stress was derived.
Dowrick(3l) presented a method for rein forcing prestressed
concrete anchorage zones, 'based on experimental work. Design curves
were included to simplify design procedures. The effects of
variations in concrete strength and anchorage size were discussed.
Okada et al. (32) analyzed the stress concentration in a
notched anchorage zone, using photoelastic methods, concrete model
tests, and measurements on a prototype.
Gardner (33) experimented with concrete cylinders pre-
stressed with high-strength steel spirals, subjected to concentric
and eccentric loadings. By varying the percentage of spiral re-
inforcement, the initial spiral prestress and its eccentticity, it
was shown that the initial prestress altered the initial stiffness
of the cylinder, but had no significant effect on the ultimate load
capacity, and that the change in lateral restraint produced by chang-
ing the percent age of spiral reinforcement allowed the concrete to
develop its strength to different degrees in a gradual type of com-
pression or tension failure. Modifications of the AC! ultimate load
formulas were proposed to include the contribution of the spiral rein-
forcement.
. (34) f d' . d' d 1 E~mer et al. per orme exper~ments on ~n ~rect mo e s
to determine stress distributions in reinforced anchorage zones under
concentric and eccentric loads. Photoelastic models were used,and
the influence of the reinforcement on the state of stress was investi-
gated by reinforcing the models with brass rods.
29
.~
'd k' d Wh'th d(35) 1 ' Ri zews ~ an ~ rea proposed an ana yt~cal
method for evaluating the tensile stresses in I-beam anchorage
zones, and a simplified design procedure based on the results.
Tests on mortar models of scale 1/12 and three-dimensional photo-
elastic models confirmed their analysis.
. (53) d' d h d' 'b' 'th Huang stu ~e t e stress ~str~ ut~on ~n e end-
blocks of a prototype I-beam, using internal and external strain
gages. A numerical two-dimensional investigation of the problem
was also performed. The paper emphasized stress concentration at
the junction of the end-block with the beam section. Within the
end-block, it was found that Magnel's method considerably under-
estimated the principal tensile stress, while vertical tensile
stresses obtained by Guyont's method were higher than observed values.
Ban et al. (26) studied the distribution of anchorage zone
tensile stresses and cracking and ultimate loads in a specifie post-
tensioning system. The influence of the following factors was
investigated:
(1) Ratio of bearing plate to bearing block dimensions, b lb ; p c
(2) Bearing plate thickness;
(3) Anchorage nut dimensions;
(4) Concrete strength;
(5) Percentage ana location of lateral reinforcement •.
Test results agreed with sievers' solution, but deviated considerably
from those of Guyon's and Magnel's analyses. The cracking load
remained approximately constant regardless of the bearing plate area,
but varied with the bearing plate thickness and the size of the
30
anchorage nut. Cracking and ultimate loads exhibited linear
relationship to the concrete strength and tothe anchorage plate
thickness, and were greatly influenced by the amount of lateral re-
inforcement.
Zielinski and Rowe(3 to 7) conducted an extensive study
of all types of anchorages used in British post-tensioned concrete
practice in 1960. In the first part of the experimental program,
the test blocks had a square cross-section and were loaded symmetri-
cally. The parameters investigated were as follows:
(1) Post-tensioning system, including the transfer of force through
cones or plates, both external and embedded, and resting on
circular or square bearing plates;
(2) Ratio of bearing plate to bearing block dimensions, b lb ; p c
(3) Tendon duct form;
(4) Amount, position and form of reinforcement, including the
mat and helix forms.
The principal conclusions reached were as follows:
(1) "The distribution of transverse stress and the ultimate load
of an end block are not significantly affected by the anchorage
being embedded or internal, by the material of the anchorage,
or by the method of anchoring the wires".
(2) The b lb ratio was the dominant factor in the distribution of p c
transverse stress and ultimate load, but it had little effect
on the position of the points of maximum and zero transverse
stress.
31
(3) All existing theories underestimated the tensile stresses and
the total tensile force. The experimental results of
Christodoulides and the analytical results of Sievers gave the
closest agreement, while the solutions of Magnel and Guyon
underestimated these quantities by a factor of 2 or more.
(4) The following tensile forces, maximum tensile stresses, and
ratios ~f uniform compressive stress at cracking to concrete
strength were observed:
at b lb = 0.30: f = 0.73 f cm' (f) = 0.16 f cu' Z/p = p c cz cm max
at b lb = 0.70: f = 0.40 f cm' (f) = 0.28 f cu' Z/p = p c cz cm max
(5) For ratios of contact stress to concrete strength, f If , cp cu
0.36;
0.20.
up to 1.9, the amount of reinforcementohad a significant effect.
Beyond this point, no increase in beari~g capacity could be
achieved by increasing the reinforcement.
(6) The important stress gradients occurred in a region extending
from 0.2 b to b behind the bearing plate. c c
(7) Helical reinforcement proved more effective than mat rein-
forcement.
(8) A practical design procedure was formulated.
The second phase of the experimental work(4) investigated
the behaviour of groups of anchorages, and resulted in the fOllowing
conclusions:
(1) The validity of Guyon's symmetric prism method was confirmed;
(2) Tensile zones owere observed between the applied prestressing
forces and near the loaded face of the end block; these zones
differed from those suggested by Guyon;
32
(3) The analysis and design methods recommended by Zielinski
( 3) and Rowe were suggested for application to the conceptual
symmetric prisms;
(4) An increased strain capacity prior to cracking resulted from
the complex stress state; an increase of the tensile strength
over the splitting tensile strength was recommended.
2.4 DESIGN OF BEAM ANCHORAGE ZONES
The current design practice is to provide sufficient
lateral reinforcement to resist the total calculated lateral tensile
force in the anchorage zone. The steel must be distributed in
accordance with the tensile stress distribution.
Curves representing the transverse stress distribution,
the coordinates of maximum and zero tensile stress, and the total
splitting force, for various plate width ratios, are shown in Fig. 2.10.
The Z/p vs bplbc curve determined by Iyengar falls above that obtained
'h d (27) h h b . h' h by Guyon. Leon ar t ,on t e strengt of data y Sarg~ous w ~c
falls somewhat above that of Iyengar, at b lb = 0.2, suggested the, p c
conservative linear envèlope
Z/P = 0.3(l-b lb ). p c
Morsch's simple analysis, evolved from statics under reasonable
assumptions, agrees remarkably weIl with the more refined analytical
solutions; for b lb ratios between 0.2 and 1.0, the formula p c
Z/P = 0.25(l-b lb ) p c
33
ru
a~
5 0.5 Point '4-1 Q3 ..... , av x/bc
N , 0 O.Z , 0.3 for
4-1 , o.z o.z max fcz
al ~l' 0.1 0.1 f'cz = 0
D 0 0 o.s f.o
(a) 7ensile stress distribution (left) ; ·position of ~ximum and zero tensile stress, and total tensile force (right).
[From Leonhardt(27) after Iyengar]
5'0" r--r--r-,---y---y-.----r-----r-"T--. '1-1 ..... ·o·'t---'t---il--i-..:::....t:~~--+--...f .. ~ .
'1-1 0·4t--+-+-i~~r.-.pa.~~~::--.t-~-=~
m 0'2 t--'i--I---I---+-~~ 0~ ..... ~~0.~2-~~0~.4~-L-~OL·,--~--0~·-.--1-~,.0
bp/bc
..-- ---H.cne'
........ : •••• - H.end (modifie<!) • D'eith e-- -GuJon ----E.;>erlmenll • • --- •• - DI.'th·Sleoen e_ • - • _ Honth
O"I--I--+---+--:~o--I--+--lH-~ •
'1l4 . 0·21--f--I--4r:--+:::......::-+-~-=+-+-+---l ..... N
o 0'2
.-.-: - -- H.cne' _ ........... H.cne' (modlfied) • D'elth -- --' Gu1on • hperlmenll' _.-.- DlcI,h·~lcven
.-.-.- HOII,h
0'11 1'0
(b) Magnitude of maximum tensile stress (top), total tensile force (bottom): comparison of several theoretical solutions with Zielinski & Rowe's experimental results.
(3) [From Zielinski and Rowe '. ]
P = bearing forc~
Z = total tensile force in concrete
f = tensile stress in cz cancre te
f ~ P/bcb cm = mean pressure in
concrete block
bp = bearing plate width
bc = concrete block width
b = transverse width of plate and block
Figure 2.10 Tensile stress distribution and total tensile force
34
is a close approximation to the results of Iyengar and Guyon.
Standard details for the anchorage rein forcement of beam
end zones can also be obtained from prestressed concrete design
handbooks, such as the handbook of the Canadian Prestressed Concrete
Institute(59) which provides standard designs based on conservative
application of Guyon's solution.
2.5 DESIGN OF'BUTTRESS ANCHORAGE ZONES
A finite element analysis of the anchorage zone in a
prestressed concrete containment vessel was reported by Kulka and
(51) . (62) Wahl, and Wahl and KosJ.ba • A cylindrical quadrant wi th one
buttress at its center was analyzed under a plane strain condition
which approximates the three-dimensional nature of the problem. The
most significant result of the investigation was that very little
tensile stress is created in the buttress area, as a relatively uni-
form compressive stress arises in the buttress except immediately
under the anchorage bearing plate. The provision of rein forcement
to resist these tensile stresses is not discussed in either paper,
although the buttress detail drawing in the paper by Wahl and Kosiba
shows rein forcing bars bent to run along the side of the buttress and
along the bearing plate (c.f. bar A in detail l, Fig. 3.2, p. 42 of
this thesis).
The most current design method consists of obtaining a two-
dimensional stress distribution in the con crete under the bearing
plate, using either a finite element analysis such as the above, or
35
the generally more conservative solutions of Leonhardt or Guyon,
and providing sufficient lateral rein forcing steel to resist the
total tensile force over the section. This steel is then dis-
tributed over the region where tensile stresses in the concrete
h l abl '1 'f' d' th d (2,45) exceed t e a low e tens~ e stresses spec~ ~e ~n e AC! co es.
CHAPTER 3
DESIGN AND FABRICATION OF TEST SPECIMENS
3.1 BUTTRESS DESIGN
3.1.1 Design Loads
The containment structure whose buttress is being examined
in the present investigation, requires an effective circumferential
prestressing force, at design working loads, of 730 kips per foot
of height (10.65 MN/m). Such a force can be provided through the
use of tendons at 2-foot (61-cm) intervals over the height of the
wall, consisting of 163-7 mm. wires with a specified ultimate tensile
strength (f ) of 250 ksi (172 MN/m2) , stressed to 0.6 of this ultisu
mate capaci ty •.
The bearing plate and anchorage zone must be designed to
resist an additional increment in the prestressing force required
to counter the effects of shrinkage, creep and elastic deformation
in concrete, relaxation in prestressing steel, and friction and
anchorage losses. Such temporary overstress in the course of the
post-tensioning operation may lead to stresses in the proximity of
0.8 f which are permissible under temporary overloads. su For the
tendons under consideration, which have a cross-sectional area of
9.75 in2
{62.8 cm2
) , and thus a specified ultimate tensile capacity
of 2430 kips (10.81 MN), the anchorage design load works out to be
1945 kips (B.67 MN).
36
37
3.1. 2 Bearing stresses
Figure 3.1 shows the buttress design which was chosen for
the present studYi it has a length of 12 feet (3.66 m) and a width
of 32 inches (813 mm). The tendons run in straight lines over the
length of the buttressi the vertical spacing between centerlines
of opposite tendons is 1 foot (305 mm) at their point of crossing.
The bearing ~lates are not centered on the face of the buttress, as
can be seen from Fig. 3.1.
Two bearing plate sizes are proposed, the plate width in
each case being twice the diameter of the central hole required for
the circular conduit, through which the tendon wires go from the
compact sheathing duct to the wider external anchor.
Both plates satisfy the compressive stress requirements of
the ACI Building Code 318-63 at working loads(2). The Code pro-
vision dealing specifically with bearing takes into consideration
the confining effect of the surrounding concrete, and limits allow-
able bearing stresses to
f = 0.6 f . (A lA) 113 • cp c~ cp p
The symbol A denotes the net bearing plate areai A is the net p cp
area of the underlying concrete surface which is geometrically similar
to the plate and concentric with it, i.e., in both present cases, a
square whose width is twice dimension marked b 12 = 17.9 inches (455 mm) cp
in Fig. 3.1. The second restriction imposed by the Code is to limit
the compression stresses at working loads to a value of 0.45 f • cu
38
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j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
j
39
The affected area is the ~ase of the pyramid obtained on the assump
tion that compression trajectories spread at an angle of 30 degrees
(~/6 radians) from outer edges of the bearing plate. This assump-
tion appears conservative when compared with the trajectories ob
tained by sargious(23) and reported by Leonhardt(27) for an eccentric
prestressing force.
given in Table 3.1.
A summary of bearing stress calculations is
The smaller bearing plate is noted to be just
adequate for the design prestressing force, while the larger plate
is quite conservative in this respect.
The principal advantage of a 24-inch square bearing surface
is that it occupies the full distance between the tendons. Thus
fabrication costs can be reduced by making the bearing plates con-
·tinuous in the vertical direction. The l2-inch diameter ho le is
designed to accommodate an alternate type of tendon consisting of
l86-6mm wires. If used in conjunction with the proposed l63-7mm
wire tendon, the 24-inch bearing plate with an Il-inch diameter ho le
becomes excessively conservative in bearing, and the use of individual
22-inch plates may be more economical. In addition, the increased
concrete cover beyond the edge of the 22-inch plate can be significant
in providing spalling resistance. For these reasons, the smaller
plate was considered in the design and experimental study of ahchorage
zone rein forcement. The continuous bearing plate was used only in
the complete buttress model, where the effect of its continuity could
be appraised.
)
40
TABLE 3.1 nfARJ~G STRESS CA(CULATIONS
PLATE SIZE bp IN HM
HOlE DIAMETER dh IN
AREAS;, BEARJf-IG PLATE~ GROSS b2 P
Hl2
CENTRAL HOLE Tl d~ Ah :"4 h IN2
BEARHlG Pt"ATE; NET Ap = b~ - Ah INz ,
CONF.NI~G CDNCRETE; WIDTH bep IN
GROSS AREA b~p IN~
NET ÀRI::A '1.' Acp ::' bcp - Ah IN~
BEARING CAPI\C ITV COEFFICIENT 0(. c ~ Acp / Ap
ALLOWABLE BEM~ Wc. PRr;SSURE +cp 0= 0.6 ex rci
ALLOWABLE BEARING FO~CE
R~GION OF UNIFORH COMPRESSrON~
KS l , MPA
KIP MN
DISTANCE h = ~(bc·-bp)cot.30· .... i, (bc-ocp)cotlô,go IN
BASE OF CONCRl:TE PVRf\MIO . bc GROSS AREf\ lt NET AltE:A Ac .. b~ - Ah
ALLDWf\BLE C[JI·lpr·:r:SSIvE FORCE p ... 0,45 +cu.Ac.
CURVATURE OF BUTiR~SS wALL NOT CONSlnEREOJ CO~PRESSJON TRAJfC10RIFS ASSUMED TO srREAD AT 30 ~EGRFES TO PRESTRESStNG FORCE prRECTIDN; ÇCi .. fc.u. ~ 5 Ksi ... 3.45 MPa
KIP ~IN
INOÎVIOUAL CONTJNUOUS PLATES PLATES
22 24 560 6io
11 12
4B4 .. 576
95 1Î3
389 463
35.8 35.8
ï282 1Z82
i187 1169
1.ft5 1".36
4.35 4~O8 3.00 2.81
ï6'90 1690 7.51 8'.40
23.4. 2i.s ,
51.0 48.6
2601 238ï
2506 2268
56/.0 5ioo 25.0 22.7
41
3.1.3 Anchorage zone rein forcement
The seven different anchorage zone details illustrated
in Fig. 3.2 provide various amounts of lateral reinforcement, while
being based on a common arrangement of the reinforcing steel, shown
in section a~d elevation in detail IV of Fig. 3.2.
AlI of these details make use of horizontal and vertical
rein forcement parallel to the bearing plate. Other means of confining
the concrete under the bearing plate are available: spiral reinforce
ment wound around the axis of the tendon, for example, is often used.
However, the chosen arrangement of lateral steel is believed to
utilize the reinforcement most efficiently in its multiple functions
ofconfining concrete in compression, sharing tensile forces with the
concrete, and maintaining the structural integrity of the wall in case
of cracking, due to the action of prestressing force under the bearing
plate, moment and shear at the junction of the wall and the buttress,
and ther~al, shrinkage, creep and other effects in the containment as
a whole.
Consider, as an illustration, bar A in detail IV: along the
side of the buttress, it acts as thermal reinforcement; in the bend,
it aids in resisting spalling; along the face of the bearing plate,
it serves as transverse tensile reinforcement, while beyond the wall
buttress joint, it reinforces against cracks initiating'at the re-
entrant corner due to shear and flexure. Bars marked A, F, Gare
elements of an orthogonal grid which provides No. 10 bars horizontally
and vertically at 1 foot (305 mm) on centers over the entire height
.42
BA
, !
1 1
A iNo8 A 1 No8 il BI NoB.4 B 1 t-lo8a3
C No8 C No8
i i
-
B BA A
1 1
Al No10 la A 1 No10 rra BI No8'a4 B 1 No8a3 C No11 C No11
i i i 1
i i 1 1
Figure 3.2 Anchorage zone' details
-
L-~~~~--------B
~~~~~---------A
m
~~~~~-------B ~4-~~~--------A
A B C
IDa
Fi~ure 3.2 (continued)
43
w""'+~~~----8 A
====rl ~ .... ---- -A -----8
63 l lie
A 8 C D E F G
No 10at12 hor No 8 at~ No 11 No8 No 8 \VIth "hairpin" tles No 10 at 12 hor No 10 at 12 vert
B
Bars E,F,G, Elevation view and Dimensions indicated in plan, are common to ail designs. (Dimensions & spaclngs in inches )
Anchorage zone details
44
and circumference of the containment structure, and is intended
mainly as rein forcement against thermal and inelastic effects prior
to prestress application. Bars F, like bars A, extend beyond the
re-entrant corner and resist diagonal tension cracking at that point.
Spalling te~dencies at the corner of the buttress are counteracted
by bar E, anchored at lB-inch (457 mm) intervals along its length by
"hairpin" bars (shown only in detail IV, but pre.sent in all others).
Ties marked B are provided throughout the buttress as creep
and thermal reinforcement, and are concentrated in varying numbers
in the anchorage region. Their extension into the wall serves the
dual purpose of embedment and diagonal tension reinforcement, as was
the case with bars E and F. Vertical bars C and D are continuous
over the height of the buttress, and perform the same role as the
ties, in the perpendicular direction.
Details la, lIa, III, IlIa, IV are practical examples of
considerably varying amounts of lateral anchorage zone reinforcement.
Cases I and II are simplified designs for laboratory investigation,
in which all bars are of the same size; they do not differ markedly
from cases la and lIa in terms of the total steel area provided.
Table 3.2 summarizes the amount of rein forcing steel, in
terms of areas and weights, for one tendon, according to each of the
seven designs. For the purpose of this compilation, ~nly bars which
ran parallel to the bearing plate were considered to be placed speci-
fically as anchorage reinforcement. Thus in cases l, la, III, IlIa,
the first inclined bar was considered as thermal reinforcement, al
though it may still be in the region of varying stress gradients, being
T!:.E.L:: 3,2 A!.OLNT CF REINFDReING STEEL P~R TENDON ANCHORAGE ZON~ FOR VARIOUS END DETAILS
1 1 at.;;.s AREA CSQ. IN.> AREA WEIGHT CPOUN!)S) ZS CKIPSi i (SQ. CM) !Hû?'I LJ~;T ~l. 3~~S :~~Ri( A /-lM!': B MA?K A MARK B TOTAL TOTAL MARK A :-1ARK 3 TOTAL
l~ëïAtL l 2 ~::. a 6 NO. 6 l,58 4.74 6.32 41.8 32 96 128 126 1 lA 2 N::.10 6 NO. 5 2.34 4.74 7.2B 47,0 52 96 14B 146 1 ! l ? t,~. a 14 NG. 3 1.5B 11.0b 12,64 61.7 32 224 256 253 ;
1 lA 2 N~'olO 14 NfJ. 8 2.54 11.06 13.60 67.9 52 224 276 272 1 1
1
III 2 N:JolO 2 N[1. 4 2.54 0.40 2.94 19,0 52 fi 60 59 IlIA 1 2 ND.I0 2 NO. 8 2.54 1.:;8 4012 26.6 52 32 84 82 IV . 1 2 1\:1.10 10 ~lD. 6 2.54 7.90 10,44 67.4 52 160 212 209
! ('dH:) 1 1
U.l (ë) - - - .. (e) (0 ) - (F)
i l'iE~TIe.lL dARS H:.RK e :~.\RK 0 MAj{K C ~lARK 0 TOTAL TOTAL NARK C 11ARK 0 TOTAL
I;:>!:TAIL l 1 Il :"/:;. a " 6.32 - 6.32 41.8 43 - 43 126 lIA 1 2 N::1.11 - 3,02 - 3,02 19,5 21 - 21 60 1 II 116 NJ. a - 12.64 - 12,64 81,1 86 - 86 253
q' ,a :~:.ll .. 12,48 - 12.48 80.5 a5 - 85 250 1 il î . 2 N·.Jo 8 " 1.38 - J.,58 10.3- 11 - ,Il 32 .
IliA J-l NO'. , 1.58 1.58 10,3 11 11 32 1 - - -1 IV 2 NL!.l1 6 NO, a 3,02 4.74 7.76 50,1 21 32 53 155, 1
1
1 (i.iJTE) - .. - - ~ .. (E) (E) - CF) 1 - ---------- --
~DTES: Cl) Ci:e BAR JN rc A~D ~~E "AR ON snTTOH OF TENDON, (5) ~AC~ rIE MA~K n B P~DV!DES O~E 6AR ON TQP AND ONE BAR ON BDTTOM OF'TENDON.
~~LY Trl~SE Ti 5 WHICH RUN PA~ALLEL TD THE SEARING PLATE FACE WERE CONSIDERED TD BELON~ TD THE ANCHORAGE ZONE DETAIL. (e) A LENGTH OF 6 F~El (TOP & BOrTOK) PARALLEL TD THE BEARING PLATE WAS CONSIDERED TO BELONG To THE ANCHORAGE Z~NE DETAIL. (0) W~IGriT P5R ;~o, ~ TIE SASEO ON T~TAL TIE WEIGHT OF APPROXIMATEL~ 32 POUNDS. (~) VE~7!eAL BAAS srA~ 2 FEET Ir! ANY TENOON ANCHORAGE ZONE.
152 li5 304 ~27
71 99
251
CG)
152 72
~~4 Z'?9
33 .33
1do
CG)
1
1
J 1
1
1
,
CF) ~s = TOTAL LATEiAL FGRCE IN AI1CHORAGE ZJNE REINFORCEHENT ASSUMING ALL BARS ATTAIN SPECJFIED WORKING STRESS 20 KsI (13.S~ HP!). (G) ZS = T~TAL LATERAL FORCE IN ANCIIORAGE ZONE REINFORCEMENT ASSUMING ALL SARS ATTAIN SPECIFIED WORKING STRESS 24 KSI C16.5~ ~PA).
~ ln
46
at a distance just greater than the plate width.
The lateral force in the anchorage zone reinforcement,
assuming aIl bars to have attained a uniform stress equal to the
allowable working stress in intermediate or hard grade steel, is
also included in Table 3.2. The results are tabulated in order of
increasing lateral force in Table 3.3, for each of the details in
both grades of steel.
providing less vertical than horizontal rein forcement
would seem appropriate considering that the structural weight tends
to counteract the vertical splitting forces. However, the concrete
surface under the bearing plate is continuous in the vertical direction,
and hence the stress gradients and resulting tensile stresses may be
higher than those in the horizontal direction and cannot be ignored.
It was, therefore, decided to use equal steel areas in both directions
for designs l and II, and only slightly less vertical than horizontal
rein forcement in the other details.
Working stress design considerations require that the total
force in the reinforcement, Zr at working stress, be equal to the re-
sultant of the integrated tensile stresses over the anchorage zone
cross-section. In current design practice, the required lateral
force is calculated from a formula Z = P (l-b lb )a, where the constant p c
a is 0.3 in Leonhardt's method, and 0.25 in Mërsch's method. The
most adverse ratio of plate width to anchorage zone width (b lb ) must . p c
be used. Calculations for various assumed concrete areas are summa-
rized in Table 3.5.
47
TABLE 3.3 END DtTAleS IN URnER OF INCREASING TENSILE CAPACITV
OeTA IL FS ZS ZS/PD (KSI; (KIPSI
---_ ... HORIZUNTAL Rf,RS III 20 .59 0.040
III 24 71 '0.049 II lA 20 82 0.0.56 IlIA 24 99 0.068 1 20 126 0.086 lA 20 146 0.100 1 24 152 0.104 lA 24 175 0.120 Iv 20 209 0.143 IV 24 251 0.172 Il 20 253, 0.173 liA 20 272 0.186 II 24 304 0.20B liA 24 327 0.224
VERTICAL BARS 1 j 1 20 32 0.022 JI lA 20 !12 0.0?2 III 24 3B 0.026 IUA 24 38 0.026 lA 20 60 0.041 lA 24 72 0.0'.9 1 20 126 0.006 1 24 152 0.104 IV 20 155 0.106 Iv 24 166 0.127 lIA 20 250 0.171 JI 20 253 0.173 liA 24 299 0.205 1 i 2'. 304 0.208
FS :: HllRKING STRr;ss IN LATERAL REWFflRcErlEtJT ZS:: T[JTAL LATER/d, FORCE HI flNCHORIIG!; ZO~IE REINFORCEMENT;
ASSUHING AL( nARS ATTAIN AL(OWABLE WORKING STRESS
ZS/PI
0.030 0.037 0.042 0.051 0.065 0.075 0.078 0.090 0.108 0.129 (1.130 0.140 0.156 0.168
0.016 0.016 0.020 0.020 0.031 0.037 0.065 0.078 0,0110 0.096 0.128 0.130 0.154 0.156
PD = 1460 KIrs c 6.50 HN I DESIGN PRESTRESSING rORCE AT FSP :: 0.6 FSU Pl = 1945 KIPS 1:1 6'.65 ~INï INITIAL PRF.STRESSING FORCE AT r-sp = 0,0 FSIJ
TABlF. 3.4 END DETAl~S IN OROER UF INCREASING STEEL WEIGHT
DETAIL STEEL ("EJGHT PER ANCI/ORAGE FRACTION OF LARGr:sT VALU!: (POUNDS)
HOR'. VE:RT, TOTAL HOR. VERT. TOTAL
!II 60 11 71 0".21'7 0.1;>8 0,197 IlIA 84 il 1)5 0'.304 0,128 0.263
Il 128 43 171 0.464 0.500 0.474 Il A 148 "21 169 0.535 0.2't't 0.,,69 IV 212 53 265 0.7fl8 0.616 0.735 Il 256 66 342 0.926 1.000 0.947 UA 276 85 361 l~OOO 0.989 1.000 --_._--"- - --
48
TABll: 3.5 REQUI)ED LATERAL REINFORCEMENT CAPACITY
tA5E .. Be/r; SP/BC REQUIRëD ZS/P
LEONHARDT MORseH
VëRTlC:\L PL~N'.: (l) 1 S!JLtlTEü Tn·l)l.1i~ 00 0.000 0.3:>0 N.A. , 2) EVERY 4 Tri T~NDf)~1 rF.IJS IO'!ED 3.00 O.~O5 0.20B 0.174 ( 3) HERY 3RO Tr.:"IDQ:.; TENSIO'JED 2.00 0.456 0.163 0.135 (4 ) EVERY ZN') i'rr~DDI-,j TEtlSJQI':ED 1.50 0.610 0.097 (J.1l7
HUR 1 ZO:JT AL p L AI~L= (5) USINe. aep (FIG. 3.1) 1.49 0.615 0.115 0.096 (6) USING Be (FIG. 3.1) 2.1i! 0.432 0.170 0.142 (7) USING \HOTIi or r.uTTRESS FACE 1.33 0.686 0.094 0.078
-
3 4
---= -.....:
I~ 1 1
1
6 7 1 . 1 1 / 1 1 . 1 1 ! / 1 / 1
1
49
According to present design methods, none of the chosen
rein forcing details is conservative. The aim of the present
investigation will be to de termine whether the transfer of force
to the reinforcing steel at working stresses does occur to the
extent stipulated. Moreover, the load factors at first yield of
steel and at crushing of concrete will be sought, and the feasibility
of relying to sorne degree on the tensile capacity of concrete will
be examined.
The economic significance of making optimum use of the load
bearing capacities of anchorage zone steel and concrete can be de-
duced from Table 3.4. The lightest reinforcement pattern requires
one-fifth the weight of steel of the heaviest one; the corresponding
saving of 290 pounds (131.5 kg) of steel per anchorage in a structure
that can incorporate up to a thousand tendon end zones cannot be
neglected.
3.2 'BLOCK MODEL DESIGN
3.2.1 Model details
The prismatic rectangular specimen with a concentric cyl in
drical cavity, termed "block model", is illustrated in Fig. 3.3. The
block model constitutes a simple laboratory specimen from which a pre
liminary estimate of the behavior of an isolated anchorage zone can
be obtained. Scale models of this specimen are simple and relatively
inexpensive to fabricate. A test to destruction which reproduces the
1 -----, \r--Yt~---~ \ \ \ 1 Il \ \\ 1 1 ___ \
\ \ ~ \ \ \
A4J _ , .JA ~ \ Sr\-;: 1 :' ---lB \
\ ", 0 i .• ,_______ ___ \
\. L[~l.~_ ____ \
l, \. \
9-2 \ \
~~.1_ \ . (1 \ i \ \
." ! ,1 \ \ "~H' L ________ ~
-!3El· ~l~ 1 "f ! 1 1 .
,1..) i ,-1 E lev at ion
1 16 i 16
! 1
C 1 nl~
-+-. -Ejj- -- - ~ ~
1 --~I----J
14 i 14 1
ie
Section 8-B
50
l a 11-0D Duct b éX3 Crack Initiator
bd\ c 1 x1 Crack Initiator ---.~ .. -r
i '1
Section A-A
1 --.-------, rr---f'~ \ \ \ 1 i 1 \
\ \ 1 1 1 \ \ \ 1· 1 \ \ \ \
AtU A \
D \ Bq ! 8 \ \ \ \ ------- \ \ \ l \
\ \ , \ \
\ JI \ \ \
LI i
- · · 1 · · 1 ft · · · 0
\ · 0 , · · · · ---1
\ \ L ________ ..1.
Figure 3.3. Block specimen details
51
action of the prestressing tendon and anchorage devices can easily
be performedusing a universal hydraulic testing machine, without
the need for post-tensioning jacks or other hardware.
The shorter dimension of the prototype was 32 inches
(813 mm), which is the width of the buttress bearing face. The
duct and plate were placed concentrically within this width, for
simplicity apd to avoid the arbitrary selection of dimensions for
the zone of influence of the mass of concrete in the wall. The
approximations introduced by truncating the anchorage zone would
expectedly be conservative from the point of view of the ultimate
capacity, as the confining effect of the thick cover and of the
wall mass are reduced; the stress distribution, on the other hand,
would be less severe, as the narrow anchorage zone results in smaller
gradients.
The longer dimension of the block face was 4 feet (1.22 ml,
which corresponds to the distance between the centerlines of tendons
above and below the anchorage under study. This wide dimension
allows for the more pronounced stress gradient and the resulting ten-
sile stresses which occur in the vertical direction when a single
isolated tendon is prestressed.
Two patterns of anchorage zone rein forcement were selected
for investigation, viz. details l and II explained and illustrated
in the previous section. The same number, type and spacing of bars
are provided in the horizontal and vertical directions. Detail l
gives 8 - No. 8 bars within a distance of 0.81 times the width of the
plate b from the face of the plate, while Detail II distributes twice p
52
this amount of steel in a depth of 1.15 b • P
Bar embedment was accomplished through standard right-
angle hooks. Supplementary reinforcement, such as the lateral
gr id and corner bars, could not be introduceQ, although their con-
tribution to spalling resistance would be beneficial in the buttress
structure. 'Closed stirrups were placed in the lower half of the
block models, to guard against sudden failure, but were not expected
to contribute to the bearing resistance.
Crack initiators were molded at the center of the faces,
and along the surface of the duct (Fig. 3.3) so as to impose the
greatest proportion of transverse force on the lateral rein forcement.
The efficiency of performance of the reinforcement was
evaluated from strain measurements on the rein forcing bars.
3.2.2 Choice of scales
The choice of appropriate dimensional scale factors hinged
simultaneously on the availability of mOdeling materials and on the
requirements of the concurrent buttress model investigation.
For the latter project, a scale factor less than 1/5 (0.2)
was recommended to avoid overtaxing available material supplies, usable
space, weight handling facilities and loading devices. On the other
hand, a scale less than 1/8 (0.125) would have resulted,in complications
in sorne areas of fabrication, strain gage instrumentation, reinforcing
cage fabrication and uniformity of concrete compaction, while demand-
ing excessively precise dimensional tolerances. From overall
similitude considerations, a scale of exactly 1/6 (0.1667) was selected
53
ta give the smallest workable specimen for the buttress model.
A similar length scale factor of 1/6 would, therefore, have
been appropriate for the block models. However, since the rein-
forcing bar diarneter was the only dimension of the blocks which could
not be accornrnodated to any chosen scale by a simple adjustrnent of
formwork, it was preferred to choose a scale factor whereby the 0.79-in2
2 (5.10 cm ) area of the No. 8 reinforcing bars was scaled exactly.
The closest available size of deforrned model rein forcing bar being of
type D2, with nominal area 0.0200 in2 (0.129 cm2) , a scale factor
equal to , 0.02/0.79 = 0.1600 was therefore used for aIl dimensions
of the block model.
In order to justify the confidence placed in any given model
study, a comparison of results obtained on models of different sizes
is worthwhile. The present investigation consequently included
models to a scale 0.375, which is approximately the median between
the small scale"and prototype sizes (viz.0.375/0.l6 = 2.35,
1/0.375 = 2.67). The sarne consideration which had led to the choice
of the scale 0.16 for the small models governed the selection of the
length scale factor of 0.375. The steel could best be modeled
using the standard No. 3 reinforcing bars to simulate the prototype
No. 8 bars, whence the use of the ratio of their nominal diameters,
0.375/1.00, as the dimensional scale factor.
The model test results were also compared with sorne data
available from independently conducted tests on two full-scale proto-
type block specimens, similar in aIl respects to those described in
the previous section.
54
The relative dimensions of models in 'the three scales
are shown in Fig. 3.4.
3.2.3 Specimen designations
Block models with reinforcement patterns l and II, detailed
in accordance with Fig. 3.3, were cast in both scales 0.16 and 0.375,
and were labelled I-160, II-160, 1-375 and 1I-375.
type blocks were identified as I-l and II-l.
The two proto-
The presence of crack initiators was intended to stimulate
the building up of force in the transverse reinforcement from the
earliest loading stages. Their effect on ultimate behaviour was,
on the other hand, uncertain. From test results it was seen that
failure of the specimens, for the given bearing plate and end block
configuration, was accompanied by the opening of cracks not occurring
at the center of the block faces. It therefore became interesting
to observe the mode of crack development in the absence of artifi
cially induced stress concentration at preformed cracks, and for this
reason two specimens designated I-160N and 1-375N were cast without
crack initiators.
Unreinforced specimens were also fabricated to complement
the series by verifying the load factor implicit in the allowable
bearing stress, for the given geometrical arrangement.
Models were designated 0-160, 0-375, 0-160H, 0-375H, 0-160L
where the letters H and L refer to high and low concrete strengths
respectively.
55
v9i
56
3.3 BUTTRESS MODEL DESIGN
3.3.1 General features
The sixth-scale buttress model (Fig. 3.5) was constructed
to simulate as closely as'possible the features of the prototype
buttress schematically represented in Fig. 3.1. The model consisted
of two full buttresses jointed at the inner face of the containment
wall. The. curvature of the wall and buttress was not reproduced,
outer surfaces being cast parallel to the central plane of the model,
but the angle between opposite sets of tendons and the inclination
of the bearing surfaces were maintained.
Four tendons were anchored at each of the bearing surfaces,
three of them bearing on a continuous plate, while the topmost tendon
rested on an individual bearing plate, both plates conforming to the
two designs shown in Fig. 3.1. In the forthcoming sections, tendons
and their associated anchorage zones are designated by a digit- and
letter code (eg: 3A), where the bearing surface is identified by one
of the numbers l 'to 4,'and the tendon is labelled A, B, C, or D
(Fig. 3.5).
Four additional tendons, sloping at a tangent of 0.1 to
the horizontal, were provided along the central plane of the specimen
to reproduce the self-weight of the containment, structure. The
designations DA through OD are assigned to these tendons.
Crack initiators of the same design as those used in the
block models were formed on one of the outer surfaces of the buttress,
at locations opposite tendon centerlines and midway between these.
57
/+7+7+7+/ ..
58
Corner crack propagators were also included to induce separation
of the buttress from the wall.
Recognizing that thermal gradients would arise in the
buttress, firstly during mass-concrete hydration, and secondly in
the advent of accidental release of steam into the containment
structure, provision was made for the creation of a thermal gradient
from approximately the temperature of boiling water at the model
center-plane, i.e. at the inner face of the containing wall, to the
ambient temperature on the outer faces. Two pipe coils running in
planes parallel to the center plane of the buttress were placed as
shown in Fig. 3.6 to carry boiling water through the interior of the
wall.
3.3.2 Reinforcement
The two buttresses constituting the model were d~tailed and
reinforced identically, the only distinguishing feature between them
being the presence or absence of crack initiators. Reinforcement
for the specimen is detailed in the plan and elevation views. (Figs.
3.7 and 3.8). For clarity,dimensions and bar designations are ex-
pressed on the basis of prototype dimensions; conversion of di
mensions to sixth scale, along with equivalent model reinforcing bars
and their properties are listed in Table 3.6.
Anchorage zone details were of three types, identified as la,
III and IlIa in Fig. 3.2 Detail la was used under aIl anchors of
buttress bearing surfaces 1 and 3. Anchorages under the continuous
59
1601 96
Trim bar grids NO.10 at 12 "'.
\ End Anchors NO.11 vert. &
1 NO.8 at 42 hor .
. . . . ~ .. -.---.------.- .-.•. - .-.. --.. -.. - .... - ..... -... ------.--.-.----.--.-. . . . . .. . ...... . 1 ........... ., ........... -.................... .
360
Figure 3.7 Buttress model - plan
0\ o
Anchorage Anchorage
surface surface Inclined
1 2 middle
1 plane
Detail 111 (No 4 bars) at anchorage LA Detail 1 (No .8 bars) tendons
Detail llla(No 8 bars) at anchorages IB,lC,lD at ail anchorages OA to OD
w--. ----. --IIij::-,llloA 0\ ....
OA 1 :11-i-t11 : " . ===-1-+1- -=-:-- :. -:-.: ::1- -1--1- -I-HII-HÎ 2A :~ ~ _ :
OB IIH--+JII ". ---~- -J-H- -= ~ .. ----~-ll--I--J--I-tl!I-I,IT~B " H'l illlOB
1 • 1 1
---- li~Il'0C "
OC liiHHt -.--- _. _·=+l~·:"":-":'·_·_:----4---I--I--l--IfI~II·112C .... 1 1
.. ~IIOD OD IIIPlll===: - ~' " ! 1 ~ tt~ - fUel!" HU •. - -~~~~--~ .. ~-~- d2D
Figure 3.8 Buttress model - elevation
~ ------~ .. _._--_ .... ------------_ ...... ,
62
TABLE 3.6 P.UTTRF.SS HODEL DIHENS;ONS AND REINFORCEHENT
CA) DHIENSIONS PROTOTYPE MODEl
IN. S.I. IN. S.J.
BUTTRESS MaDEL, CVERALL LENGTH 360 9.14 M 60.0 ' 1.52 M WIDTH 160 4.06 H .26.7 0.68 H HEIGHT 128 3.25 M 21.4 0.54 M .
BUTTRESS PROPER, LHIGTH lA4 3.66 H 24.0 610 MM THICKNESS 32 812 MM 5.33 135 HM
BEARJNG PLATE THicKN~SS 3 76.1 HM 0.50 12.7 MM CONTINUOUS PLATE; WloTH 24 610 MM 4'.00 101.5 MM
~IOLE DIAHETER i2 305 MM 2.00 50.8 HM INDIVIDUAl PLATE; WloTH 22 560 MM 3.67 93.3 HM
HO LE OiAMETER ïl 280 HM 1.83 46.6 MM
BAR. SPAC ING, NO. 10 TRIM BARS ï2 305 MM 2.00 50.B HM NO. li INNER GRJDS 6 152 HM 1.00 25.4 HM
(B) REJNFORCING B~RS
PROTOTYPE BARS, DESIGNATIOI~ NO. 4· 6 B 10 11
DI AMETER IN 0.50 0.15 1.00 1'.27 1.41 MM 13 19 25 32 36
AREA iN'1 0.20 ·0.44 0.79 1~27 1.56 MM'1 129 284 510 820 1005
MaDEL BAR AREA REQID FOR SIMILITUDE IN'- 0.0055 0.01?2 0.0219 0.~353 0.0434
MaDEL BARS, DESIGNATIOt~ NO. on-LC 01.5 02 09.5 04.5
AREA HI2. 0.0055 0.0150 0.0200 0.0350 0.0450 foIM2. 3.5 9,7 12.9 22.6 29.0
'.
63
bearing plate of surfaces,2 and 4 were of type IlIa, and those
under the individual top plates were of type III.
Investigation of the heavier reinforcement patterns, de
tails lIa and IV, was considered redundant, as detail la was esti
mated, from previously acquired data, to provide adequate resistance,
with a large margin of safety for normal working loads. The lighter
reinforcement, détail IlIa, was exarnined for the efficiency of the
sirnplest anchorage zone detail.
The upper tendon anchorage was'to be loaded until failure
by progressive increase of the bearing load. Taking account of
the 'capacity of the available post-tensioning system, it was assumed
that only the lightest reinforcing pattern that could suitably be
modeled, viz. detail III, would allow destruction of the anchorage
zone concrete. This light detail was therefore used under bearing
plates lA and 3A.
End anchorages at wall extrernities were detailed conserva-
tively. Square orthogonal grids were placed on outside faces of
the wall and the buttress, and on opposite sides of the central ten
dons, to simulate the presence in the containment structure of the
reinforcement required for loading conditions prior to prestressing.
Suitably anchored vertical bars were provided at every corner to re
sist spalling of concrete.
64
3.3.3 Post-tensioning system
The post-tensioning tendons of the model, like those of
the prototype structure under investigation, were fabricated from
7-mm wires with a specifi~d ultimate tensile strength of 250 ski
(172.5 MPa) '. Exact dimensional scaling of the prestressing steel
area would require an area equal to 4.53 times that of the single
wire, to simulate the 163 wires of the prototype tendon. Since it
was possible, within the dimensional restrictions of the anchors,
bearing plates and ducts, to utilize a tendon consisting of seven
wires, its use made available the application of prestressing forces
in excess of the prototype design bearing load.
Details of the post-tensioning system are shown in the
photographs and the schematic diagrams in Fig. 3.9. The BBRV
system was employed, whereby the individual wires are slipped through
holes in an anchor bolt, and held there by button heads formed through
a patented process. Threads on the anchor boIt allow the attachment
of a tie-rod through which tension is applied to the tendons, and
provide for exact shimming achieved by positioning the anchor nut
against the bearing plate under full tension. Load levels were
ascertained from a load cell containing eight electric resistance
strain gages. Tendon elongations and hydraulic jack pressures
were measured as tension was applied, but not used for exact load
measurements.
Anchorage at the passive ends was provided by fIat, un
threaded anchor bolts resting on distinct 4-inch (102 mm) square
bearing plates. Plates at the buttress surfaces had the dimensions
specified in Table 3.6.
65
(a) Construction phase
(h) Anchors (c) Tensioning system
Figure 3.9 Post-tensioning system
-" -
'.
\ \ \ , \ \
\
66
\ \ \. \
(d) Passive end anchorage
"-" ,
------\
1
2
3
4
5
6
7
\ . \ "\ \ \ '\ \'
(e) Active end anchorage
ELECTRICAl lOAD CELL
JACK
SEAT
SUTTONHEADED WIRE
ANCHOR 'SOlT
AN:HOR NUT
SEARING PLATE COVERING RING
Post-tensioning system
67
3.4 MODEL FABRICATION
3.4.1 Materials
Model mate rials for reinforced concrete simulation have
been under study for several years at McGill. (61) The work of
syamal(47) on the similitude analysis of different scale models of
concrete members in combined loading investigations is especially
worthy of mention. The bond between the small-sized reinforcing
bars and concrete was studied by Hsu. (48) The test results from
these and other experimental programs tend to confirm the feasibility
of modeling reinforced concrete structures with the scaled concrete
and reinforcing steel described hereafter.
A mix termed "micro-concrete" containing graded small-size
aggregate in a distribution analogous to that of aggregates in proto-
type batch concrete, was used in the fabrication of models of both
scales. Recommended proportions, from a report by Tsui and Mirza(49) ,
are shown in Table 3.7, along with partial data from a forthcoming
t b Labonte~ and Ml.'rza(50) on t' , ff t d repor y compac l.on, Sl.ze e ec s, an
splitting tensile strength. The mix employed throughout this
project made use of high-early-strength cement and crushed dry silica
sand in a water:cement:aggregate proportion of 0.55:1:2.75 by weight.
Under various curing conditions, it could attain compressive strengths
at 7 days of 3.3 to 5.2 ksi (2.28 to 3.59 MPa) and splitting tensile
strengths fct of approximately 0.1 f or 6.45 qo- (where f is cu ~LCU cu
expressed in psi) . InternaI compaction applied to the cylinders by
means of an internaI vibrator increased the strength by an average
factor of 1.158. A size effect was noticeable, viz. strengths
TABLE 3.7 H!CRO-CO~CRETE PRnPERTIES
lAI PROPORTIO~S ~Y WEICHT
SPECIFIEO f,u RATIOS
PSI HPA W/C A/C
2500 1.72 n.1l3 4.00 3000 2.07 'l.72 3.75 4000 2.76 0.60 3.25 5000 3.45 0.55 2.75 6000 4.14 1).50 2.50 7000 4.S3 0.45 2.25
(BI ACCREGATE OIS~RIAUTION
AS TH PROPr,RTIr.N SIEVE NO. RETAlt/EO CU~ULhTIVE
10 0.20 0.20 16 ,0.20 0.40 24 0.25 0.65 40 0.25 ' 0.90 70 0.10 1.00
--- -- --------
ICI SIlE EFFECT lOI 5PLITTING TENSILÈ STRENGTH
~,,& ~t csi RATlf1 1
CYL. OIA/1. 3 I~
3 IN, 6 l'l 'bï\
feu +ct P.ATlO~
.h çc.t psi psi çc.u !Ç
4722 426 0'.0907 6.23
1
51B3 5164 i .001 4739 4527 i.C>45 5471 4445 i.Z30 3919 4191 0.936 3774 3714 1.015
4191 437 0.1041 6.75 5164 487 0'.0944 6.79 5163 433 0'.0835 6.01 4527 414 0~0915 6.15 4739 436 0:0920 6.34
4325 3669 L1S0 3494 3298 i.06C\ 3894 3448 i.129 4166 3918 1.062 4188 3767 i 0111
550 539 1.020 ,
433 487 0.B9(1
1
436 414 1.051 345 3B4 0.n99 412 352 ï.170 1
4191 384 0.0916 5.94 3919 345 0:0880 5.51 3669 376 0:1025 6.20 4325 507 0'.1170 7.70 3298 388 0.1177 6.75 3494 416 0'.1195 7.07'
MEAN 0:1000 6.45
507 376 i.350 418 388 1.079
MEAN ;'.072
NOTES (lI EACH STRE~GTH VA(UE IS THE hVERAGE OF 4 CYLINDER TESTS (21 SPECIFIEO STPENGTHS IN CAl ARE AT 7 OAYS UNOER FOG ROOH CURING (31 STRENGTHS IN Irl; lOI, CFI WFR~ nSTATNED FROM ONE HIX
(El COHPACTION EFFECT
~u F~\ ~ATlr'
1. V. A5TH 1. V. AST"
3300 3670 ï .11\ 3490 4320 ' i.239 4530 5160 1.141 4740 5160 1.092 4190 4120 1.127 3050 3760 ï.240
iolE AN ï.i58
(W/CcO.S5, A/C-2.751 THRnUGH VARIOllS TESTING AGES AND CURING CONDITIONS (41 ABBREVIATIONS:
~/C • WATER/cEMENT RATIO, SY WEIGHT; A/C " AGCR./CE~ENT RATIO, BV WEICHT; ASTH • STANDhRO TAMPING PROCEDURE FOR COHPACTION OF TEST CYLINDERS; I.V, • INTERNAC VIBRATION APPLIED wITH AN ELECTRIC VtSRATOR,
151 RATIO Pc.~/.JÇ = 6.45 IS FOR PSI UNlTS ONLY
1
i
0\ 00
69
determined in the testing of 3-inch (76-mm) diameter cylinders were
7.2 percent higher than those obtained from 6-inch (152-mm) diameter
cylinders, therefore the diameter of the coupons used for strength
evaluation \'las selected to correspond ta the significant dimension
of the parent specimen. Sample stress-strain curves obtained from
measurements with strain gages on 6-inch diameter cylinders, are
shown in Fig. 3.10; gage type, strain rate, concrete compaction
and curing, were the same for these cylinders as for the block speci-
mens. The average value of Poisson's ratio was noted to be 0.175,
and the initial tangent elastic modulus could correctly be predicted
by Pauw's equation:
E c
33 .; w3 f cu
(Units: psi, Ib/ft3)
taking the unit weight w as 140 Ib/ft3 (2240 kg/m3).
Steel for model rein forcement consisted of either Grade 50
No. 3 standard reinforcing bars or cold-drawn deformed D2 rein forcing
bars (area 0.02 in2 = 0.129 cm2) , neither of which perfectly simulates
the behavior of the Grade 60 steel used in the full-scale block models.
Experimental stress-strain curves for the D2 and No. 3 bars (Fig. 3.11)
(52) and the characteristics of Grade 60 steel according to Hanson ,
show the Grade 50 steel ta have a more sharply defined yield point and
greater ductility than the prototype Grade 60 material, while the con-
verse is true of the cold-worked bars. Nevertheless, use of these
materials \'las dictated by their availability, and so they were
employed to model the reinforcement. In order to increase the
ductility of the reinforcement for the buttress model, aIl the
Stress (ksi)
4 1400~
3
2
l
o
2
l
Stress (MPa)
Figure 3.10
1000~
Ultimate 3900 psi 2.59 MPa
o ~ateral strain vs Longitudinal strain v = 0.178
o Stress vs strain
Experimental
Theoretical
(using E c
E = 3560 ksi = 2.46 GPa c
E = 3420 ksi = 2.35 GPa c 33~ = 33/140x3900)
cu
2000~
L,ongitudinal strain
Micro-concrete load-deformation characteristics
Lateral strain
300~
200~
10011
3-000~
-..J o
90
Stress 160
(psi) 80
1 50 70
60
140
50~ 30
40
30
20
la
a
Stress (MPa)
lOOOIl
Figure 3.11
--0- D2 bar, co1d-worked (as fabricated)
-0-- D2 bar, annea1ed (925C)
--Q-- No.3 bar, Grade 50
No.8 bar, Grade 60 (assumed characteristics)
2000ll 3000ll 4000ll
Reinforcernent load-deforrnation characteristics
. 50001l Strain
"'-l 1-'
72
rein forcing steel used in its fabrication was annealed at 925C
for a period of 30 minutes; the resultant lowering of the yield
point and the ultimate strength are evident in Fig. 3.11. The
three exper:i.mental stress-strain curves were obtained using strain
gages of the type used in the instrumentation of the models. The
elastic modulus for aIl coupons had the same value of 29.4 x (10)6
psi (20.3 G.Pa) •
Anchorage plates for both block and buttress models were
machined from high-strength steel specified to conform to the norms
for post-tensioned construction. The prestressing tendons con-
sisted of six 7-mm wires, specified to have a yield point of 250 ksi,
and thus capable of developing 100 kips (445 kN) at ultimate. Since
tendon loads were to be measured from a precise electrical load cell
rather than from deformation characteristics, no material tests were
deemed necessary in this instance. , I
3.4.2 Instrumentation
Instrumentation for aIl models consisted of electrical
resistance strain gages on the reinforcing bars. The gage type
and mode of application were uniform, as described in the following
specifications:
Manufacturer Tokyo Sokki Kenkyujo Co. Ltd.
Brand TML
Type PL-ID-Il
Backing f.laterial Polyester
Filament grid length
Coefficient of thermal expansion
Resistance
Gage factor
Adhesive
Moisture coating
Abrasion coating
Con crete pre-coating
73
10 mm
Il ll/C
120 ± 0.3 ohms
2.07
Micro-Measurement Eastman 910
Micro-Measurement Polyurethane M-Coat A
silicone Rubber 3145 RTV
TML Type PS
Application of the gages required removal of bar deforma-
tions over al-inch (25 mm) length; lead-wire connections, water-
proofing and other disturbances to bond action were confined to this
narrow region.
The readout instrument was a 3D-channel Budd Datran printing
strain recorder. Thermal effects in the lead-wires were compen-
sated by the use of a 3-wire circuit.
Gage locations in each model will be reported along with the
experimental results.
3.4.3 Block Model Fabrication
Reinforcing cages for the block models were assernbled by
tieing the individual bars to transverse bars of nominal size. The
cylindrical duct was formed from cardboard molds; stiff cardboard
strips were used as crack-initiators. The models were cast in ply-
wood forros, with the duct running horizontally as would be the case
74
in the buttress construction. The models were cast in three
lifts, with the first just reaching the level of the bearing plate
and the second just covering the top of the plate.
InternaI vibration was applied during the pouring of rein-
forced models 0-375, 0-375H, 0-375L, 0-160H, and reinforced models
1-375 and 1-375N. The small bar spacing in blocks 1-160, I-160N,
11-160 did not permit the use of the internaI vibrator; compaction
was achieved through external forro vibration for these three models
and block 0-160, aIl four of which were cast simultaneously from
the same batch of concrete. It was anticipated that model 11-375,
if fabricated like the other specimens, would exceed the testing
machine capacity, hence no compaction other than the rod-tamping pro-
d 'f' d b th S f l' d f'ab' , (41) ce ure spec~ ~e y e A TM or cy ~n er r~cat~on ; model
0-375L was also cast in this manner to give,a measure of the strength
of an unreinforced anchorage compacted without recourse to internaI
vibration.
Forms were stripped after 24 hours, and the specimen then
left to cure in the dry air of the laboratory until test time. AlI
0.16-scale models and specimen 0-375H were tested 14 days after casting,
at which time the concrete compacted by internaI vibration would have
attained a strength of approximately 5 ksi (3.45 MPa). AlI other
models were tested at an age of 7 days, so that the concrete would
have a strength of 3.5 to 4.5 ksi (2.41 to 3.10 MPa) depending on
compaction.
Concrete coupons were sampled in the process of casting and
tested on the same day as the parent specimens. The compressive
75
strength evaluation is based on the average of four cylinders of a
size comparable to that of the specimen and subjected to the same
compaction and curing conditions. For scale 0.375, 12 x 6 inch
(305 x 152 mm) cylinders were representative of the distance bet~een
faces exposed to moisture loss (outside surface to inner face of
circular duct) i 6 x 3 inch (152 x 76 mm) cylinders were appropriate
for thé small-scale specimens.
3.4.4 Buttress model fabrication
Reinforcement for the buttress models was assembled in two
steps: Grids were first formed by spot-welding the bars at aIl
intersections, following which the anchorage zone ties and vertical
bars were tied to the preformed cages. Final assembly of the rein-
forcing and prestressing steel took place within the formwork to
insure proper placement and tolerances. (Fig. 3.9a).
Sheathing for the prestressing tendons was made from slit
rubber tubing covered with watertight adhesive tape. The thin steel
cylindrical sleeve was welded to the back of the bearing plate, and
had a length of 6 inches (152 mm). Cardboard crack initiators were
placed in the forros where specified. (Fig. 3.5).
The buttress was cast in eight lifts, and internaI vibration
was applied throughout. Control cylinders were made ~ccording to
ASTM procedures. The specimen and the cylinders were cured under wet
burlap for seven days.
CHAPTER 4
BLOCK MODEL TESTS
4.1 TEST PROCEDURE
The block models were loaded incrementally to des truc-
tion under a concentric bearing load applied through a steel ring
designed to simulate the anchorage nut of the post-tensioning
system.
Two hydraulic testing machines were used: small-scale
models were tested in a 300-kip (1.335 MN) capacity cylinder press,
while blocks of scale 0.375 were loaded in a 400-kip (1.78 MN)
univers al testing machine. The load indicators of both machines
h d l 'b d' d 'th d (43) , a been ca 1 rate 1n accor ance W1 ASTM proce ures pr10r to
the tests.
Strain readings were recorded for each of the fifteen to
twenty approximately equal load increments until failure. Loading
between successive increments was applied at a continuous rate of 2
percent of the final load per minute. Application of load was
discontinued when damage to the blocks was such that the load-bearing
capacity was reduced to less than one-half the ultimate capacity.
4.2 EXPERIMENTAL DATA
The experimental data for the block models consists of
the ultimate loads for all specimens, and the transverse steel
76
77
strains for aIl reinforced specimens.
Data for the prototype blocks is limited to strain gage
readings at five load levels between Pd and 2.05 Pd. No strength
or deformation characteristics are available for the materialsi
it is known, however, that Grade 60 steel was used as reinforcement, ~
and that the concrete had probably not attained its 28-day specified
strength of· 5 ksi (3.45 MPa) at the time of testing. The specimens
carried without failure a load of 2.05 Pd' which represented the
maximum capacity of the testing machine. No qualitative assessment
~f the extent of structural damage is available.
Ultimate loads P and con crete compressive strengths f u cu
for the block models are reported in Table 4.1. The ratio P If A, u cu P
which provides a non-dimensional parameter accounting for the strength
of concrete, is included for subsequent discussion.
Strain gage readings for aIl prototype and model block
specimens are reported in Table 4.2. AlI strain values are recorded
-6 in dimensionless ~ units, i.e. (10). Gage locations are indi-
cated in the accompanying sketches. In model II-375, gages were
placed at the center of every bar in the wide and the narrow direction,
and at points aligned with the outer edge of the bearing plate,
selected to reproduce homologous points in the prototype II-I. Gage
locations in model I-375 were identical to those in prototype I-li
four additional gages were placed at the center of bars on the second
wide face of the specimen to obtain a complete set of data involving
every reinforcing bar in the more critical direction. In model
78
TABLE 4 el BLOCK MOOÉL ULTIMATE STRENGTHS
HOOEL REINF. MOOEL CRIICK Pu fcu Pu TVPE SCALE INIT • KIPS KN PSI MPA ÇUAp
0-160 NONE 0.160 YES 75.2 335 4940 3.41 1.530 Q-160H NONE 0.160 YES 72.8' 324 5183 3.58 1.410 0-375 NONE 0.375 YES 365.0 1625 4720 3.26 1.417 0-375H NONE 0.375 YES 396.0 1762 5164 3.57 1.400 0-375L NONE 0.375 YFS 282.0 1255 .4191 2.89 1.230
1-160 J 0.160 YES 88.0 392 4940 3.41 1.789 1-160N 1 00160 NO 80.0 356 4940 3.41 1.630 1-375 1 0.375 YES 360.5 1605 3669 2.53 1.800 I-375N J 0.375 NO 362.0 1610 3980 2.75 1.670
II-1bO Il 0.i60 YES 92.0 409 4940 3.41 1.1175 11-375 II 0.375 YES 400.0 1780 3714 2.56 1.970
MEAN VALUES OF P/fcuAp AT UL TI MATE LOIIO
REINF. CRACK SCALE SCALE GENERAL
TYPE INIT • 0.160 0.375 AVERAGE
NONE YES 1.470 i.369 1.397
J VES i.789 1.800 1.795 ND 1.630 1.670 1.650
JI VES 1.875 1.970 1.922
TABLE 4.2 TRA~SVERSE STEEL STRAtNS lAI BLOCK 1-160
P \i.:p) P /f~u.A,,1 0 l 2 3 4 5 6 7 8 9 10 11 1
2.0 1
005 -0', 000 004 000 -la -04 -05 1 0.041 1 .. 06 005 -Ob -04
4.0 0.082, .. 04 016 009 1)10 014 016 014 008 010 010 on4 tll4 , e.o 0.1631 006 n30 047 047 05!) 044 039 034 027 020 039 030 ;
12.0 0.2441 014 Cl6lo nno n90 079 060 ObO 054 050 050 OnB Cl78 1
16.0 0.3261020 n94 134 l'iO 175 07n 079 084 079 oaR 118 135 1
20.0 O,lo08.046 116 179 220 174 090 098 109 144 154 216 23ft ' 24.0 0,4901 080 14', 230 279 27.6 11<) 114 130 165 194 275 196 !
28.0 0.570, 14!l 1 ~', 270 34<) 290 15R ua 1!l0 234 240 335 360 32.0 0.6511 2111 22il 324 415 368 210 174 174 269 290 404 430 36.8 0.749 i 29/; 7.1l9 3'15 500 448 268 214 199 336 340 470 510 1
40.3 0,620 3(,4 336 4 /,4 560 506 32'1 2!>4 225 390 386 520 569 45.5 0.927' 499 44/, 569 690 !ll7 1,20 319 300 496 478 605 676 50.5 1.030 600 534 674 R04 686 490 370 354 578 556 679 764 55.0 10119 744 f:5A 818 950 7b5 59') 4;0 428 710 680 765 678 59.0 1.200 855 737 945 In92 846 655 469 484 814 788 837 968 63.0 1.262 1034 640 10118 1240 948 737 524 564 960 897 929 1120 67.0 1.3é311448 94:1 1174 1280 1032 816 595 650 10S8 992 960 1180 71.0 1.444 !1728 In5f1 1238 1;50 1128 900 644 729 1170 1100 1054 1270 75.6 1.538 i2356 1320 1480 1548 12613 1068 774 aa5 148B 1360 1348 1570 80.0 1. 6 26\3660 2Ob4 2000 1960 1660 1470 996 1100 1956 1530 1900 1976 84.0 1.710 4460 3070 2680 2500 2340 2150 1200 1228 2400 2720 2000 2280 88.0 1.795
1 - - ; 3600 7160 3576 1520 1100 1034 3316 1200 1232
TA8LE 4.2 TRA~SVERSE STEEL STRAiNS (BI BLOCK 1-160N
P (t.;?) P/Ç,ut'\ 0 1 2 3 4 5 6 7 8 9 10 11
2.0 0.041 019 015 020 027 oïo 010 009 010 025 004 014 017 4.0 0.082 046 03<- 037 045 020 026 020 020 035 015 020 025 1 B.O 0.163 060 1157 059 064 029 044 035 034 047 030 040 M7
12.0 0,244 086 ('\90 O'JO 104 048 069 059 045 058 050 064 059 , 16.0 0,326 110 124 128 140 074 104 050 067 064 060 OR4 076 1
20.0 0.408 149 16f- 167 IB4 114 144 117 097 075 079 . 104 095 , 24.0 0.490 196 224 214 218 154 177 167 139 066 109 135 114 26,0 0,570 255 ,et.; 267 '-66 205 2211 214 184 104 140 174 148 1
32.0 0.651
1
32b 3f:9 330 319 264 267 260 228 ! 127 180 229 106 36.0 0.732 420 47r) 430 ;64 330 320 304 295 : 170 247 300 264 40.3 0.8201 518 57~ 534 460 408 37n 340 364 240 346 436 390 1
44.0 0.896 659 726 6'19 bl0 510 439 3P,4 466 438 600 n8 ft65 48.0 0,978 790 A50 844 754 620 4<;5 40B 544 545 714 {l44 798 50.5 1.03e 924 954 937 1;55 700 536 440 620 630 799 944 867 55.0 1.119 1068 In51 1032 972 RI0 600 475 689 739 'lOO 1068 1000 59.0 1.200 1216 112P. 1120 1('\80 958 677 517 745 876 1032 1040 1114 113.0 1.262 138a 122q 1212 1236 1136 77R 569 fl14 10b8 115a 1096 lC96 67.0 1.363 1574 133~ 1348 1416 1320 916 62B 928 1254 1280 1168 1170 71.0 1.444 '1840 14bH 1528 16601150a 1070 729 1066 144B 1510 1334 1320 75.6 1.53a j2180 1624 1856 201611734 1300 840 118011748 2016 1768 1708 : GO.O 1.b30 u~T ~ 1
- __ 1
-Cl 1... ___ .;-j ____ ..J
=====~:::~:~::_~~~==== -----~---~~-----------------~---'l~-----------
i 1
1·
_u '- ____ l ____ -.J
41
=====~===~::= ______ .=1 ______ _ 'lI -------ç-----1
i 1 i
~
I...---ëi----..J ---------~Ï--------------~----~J---------------------r---------
--------~l------------i 1 j.
-.J 1.0
TABLE 4.2
PC<.l!') P/LAp
2.0 0.041 8.0 0.163
12.0 0.244 16.0 0.326 20.0 0.408 24.0 0.4 90 28.0 0.'510 32 .0 0.651 36.8 0.749 40.3 0.820 45.5 0.926 50.5 1.030 55.0 10119 59.0 1.200 63.0 1.282 67.0 1.363 71.0 1.444 75.6 1.538 80.0 1.628 84.0 1.710 88.0 1.795 92.0 1.875
TRANSvERSE STEEL STRA1NS (C) BLOCK 11-160
0 1 2 3 4 5 6 7 8 9 la 11 12 13
020 OZ:l 024 n20 020 02'+ 014 010 005 006 017 -05 014 004 039 ::5'. 056 058 054 057 044 030 024 054 057 044 054 050 054 077 100 089 075 0"8 064 0't8 036 087 080 080 087 075 059 n99 100 119 100 100 OCB 067 047 124 ln 124 117 106 074 17.(; 128 147 125 12R 110 086 1 064 159 165 170 150 149 0')4 \ 5'. 155 180 154 154 135 106 078 188 210 208 190 176 125 1C):) 1 ')9 234 198 19B 168 145 109 210 228 244 217 204 170 245 269 327 300 289 245 205 146 249 266 284 255 248 230 304 350 434 405 39f1 340 310 210 310 306 339 310 296 267 347 374 470 455 447 384 364 256 357 349 376 344 334 329 39il 427 548 534 524 459 459 345 436 398 426 387 398 416 455 480 618 610 600 524 548 446 53/. 464 470 414 474 528 520 540 697 687 669 578 630 569 640 534 534 449 520 637 57,-+ 604 766 748 734 618 689 680 734 600 580 489 580 729 645 664 830 806 789 650 748 789 824 659 620 520 640 878 74e; 790 940 904 877 6130 814 959 949 715 684 570 724
106$1 ë5f> RH6 11)80 1012 998 7(.4 880 1148 1120 7130 754 610 778 1250 1020 1020 1214 112B 1112 815 978 1336 1292 859 825 656 816 1620 122~ 1276 1520 1318 1300 976 1180 1514 1510 952 908 688 830 2024 1~16 15RB 1956 1660 1608 1180 1468 1940 1768 10 BO 1108 748 857 2308 1730 1874 2280 2000 1900 1348 1640 2944 1944 1256 1540 844 878 2616 1792 2388 3152 2240 2136 1600 1832 3628 2428 1104 2916 ï428 1010
------ -
Q
---------~ ----------L----i·----...J < ----------- ----------
----------~ ----------
=--===~=~l-=-~~~----~~-=-= =-:.=-.::-:.~4====:: __ .-------2 f ----------
--
_.0_ L----
t' ____ ...J
------~ ------q ------
======;~------------~t----~:~~ ===~~~~-----------~~. -----
15· _____ _ ------
1
i
14
014 050 064 039 120 144 164 197 260 289 344 410 456 509 564 648 694 727 734 7,+5 754 840 -
15
005 034 059 084 Ï14 ;35 157 245 l38 766 324 380 438 480 514 584 634 677 714 745 760 840
CD o
TABLE 4.2
P(\..d P/;CJAp!
Il 0.055 1 44 0.Z22 ,
67 0.334 89 0.443
112 0,560 134 0.668 156 0.777 178 0.887 Z05 1.021 235 1.171 264 1,315 281 1.400 306 1.527 329 1.640 351 1,750 360 1.800
TRANSvERSE STEEL STRAtNS ( Dl
0 1 2 3 10 Il 12 13
006 r.O:l ooa -14 006 010 ~16 ;'04 004 r.15 010 008 076 055 020 017 020 C'l4t) 037 0Z9 110 OflO 05', 046 058 074 078 060 14'1 115 080 068 088 1ln 109 100 190 137 096 085 214 275 264 25'. 320 156 104 094 295 360 3'.7 340 314 170 120 100 396 474 454 449 384 195 170 120 494 576 548 538 484 225 230 159 734 75" 674 590 650 364 559 287 900 1128 776 654 859 454 675 380
1090 Q14 916 732 1036 50R 719 lt36 1240 1080 1078 609 1272 57A 879 499 1354 1200 14i2 1148 1498 654 947 574 1590 1652 1848 1518 2056 770 1270 704
ULT --- -_. --- - -
o ~----r---~
----:----~T---~~----------~----~r_---g-----
----~-----~---~6_-~-------~-~-~lr-~---
i 1
\-
BLDCK 1-375
20 21 22 23
020 020 016 -15 088 080 OilO OZl 144 13 l • 158 090 366 440 '.79 480 475 526 568 565 614 635 685 670 754 739 809 774 899 R56 964 R86
1130 1008 1140 960 1400 1228 1330 1154 1696 1520 1568 1320 1952 180B 1760 1476 2340 2360 2016 1578 3072 2992 3864 1792 5400 4200 4600 2136
_.-
-c> L. ____ ~----...J
1~~---~:1---:~-=:===~;è===~;:= 1:s----'lè----:~-
1
1
i i
4 5
004 -10 006 -06 018 008 047 029 080 069 164 060 210 084 268 117 274 175 300 460 324 529 338 560 318 586 334 580 344 551
- -
6 7 8 -9 1
14 15 16 17 18 19 1
-09 :15 -06 Ooo·l':'iO 004 -14 -07 ':'09 006 1 -10 :'15 -10 Ono. OM OIS -05 009 -07 ':'37 1 016 :'04 -07 000 ! 007 036 014 035 010 ':'50 ! 030 010 010 000 016 068 060 084 024 ':'30 !
045 045 017 000 020 117 127 144 075 ':'30 070 080 040 000 040 206 186 218 135 ':'19 104 i04 074 000 069 278 254 314 198 .. 10 160 ï34 118 000 087 357 338 405 250 018 226 ï60 178 000 120 495 467 529 304 070 487 i08 327 000 244 678 735 808 435 i67 510 088 304 000 354 884 1003 1072 735 294 488 ïoo 452 000 480 1076 1232 1338 960 435 440 094 445 000 646 1256 1498 1650 i268 629 398 078 435 000 800 1420 1760 2040 ï656 788 008 :'27 408 000 1056 1780 2160 2976 2356 1200
- -_. _.
<> . ,
~----~---~ ---------;y~------------------;J-------------------- ---------------~---!3T--------·---. -
i 1
1-
- -
Cl) 1-'
TABLE 4,2 TRANSVERSE STEEL STRAINS (E) BLOCK 1-37.5N
P(l--iF) P/\c.u,\ 0 1 2 3 4 5 6 7 B 9 lÔ 111 1
11 0,050 ... 06 -04 014 008 . 004 -09 037 000 009 005 010 0061 45 0,207 117 030 056 044 056 016 055 014 024 034 037 040 1
b8 0.313 165 05'. 0116 074 180 037 065 028 044 060 064 074 90 0.445 215 f180 115 105 350 OSA 087 040 066 089 090 104
112 0.516 296 138 194 176 384 080 104 055 094 134 144 . 174 133 0.613 397 217 340 309 415 110 139 074 284 394 450 544 154 0,710 490 7.94 449 420 455 134 150 104 334 475 540 640 178 0.820 550 38:1 558 53'. 490 160 184 124 395 554 624 734 205 0.945 738 506 697 670 540 196 220 145 477 638 716 816 235 1.085 890 644 878 829 606 240 254 174 574 740 826 914 266 1.225 1070 RO~ 10n 1068 690 292 300 200 694 848 916 lC120 2Bl 1.295 1252 952 1318 1232 774 380 372 216 828 950 1000 110B 306 1.410 1250 1134 1608 1320 910 490 500 268 1030 1148 1168 1292 351 1.615 1740 1572 2600 1944 1036 820 800 476 2216 2152 1864 2216 362 1.710 1956 2456 5090 8780 2080 1550 1240 1000 2970 2640 2270 9999
..!.J. 1... ___ ;;-j ____ ...J
----------~----------Ii.~
==~~~=:===~~=~==~= ---------~T-----------..
i 1
1·
--
D
_~:::~4-----1
=====--~~===~~~~ ---'I=--=
1
1 i
D I...---ëj----...J
----------~-------------------~~----------
• 1:>
=========~JC=~~~~~== . '. l 1
1
1·
CP 1\)
TABLE 4.2
P (~:::) P/~::J.A?
11 0.05'.
1 45 0.222 67 0,330
1 89 0,438
112 0.552
1 t;~ 0,600 0.760
178 0.876 205 1.010 235 1.156 264 1,299 281 1.362 306 1,506 329 1.619 351 1.728 3ï3 1.835 396 1.950 400 1,970
P(i-.;;:) P/Çc\.\A,
11 0,054 45 0,222 67 0.330 89 0,438
112 0,552 134 0,660 156 0,766 178 0,8Bb 205 l,DIO 235 1.156 264 1.299 281 1.382 306 1.506 329 1.619 351 1.728 373 1,835 396 1,950 400 1,970
TRANSVERSE STéEL ~TRAiNS (F) BLaCK II''375
0 1 2 3 4 5 6 7 8 9 21 23 25 27
... 30 014 010 017 OOR 014 004 000 -29 -09 009 000 ':'24 -10
..,27 n2'. 0?4 n17 016 O'tO 028 000 -40 019 014 014 -15 010 -06 C45 060 037 1)56 Ob4 057 000 -26 025 014 019 -16 0()4 .. 04 n6C'1 064 050 064 070 074 000 -27 024 010 027 010 024 1
104 18n 2t.9 230 254 274 - 000 005 055 036 068 015 040 . 310 24:) 364 314 34 f, 384 404 000 055 109 070 088 050 064 184 27n 419 300 384 454 490 000 078 134 105 114 090 084 234 :20 516 467 450 534 596 000 110 190 170 158 134 114 355 3Bo 614 550 530 61A 7ï7 1)00 150 254 229 208 220 114 405 444 710 640 619 714 830 "000 174 309 264 235 310 284 460 520 796 706 705 796 940 000 164 319 308 248 360 380 524 5713 870 81', 991) 858 1050 000 139 204 340 256 410 455 624 669 968 850 944 960 1208 000 104 245 370 264 467 524 868 904 1268 1120 i236 1248 1698 000 069 270 414 316 519 674 970 102('1 1308 1220 1360 1348 1652 000 066 314 447 380 624 769
108S 1150 1534 1340 1480 1430 2000 000 044 355 449 440 709 820 1228 1276 1450 1292 1618 1554 - 000 028 440 338 1,60 648 820 2336 3520 6900 9999 2544 2624 2416 000 -210 -146 020 114 398 484
10 1\ i2 13 14 15 i6 17 18 19 " 20 22 24 26
004 .. 04 018 014 014 005 014 -10 000 004 .. 30 006 -3:4 000 040 039 048 048 0'.0 029 048 014 014 018 015 004 010 026 049 07~ 0<)0 080 070 057 060 030 014 005 014 020 007 036 076 096 0<)7 099 100 076 070 027 018 024 024 027 020 039 129 149 155 139 118 110 094 067 076 060 -16 050 040 058 175 18 t• lR8 166 145 130 124 099 116 118 036 089 080 066 228 224 27.6 180 180 149 146" 108 174 164 066 124 114 095 310 -;,97 274 214 204 170 164 137 264 230 090 196 170 119 384 351:' 37.0 247 268 216 198 164 357 289 134 254 245 178 498 424 414 296 315 264 250 200 464 360 254 324 31'7 234 604 50'~ 500 364 427 285 326 290 566 480 345 418 394 304 698 524 529 ',00 497 314 396 300 620 564 380 504 440 360 871) "05 624 470 5'j8 397 4 /,0 320 714 640 504 580 510 429
1168 784 7 r}0 590 716 504 604 1,10 824 838 R08 819 694 626 1250 1140 840 640 8Q7 567 674 444 854 916 894 977 794 758 1398 916 916 705 856 610 730 489 908 978 9R4 1140 920 995 1626 115/] 10' .. 8 768 900 69Z 780 520 1048 994 1120 1918 i020 1148 1556 98C 895 1976 1620 809 1528 980 ~040 2400 9999 9280 6000 3272
--------_ .. _--
1
!
1
{:}
'-____ ; ____ J
----B----~f---;-----------~---~----e---------------~T---------------------Lf--~~---------;----: ---;;-----=--~-=-~-=-~-=-~~==~=== --~------2f---u~----
Q
L----
t" -----'
-;----~ ---;0---~---~i :----e-
\; 22 ------- -----e-a -------
19 IS 24 -t>---~~ 1 ----~
-----~~----;--------:!t----~
1 i
CD w
TABLE 4,2 TRA~SVERSE STEEL STRArNS (Gl BLOCK 1-1
P',~:;i P/.c,"A? 1 2 1 3 4 ~ - 6 1 7 8 9 10 1 1 • 1
1460 0,939 215 12t' 1640 660 630 83511 195 215 245 ~ 16::0 1,028 1240 135 705 720 bAO 890 210 235 270 -20~0 1,265 39n 22~ ~~O 925 B95 1315 375 430 4bO .. 2500 l,60S 615 56~ 1295 1470 1450 .. 665 925 B40 ~ 3000 l,9_2_~ 15~ It'):~890 9:99 91~",---~15 1670 1465 -
11 12 1 13 14 15 16 17 lB 19 201
1460 0,939 120 100 290 410 455 . - IbO 215 220 045 1
1600 1,028 130 111 320 445 490 ft 180 230 240 050 , 2000 1,285 185 165 430 bOO b40 .. 2AO 350 325 OB51 2500 l,60S 265 285 5tO 820 9B5 - 470 620 595 205 3000 1,927 390 660 1065 2385 2470 .. 990 985 2325 675 1
--
TABLE 4,2 T~ANSvERse STeEL STRAINS (Hl BLOCK ll-1
Ft',,;;» P/ÇcaA; 1 2 3 4 5 6 7 8 9 10
1460 0,939 lbS ()ln 425 500 600 591' 175 305 260 .. 16~0 1,028 180 010 470 545 650 635 190 350 280 .. 2000 1,285 260 MS 600 740 855 870 275 490 405 .. 2500 l,b05 525 170 BaS loBS 1245 1370 460 B45 770 .. 30:l0 1,927 1135 '300 1325 1755 .. 2530 990 1535 1450 -
- - ---
11 12 13 14 15 lb 17 18 19 20
1460 0,939 085 140 130 160 315 465 105 150 130 170 1600 l,02e 090 15~ 145 175 330 500 115 160 145 180 2000 1.285 150 20:1 195 235 4~5 595 155 235 210 255 2500 1,600 2<;5 265 280 330 .. 950 260 460 370 535 3000 1.927 725 715 490 490 .. 2410 410 1050 720 1485
f cu
4 ksi 2.76 MPa (assumed)
~----~---~ ----;----~i--2~~----------5-----1----&-----. . ---------- ----~-----1 1 T , -----~~lr----&-----
1
!
Q
~----i----..J
===l~~--~ï==ï~:==== ---------~--;------------------r----&----------~----; ---~-----------e---- ---~-----
===--=--=--=--=--_-_Lt-=--=--1.-;:.::.:::
1
i
D L ____ 1. ____ .J
____ 2~~---!1..;_
l~_EJ_-_...:l.s_-~!..s_ 1" .:. te _
1~::==2~r===ï~:= 1
1
1 i
D '-____ 1 ____ ..J
~;:~==-;.t~=:î:= ===_~~=1~: ~-----;~~---;;--e----q,----_e_ ------._-----
Il 1 17 -----'?----o&-
1 i
CD ~
85
1-375N, gages were placed at the center of bars only, on two
wide faces and one narrow face. Due to size restriction, strain
gages on bars in models of scale 0.16 were mounted only at the
center point. AlI the bars on the wide face and the narrow face
were instrumented in model 1I-160; both wide faces carried strain
gages in models I-16 and I-l60N.
Load increment values were set from similitude considera-
tions. In the prototype structure under investigation a design
prestressing force Pd = 1460 kips C6.50 MN) is obtained when the
tendons are stressed to 0.6 f • su
Load levels in the block models
were chosen to give approximately the same nominal pressure over
the plate area, and therefore the same P/Pd ratio, in the models
of various scales and the prototypes. For the purpose of taking
account of the variability of concrete strengths when reporting
the data, the ratio Pif A was introduced in the tabulations, cu p
where f and A are the concrete compressive strength at the time cu p
of testing and the net area of the bearing plate respectively.
4.3 GENERAL BEHAVIOR
The mode of failure of the block specimens is best
exemplified in the photographs of Fig. 4.1. Failure occurred
when diagonal cracks originating from the corners of the bearing
plate propagated down tlle faces of the specimens, which resulted in
spalling of the concrete cover. This failure mechanism was the
same for aIl specimens, whether reinforced or plain (Fig. 4.1d,
. __ .
(a) Black 11-0.375, at ultimate laad
-' ;/
1 1 1
/,
1
..i, ' ,::~}:_!. '
.--,!I..-.~-"*,,.,
(b) Black 11-0.375, past-failure
Figure 4.1
fi ~ i ,;. , ,! <
; , .. ~.
.~,(
! 1 f
l ~ l
.. 1 1
f - -
(c) Black 1-0.16
(d) Black 1-0.16, 1-0.l6N, 0-0.16
Black madel failure mechanism
co (j\
87
unreinforced specimen 0-160 at right) . In unreinforced specimens,
spalling led to immediate failure, while in reinforced specimens
the appearance of the destructive cracks closely preceded attain-
ment of the ultimate load P , occurring near 0.85 P • u u
The angle
of the shedded wedge of con crete was between 15 and 30 degrees
(TI/12 to TI/6 radians), to the vertical.
Considerable ductility and plastic flow was displayed in
reinforced specimens as load application was maintained beyond the
ultimate capacity. (Fig. 4.1b shows 0.75-inch [19 mm] settle-
ment of bearing plate in specimen II-375). Spalling of the
outer layers left a pyramid-shaped core around the duct, held to-
gether at its top by the binding steel. In unreinforced speci-
mens, absence of the restraint of lateral rein forcing steel led
to rapid crushing of this underlying con cre te pyramide The
appearance of midside cracks at early stages in the loading, at
loads between 0.4 P and 0.5 P, was of no consequence in the final u u
failure mechanism. Midside cracks appeared along the longer faces
independently of the presence of crack initiators. On the narrower
faces they occurred sporadlcally in models with crack initiators, and
did not occur in models without them.
88
4.4 DISCUSSION OF ULTIMATE LOADS
4.4.1 Ultimate Loads of Unreinforced Blocks
The five values of P If A for unreinforced specimens u cu p
have a mean value of 1.397 and coefficient of variation 0.0690.
Reformulating the ACI-3l8-63 bearing capacity equation, without
inclusion of the safety factor 0.6, yields
P If A = u cu p 31 A lA
cp P
In the present case, A is the net area of the 32 x 48 inch cp
(813 x 1219 mm) face of the specimen, and A is the net area of the p
22-inch (559 mm) square bearing plate, both having a concentric
Il-inch (280 mm) diameter hole. Introduction of these areas into
the above equation gives a value of 1.391, which is remarkably close
to the mean of the experimental results (error within less than
0.5 percent). The 7 percent coefficient of variation is no larger
than that generally found in the compressive strength of concrete
cylinders cast from a single batch of concrete, and is therefore
acceptable.
The safety factor of 0.6 used for design purposes in this
working stress equation is equivalent to the use of a load factor of
1.5 and an undercapacity factor of 0.9. This choice of load factor
is justified, since prestressing forces can only vary within a narrow
range, and can thus be taken as dead loads. The undercapacity
factor of 0.9 appears suitable because
(1) the experimental and theoretical background for the formula
. (18) have shown it to be conservat1ve ,
89
(2) erratic design assumptions (loads, areas, etc.) are unlikely, and
(3) failure of the anchorage zone during post-tensioning does not
entail structural collapse.
The ACI Code equation for the bearing capacity of plain con-
crete is therefore applicable.
4.4.2 Ultimate Loads of Reinforced Blocks
The mean values of the ratio P If A for the reinforced blocks u cu P
are given in Table 4.1. Excellent agreement was obtained between the re-
sults for models of various scales, as individual tests gave values which
deviated from the mean by approximately ±l percent. The provision of anchorage
reinforcement, in addition to increasing the ultimate bearing capacity of the
end block, had the beneficial effect of decreasing the variability of the ob-
served strengths, because the increase in lateral restraint in the anchorage
rendered the overall strength less susceptib~e to local variations in the
concrete.
The absence of crack'initiators decreased the strength of blocks
in series l by a factor close to 10 percent. A possibie explanation is
that the absence of crack initiators precipitates the early formation of
spalling cracks, rather than midside splitting cracks. However, since the
mode of failure was the same in aIl models, irrespective of the presence or
absence of either crack initiators or reinforcement (Fig. 4.1), this expla-
nation is not thoroughly satisfactory, and further study on this matter
may be necessary.
Although a definite judgement cannot be based on the investigation
of only two steel arrangements, it seems that a limit of diminishing
efficiency is approached, with the use of the steel arrangement of details
l and II. The spacing of bars in detail II gives minimum practicable
distances for adequate placing and compaction.
90
Increasing the number of bars at the remote end of the anchorage
would result in an inefficient force transfer to these bars.
Zielinski and Rowe(3) noted that the addition of reinforcement at
a ratio of local bearing pressure to cube strength of 1.9, (which
is the case here with detail II giving P If A = 1.922) did not u cu p
result in an increased bearing capacity. Detail II therefore
seems to be an upper limit to the effective amount of lateral re-
inforcement that can be used in the anchorage zone.
An ultimate strength approach to the design of beam
anchorage zones can be conceived, wherein an experimentally deter-
mined factor could be introduced in the bearing capacity equation,
to account for the percentage of lateral reinforcement. Curves
such as that of Fig. 4.2 (drawn from the three available points and
suggested as an illustration only), could be obtained from test data,
for various bearing plate, end block and lateral steel configurations,
and subsequently used to calculate the anchorage zone steel area for
a given prestressing force.
Following the reasoning explained earlier, the ratio of
the design ultimate capacity P to the maximum initial bearing load u
in post-tensioning operations P., should be attenuated by a load 1.
factor of 1.50 and an undercapacity factor of 0.9, which corresponds
to requiring P IP. = 1.66. u 1.
In keeping with CPCI requirements for
experimental determination of anchorage zone reinforcement adequacy,
the ultirnate load in a test to failure should double the effective
design prestressing force, i.e. Pu/Pd = 2.00.
2
~P! ::1 u
lI-l "-III
'{j tri s:: Q) I-l .jJ Ul
Q) .jJ 1 rd S
'r-! .jJ r-! ::l'
r-! rd s:: 0
'r-! Ul s:: Q)
S 'r-! rel 1 s:: 0 z
0
0
91
TABLE 4.3 E~TjHATED ULTIMATF BEARiNr. CAPACITIES OF END BLOCKS
REINFORCEMENT NO/JE TYPE 1
fcuAp * KIPS i945 1945 HN 8.B5 8.85
puJr,u.Ap !le * 1.397 1.795
Pu KIPS :n'~o 3490 HN 12.0 15.5
Pv./~ i.B6 2.39
Pu/ft i.39 1.79"
* DESIGN DATAi ~iÇ ~u c 5 Ksi ~ 3.45 MPA ÀpD 389 IN = 0.251 M ~ = 1945 KIPS • B.BS HNI AT ~ (1 1460 KlpS "' 6.50 MNI AT
TYPE II
1945 8.85
1.935
3780 1b.8
2.59
1.94
** EXPERIMENTAL MEANS FOR END BLDCKS WITH CRACK INITIAToRS
1
II
2
Ratio of lateral steel area to area in Detail l
Figure 4.2 Ultimate strength graph
92
As an illustration of the use of ultimate capacity data
in design, Table 4.3 shows the predicted ultimate bearing loads
of anchorage zones reinforced with Type II, Type l and unreinforced
details, assuming a specified concrete strength of 5 ksi (3.45 MPa).
Specimens with detail l are seen to satisfy the criteria Pu/Pd = 2
and P /P, u ~
1.66, without being excessively conservative.
A design on the basis of ultimate strength alone may not
satisfy serviceability at working load levels, if excessive crack
widths or permanent deformation in the steel occur. The next
section investigates these working stress considerations from
strain data.
4.5 DISCUSSION OF STRAIN DATA
4.5.1 General observations on steel strain data
The transfer of lateral force to the transverse reinforcing
bars took place from the onset of loading. The graduaI rate of in-
crease in bar strains was related to the progressive splitting mode
of cracking which initiated in the regions of high tensile stress
and propagated towards the outside surfaces, where cracks were visible
at loads of approximately 0.5 P , where P is the ultimate load sus-u u
tained by the specimen. Bars were not uniformly strained at low
loads, hence points of peak tensile stress were distinguishable.
As loading increased and cracking radiated from the point of maximum
tensile stress, the distribution of bar strains became more uniform.
93
When the yield point of the reinforcement was attained, or when
inelastic steel deformations became predominant in the work-
hardened D2 bars, cracks associated with the spalling mode of
failure became visible on outer surfaces. Failure by spalling
occurred when plastic flow in the restraining steel allowed ex-
cessive lateral deformation of the anchorage zone concrete.
On reaching failure, aIl bars paraI leI to the wide faces
yielded, in details of both types l and II. In the narrow direc-
tion, yielding took place in the group of bars closest to the bearing
plate; this yielding occurred as a r~sult of failure by spalling in
the long direction.
4.5.2 Strains under maximum initial prestressing force
From the available strain data, inferences can be drawn
regarding the strains in lateral rein forcing bars of a prototype
anchorage block having the same characteristics as the models under
study. The product fA, for a plate area A cu p p
and a concrete strength f = 5 ksi (3.45 MPa) has a value of 1945 cu
kips (8.65 MN) which is identical to the force applied by pre
stressing strands with a cross-sectional area A = 9.75 in2
(62.9 cm2
) sp
tensioned to 80 percent of their ultimate tensile strength
f = 250 ksi (172.5 MPa). su
Thus for the prototype structure at
maximum working load, P./f A = 0.8 f A /f A = 1945/1945 = 1.000. ~ cu P su sp cu P
AlI strain readings at the load level nearest P/f A = l were, therecu p
fore, extracted from Table 4.2, and reproduced in Table 4.4.
94
TABLE 4.4 TRANSvERSF. STF.EL ~TRAINS AT MAXIMUM INITIAL PRESTRESSiNG FORCE
SPECIMEN
WIDE FACE bp/bc = 0.458
11-160 11-375 Il-l
1 .. 160
1 .. 160N
1 .. 375
1-375N
1 .. 1
NARROW
11-160 11-375 11-1
1 .. 160 1-160N 1-375 J-375N 1-1
SIOE i 2
SIDE 1. 2
SlOE ï 2
SIDE l 2
FACE ~/bc JI 0.688
1 1 1
---------~.----------L.---r-.----...J ---------- ---------_____ . ____ .1 _______ ~- .
-----~--~~lr-----------
i 1 j. 1
0 1 2
416 455 480 355 3BO 614 .. 635 .. 600 5~4 674 578 556 679 924 954 937 630 799 944 494 576 548
1130 1008 1140 890 644 878 574 740. 826 890 680 720
446 534 464 384 350 320 .. 500 .. 686 490 370 700 536 440 484 225 230 606 240 254 .. 455 410
3 ft 5 6 7 P/fcuAp
618 610 600 524 548 ï.030 550 536 618 717 - 1.010 650 - 545 - 470 1.028
804 i.030 764 1.030 855 1.030 887 1.030 538 1.025 960 1.025 829 i.085 914 1.085 705 1.028
470 414 474 410 380 1.030 247 268 216 198 164 1.010 330 op 175 .. 145 1.028
354 ~.O30 620 1.030 159 1.07.5 174 1.085 290 1.028
1 . L ___ ;. ____ -'
---------~ -------------------- ----------__________ 2 _________ _
-----------Jt----------
-=~~~~~~ii=========: ~--------JI----------
. i .1.
95
Summary examination of the table reveals that detail l
is adequate for b lb = 0.688, and detail II is satisfactory for p c
b lb = 0.458. P c
In the wide direction (b lb = 0.458), maximum p c
strains in the bars of detail II were close to 680 ~, while those
in detail l could be as much as 1140 ~ and generally ranged from
800 to 1000 ~. In the narrow direction (b lb p c
0.688), detail II
is conservative, while maximum strains in detail l just attain the
permissible working stress level.
Peak strain in the wide-face rein forcement occurred near
the fifth bar in model II-160, near the seventh in 1I-375, and in
the vicinity of the third bar in models of type I. The maximum
bar strain in the narrow direction occurred in the bar nearest to
the bearing plate. These results will 'be given further considera-
tion in the next section.
4.5.3 Attainment of maximum allowable stress or yield point
in one bar
Working stress design requirements generally limit the
allowable stress in lateral rein forcing steel, such as ties and
stirrups, to a value of 20 ksi (13.8 MPa). It was pointed out
in Section 3.4.1 that aIl types of reinforcing steel used had an
elastic modulus E s
29.4(10)6 psi (20.3 GPa). Thus the allowable
stress level, at any length scale, corresponds to a strain of 680 ~.
The yield point strain in the No. 3 (10 mm) bars had a sharply de-
fined value of 1840 ~ at 54 ksi (37.2 MPa) , while that in the D2 bars
was taken to be 2000 ~, the strain corresponding to the proportional
limit of the material.
96
From the strain gage data, it is possible to determine
the non-dimensional load P If A at which one of the bars reaches cu p
either the 680 ~ threshold or the yield point. Table 4.5 lists
the load and the position of the bar at which the critical strain
level was attained, in each instrumented face of every specimen,
as found from interpolation between recorded strain values. Where
several bars were within 10 percent of the critical strain, aIl of
their positions were noted in the table, and the location of the
most heavily strained was marked with an asterisk.
From the location of the most critically strained bars, it
is observed that the highest transverse tensile stresses occur at a
distance from the bearing surface which differs somewhat from that
predicted by classical theories. Use of Iyengar's graph of
x{max f ) vs. b lb (Fig. 2.10a) gives, for the wide and narrow z p c
directions respectively:
x{max f , b lb z p c 0.458) = 0.42 b = 0.42 b 10.458
c P 0.917 b
P
x{max f , b lb = 0.688) z P c
0.47 b = 0.47 b 10.688 = 0.682 b c P P
The distance 0.917 b falls near the sixth bar of detail II and p
slightly beyond the fourth bar of detail Ii fair agreement is
therefore seen between the the ory and the experimental results in
the wide direction. In the narrow direction, 0.682 b coincides c
with the third bar of detail l and the fourth bar of detail IIi it
was found experimentally that the first bar underwent significantly
higher strains than aIl others, in almost every specimen. The strain
distribution at Pif A = 1.0 (Fig. 4.3) shows to what extent the discu p
tribution of strains in the bars is different from that in the concrete.
97
TABLE 4.5 ATTAINHF.NT OF CRITICAL STRAJN LEVEL IN ANV BAR
bp bp
CONDITION SPEC ItIEN WIOE FACE bc -OAsa NARROW FACE b=O.6BO c.
CRITICAL L CRITICAL P BARS fcuAp BARS Fc.uÀp
ALLOI/ABLE Il':'160 3.4·.5 1.14 0.1*,2 1.17 WORKING Il .. 375 2.5,6* 0.99 0 1.37 ST~ESS 1-160 2.3* 0.92 0 1.02 LEVEL 213* 0.93 IN A'IV 1-37, 0'1·.2.3 1.14 0 1.19 BAR CI, b2*,3 0.66
1-160N 0,1·.2.3 0.87 0 1.02 1 .. 2*, 3 0.118
f 1-37!1N 0 •• 2,3 0.93 0 1.21 S 2,3* 0.75
VIELD POINT ;1':'160 0* .. 3 1.70 0 1.72 OR Il .. 375 6 1.72 0.3* 1.96 PROPORTIONAL 1-160 0 1.51 0*,1 1.67 LJHIT 0*.3 1.65 LEVEL 1-37' 2 1.75 0 1.72 IN ANV 0*,1 1.37 BAR l-l60N 0*,3 1.51 0 1.63
f 1 1.55
i-~75N 2 1.50 0 1.69 Y 0.1,3* 1.55
.(} D 1
. L----l----..J
~~~~:~~~~~~~~~~~ ---------~f-------------------- --------------------~t----------~~-------2r----------
i
L.----t----....J ---------~ -------------------~ ---------_____ • _____ '1. _______ ~--
-----~----3T-----------..
i 1 i
MEAN VALUES OF P/~uAp AT CRITICAL STRAJN LEVELS
CRITICAL REINF. CRACK NIIRr.OW WIOE
CONDITION TVPe INIT. FACE FACE
f 1 VES l.io 0.91
s NO 1.11 0.86
J J VES 1.27 1.07
f 1 VES 1.69 1.'7
NO 1.66 1.53 Y
JI YES 1.84 1.71
ULTI~ATE 1 YES 1.80 NO 1.6"
Il YES 1.92
98
c) - - 0-__ 0
Wide face b lb = 0.458
P c
Bar locations 1-- correspond to
x-coordinates in the·graph
(Detail 1Πshown)
al +' al J..I U s::
1200~ . 0 1-160 ID 1-160N 0 ...... 0 0 Reinforcing U Pl
0 1-375 • 1-375.N s::~ bar strains 1-1 or! Pl
Ul 0 • 1.0 J..I 1000~ 0 Q II-160 . Ul4-l
III CI il al 0 .Q III .,
II-375 Ul t ., ID le Ul Ul
tri ID • 11-1 al s:: s:: ., " J..I 0
-ri 800~ li! 0 0.8 +' or! U 0 Ul+J ., J..I • • oc U 0 • fi concrete, -Zielinski al III
4-l • G> • r-f J..I s:: t!I x Q ~ stresses and Rowe 0.6 or! 4-l or! 600~ c v Ul al 61 0 s:: Ul J..I 8
Ox Q --Leonhardt al III 0
'Q +J s:: Q • 'tI or!
400~ v 0.4 al al Ul Ul
Ul " x J..I Ul s:: al al
or! :> J..I III Ul Pl J..I
200~ ~------- 0.2 m ~ +J -- ~--Cl) --- J..I ........ - 8 --- ------:--0 0.2 0.4 0.6 0.8 1.0
Distance from bearing plate
Figure 4.3 Strain distribution (a) Wide face
III !-I m
..Q
Cl. s::
-.-1' U !-I 0
4-l s::
'.-1 QJ !-I
S::. -.-1
III s:: -.-1 m !-I .jJ U)
99
~}-_.-r-- . ... - - - -,-- - - _. r--'- ._ -.....-- ,--- - -_ .. __ - --_. -' 0._.-
-
1200).1
1000).1
800).1
El
600).1 " 0 17
400).1 )(
200).1
0
D I-160 Reinforcing o 1-375 bar strains • I-l
Q II-160 x 1I-375 - II-l
Con crete stresses I!I
17 I!l • 0
• 17 17 Q il
Q Q
x 0
• x Il
le _.1.._-
.' Narrow face b lb = 0.688
P c
Bar locations correspond to
,x-coordinates in the graph
QJ .jJ QJ !-I
a I-160N u ....... § f'1;0!
CI 1-375N 1.0 u,
III -~. 4-l
0 III
0.8 QJ III III s:: III 0
Zielinski QJ -.-1 !-I .jJ .jJ U
and Rowe 0.6
III m !-I
QJ4-l Leonhardt r-I
• .-1 III III m @'tI 0.4 .jJ QJ
III QJ III III QJ !-I !-I
0.2 QJ ~ :> A III QJ
,..- --- ~ ..... ,/ -- !-I
8 0.2 0.4 0.6 0.8 1.0
Distance from bearing plate (expressed as a fraction of b c)
Figure 4.3 Strain distribution (b) Narrow face
100
4.5.4 Total force in transverse rein forcement
The variation of the total transverse tensile force in
the lateral rein forcement as the bearing load increases is of
interest, as it gives an indication of the extent to which the
anchorage zone steel and concrete interact in sharing the induced
transverse stresses. The total steel tensile force was calcul-
ated by summation of the contributions of the individual reinforcing
bars. Nominal bar areas and the experimental elastic modulus
values were used in evaluating the force in each bar from the strain
data. When bars on a single side of the specimen were instrumented,
the total force was obtained by doubling the force calculated from
the available data. In the absence of a reading due to an indicator
malfunction, the stress was estimated on the basis of available read-
ings. For strains above the elastic limit, the experimental stress-
strain curve was used to evaluate the stress in the work-hardened
cold-deformed 02 bars, while a constant stress equal to the yield
point was assumed for No. 3 bars. Table 4.6 lists the average force
Z in the lateral rein forcement along both the wide and narrow faces s
of the specimen, expressed in force units, and non-dimensionally as
ratios of the product f A or of the applied bearing load P. cu p
results are presented graphically in Fig. 4.4.
These
The Zs/fcuAp curves show the variation of the total tensile
force in the reinforcement with increase in the applied loads. The
curves possess two distinct portions, the first one being a straight
li ne where the steel and the uncracked concrete share the tensile
forces proportionately, while the second part is a curve of increasing
101
TA8Le 4.6 TOTAL FORCE IN TRANSVERSE STEEL (AI BLnCK r:'160
WIDE F'ACE bp/~:. 0.458 NARROW S::ACt: bp lit :. 0.6ée
p p p tm Zs Z~ ,-k b ~ z, Z$ --h. :b. klp 1<111 ' ,"CA; p.. klp k.N ~Ap P J.I Idp kN ~ ... Ap P
2,0 9. 0~041 O. 0.00 0.0 0.0000 0.0000 o. 0.00 0.0 0.0000 n.OOno 10,0 18. 0.0111 9. 0.04 0.2 0.0009 0.0106 13. 0.06 0.3 0.0012 0.0153 8,0 36. 0~163 32. 0.15 0.7 O.OG31 0.0188 42. 0.20 0.9 0.0040 0.0247
12,0 '3. 0:244 62. 0.:»9 1.3 0.0059 0.0243 63. 0.30 1.3 0.00'"'0 0.0247 16.0 71. 0.31'6 102. 0.48 2.i 0'.0098 0.0300 91. 0.43 1.9 0.008'1 0.02t>B 20.0 89. 0:407 164. 0.77 3.4 0.0157 0.0386 118. 0.56 2.5 0.0113 n.0278 2/ .. 0 107. 0.4n8 210. 0.99 4.4 0.0201 0.0412 147. 0.69 3.1 0.0141 0.02:16 28.0 125. 0':570 265. 1.25 5,5 0.0254 0.044S 184. 0.87 3'.8 0.0176 0.03n9 '2.0 142. 0:651 337. 1.59 7. i 0.0323 0.0495 232. 1.09 4.9 0.0222 0.0341 36,8 164. 0'.749 392. 1.84 8.2 0.0375 0.0501 282. 1.33 5.9 0.0270 n.0360 40,3 ,179. 0.820 446. 2.10 9.3 0.0427 0.0'21 328. 1.510 6.9 0.0314 0.0363 45,5 202. 0:926 557. 2.62 11.7 0.0533 0.0576 414. 1.95 9.7 0.0396 0.0429 50.5 225. 1:027 658. 3.10 13.R 0.0630 0.0613 475. 2.23 9.9 0.0455 0.0442 55.0 245. 1'.119 776. 3.65 16.2 0~0743 0.0664 544. 2.56 11.4 0.0521 0.0465 59.0 262. 1.200 880. 4.14 18.4 0.0842 0'.0702 613. 2.88 12'.8 0.0587 n.04R9 63,0 260. 1~282 986. 4.64 20.6 0.0944 0.0736 693( 3.26 14'.5 0.06b3 o. OS 17 67.0 298. 1:363 1135. 5.34 23.7 0.1086 0.0797 774. 3.64 16.2 0.0741 0.05103 71,0 316. 1:444 1255. 5.90 26.3 0.1201 0.0831 850. 4.00 17.8 0.0813 0.0563 T5.6 ~36. 1.538 1555. 7.31 32.5 0'.1488 0.0968 1000. 4.70 20.9 0.0957 0.0622 80.0 356. 1:628 2025. 9.53 42.4 0'.1938 0.1191 1312. 6.17 27'.5 0.1256 0.0771 84,0 374. 1~709 2491. 11.72 52.i 0.2384 0.1395 1720. 8.09 36.0 0.1646 0.0963
FIGURE 4.4 TOTAL FORCE IN TRANSVERSF. STEEL (AI 8LOCK 1-160
i Il =tJ 1
j Il 1
1
U.€88 , i
1 ! bp/bc. .. 0.458 _I! , -... -! Zs/P --0- -0-
l " 1 '-f! ! Zs/fcuAI' --x-- --Q--1 Il i---r--
i Il ,,-
1 i ! 1 '1
r-I ---1-.. ·-rr ---,-
)<j
i 1 0.2
--1 1 / l' ? ! 1 1. __
1 , / 111 f 1 1 J< 1
0.1
.// J/ " ~-L ," '7 Il )1' J L . :
..... V ~ " !
",,'" ..r/ / . ,1 ---t ,
~ ~ ~ /../-' '-1-l1?- " 1 ~~~~> Q .... .Jl. :..-u- I! -LI---1 ........ r- -r--~~kd: ,- ~r& t-- -'r----[ /. It'I'--D ;:,i.' Q_ 1:1'-
---l --~ , :4--0(· -=- -l- .. 1 g , .l",.~.t -JI 1 !i L.- '--,-,."-, _-= _.L--
o '·0 '·5 2·0
102
TABLE 4.6 TOTAL FORCE IN TRAN5VERSE STEEL le) BLOCK Iw160N
WIOE FACE bp Ibc. = 0.456 NARROW I=ACt: bp lit ::. 0.6Se
p p p t.m Zs 7~ ,~ .7::;;. e".. Z, z. -k b kip \.N , fwAp po klp I<.N ~uAp P '" Idp kN ~u.Ap p
2.0 9. O~041 i7. 0.08 0.4 0'.0016 0.0400 10. 0.05 0.2 0.0010 (h0235 4.0 18. o.onl 32. 0.15 0.7 0.0031 0.0376 21. 0.10 0.4 0.0020 0.0247 8.0 36'. 0'.163 50. 0.24 1.0 0.0048' 0.0294 35. 0016 0.7 0.0033 0.07.06
12.0 53. 0'.244 75. 0.35 1.6 0.0072 0.0294 45. 0.21 0.9 0.0043 0.(\176 16.0 '71. 0~326 98. 0.46 2.1 0.0094 0.0288 81. 0.38 1.7 0.0078 0.0230 20.0 89. 0:40'7 128. 0.60 2.7 0~0122 0.0301 110. 0.56 2~5 0.0113 0.0278 24.0 107. 0:4A8 162. 0.76 3.4 0'.0155 0.0318 159. 0.75 3.3 0.0152 0.0312 28.0 125. 0~570 205. 0.1)6 4.3 0'.0196 0.0344 208. 0.98 4.4 0.0199 0.0349 32.0 142. 0.651 250. 1.18 5.7. 0.0239 0.0367 255. 1.20 5.3 0.0244 0.0375 '6.0 160. O~ 732 335. 1.58 7.0 0.0321 0.0438 312. 1.47 6'.5 0.0299 0.0408 40.3 179. 0.820 437. 2.06 9.1 0.0418 0.0510 370. 1.74 7.7 0.0354 0.0432 4".0 196. O~895 638. 3.00 13.3 0.0611 0.0662 450. 2.12 9.4 0.0431 n.04R1 48.0 214. 0.977 766. 3.61 16.\ 0.0735 0.0753 517. 2.43 10.8, 0.0495 0.0507 !l0.5 225. 1~027 865. 4.07 18.1 0.0828 0.0806 574. 2.70 12~0 0.0549 n.0535 55.0 245. 1'.119 976. 4.60 20.5 0.0936 0.0836 643. 3.02 13~5 0.0615 0.0550 59.0 262. L2C'0 1076. 5.06 22.5 0.1030 0.0858 724. • 3.41 15.1 0.0693 n.0577 63.0 280. 1~2!12 U1l5. 5.57 24.8 0.1134 0.0865 824. 3.68 17.2 0.0789 n.0615 67.0 298. 1:3~3 1320'. 6.21 27.6 0~1263 0.0927 948. 4.46 19.8 0.0907 0.0666 '71.0 316. L444 1514. 7.12 31.7 0'.1449 001003 1094. 5.15 22~9 0.1047 n.0725 ".6 336. 1:538 1870. 8.80 39. i 0.1790 0'.1164 1262. 5.94 7.6~4 0.12e8 0.0785 80.0 356. 1:628 2300. 10.82 48.1 0'.2201 0.1352 1600. '7.53 33.5 0.1531 0.0941
FIGUP.E 4.4 TOTAL FORCE IN TRANSVERSE STEEL lB) BLoCK 1-160N
~:~~;Ap I---j---!--t--t---I--t---Ir--t--t---t--!--r--!--+--t--t---Ir--r--t-i-W ... ~. 1 1
O'~~~I~I~~~'-~-l-_J--~-4--+--+-~~-4-+--+-~~-4--+--+-i-b' lb c 0.458 O,Eë6 J
~, ~s/~ -0- -0- 1---l----I--�--~--I--+--I---I--I--+--t---t--r·I--
:=:I~~+-'_ Z-,s/_f,_uA,p--.,-r-_y-_--r--_-Q-r-_t-I--_j~~:~::~:~::-~:~~:~::.~~-~-~t--__ -lj-,_--I_f-_~--i-_'-'l~~
0.2 1-_+--'!i--!---+--I---I--1I---!---+--I--I---I---!--t--t---;/~I---l-+--t-ll= ,/ l -+--i--
1 J _jJ __ ,_ 1---t--+---t---r-i--t--+---t---t-+---t--+--J---tf-'-l--f-l"-+--l-+--+--iJ-~~~~~1--~-1--+-+-+-4--,~~~'~/~!~~ 1 1
:=:==:==:=:==:=::=:==:==:~--+1~-Y'-'~~~,'~~-~_,+~ __ '~~~v+~_/~~~ __ +-~ ~--1== /'''' i"," ...... !-o-:.JY' 1 -1-_!_-,
I--+--'I_L---~~~~i<!?"'- r---! -1--1--1---- "1>-0-- -'Dj~ •• ~:~~_.-' I-~ -+- - -- ---+--1-- ---. '~i-- .- 1--Tl'-j '---"T--: --.- ~- -r 1 1 1
o 0,5 '·0 ',5 2·0
103
TABLE 4.6 TnTAL FORCE IN TRANSV~R5E STEEL (cl BLOCK li-160
YIIOI! rACE bp Ibe. : O.4~:.e NARROW ~ACE bp lit :. O,6ne p p p E.m Zs 7~ ,-1L k: c".. Zs Zs -k b.
lo.ip I.N fc.uAp !.I. klp \c.N 6 Ap P /4 \dp kN ~",Ap P
2.0 9. 0:.0 4 1 20. o. ï9 O.A 0'.0038 0.0941 8. 0.08 0.3 0.0015 0.0376 B.O 36. 0.163 49. 0.46 2.1 0.0094 0.0576 46. 0.43 1.9 0.0088 n.0541
12.0 53. 0~244 70. 0.66 2.9 0.0134 0.0549 71. 0.67 3.0 0.0136 0.0557 16.0 71. 0.326 92. 0.87 3.A 0'.017'(> 0.0541 102. 0.96 4.3 0.0195 0.0600 20.0 89. 0'.407 115. 1.0B 4.8 0.0220 0'.0541 136. 1.28 5.7 0.0260 0.0640 24,0 107. 0~4~B 141. 1.33 5.9 0.0270 0.0553 166. 1.56 6.9 0.031B 0.0651 28.0 125. 0'.5'10 102. 1. '11 7.6 0.0348 0.0612 192. 1.Bl 8.0 n.03h7 0.0645 IZ.O 142. 0~651 281~ 2.64 11.8 0'.0538 0.0026 236. 2.22 9'.9 0.04'52 0.0694 '6.8 164. D'. '749 348. 3.2'7 14.6 0.0666 0.0890 284. 2.67 11.9 0.0544 0.0726 40.3 179. 0.820 3A8. 3.65 16,2 0.0743 0.0906 321. 3.02 13.4 0.0614 0.0749 45.5 202. 0'.926 450. 4.23 18.8 0.0861 0.0930 382. 3.59 16.0 0.0731 0.0790 50.5 225. 1'.027 532. 5.01 22,3 0.1018 0.0991 449, 4.22 IB.8 0.0859 0.0836 55.0 245. 1'.119 606. 5.70 25.4 0.1160 0.103'1 517. 4.86 21'.6 0.0990 0.08S4 59.0 262. 1'.200 671. 6.:31 28.i 0~1284 0.1070 581. 5.47 24.3 0.1112 0.0926 63.0 280. 1'.282 733. 6.90 30.7 0'.1403 0.109!1 641. 6.03 26~8 0.1227 0.0957 67.0 29B. 1~363 829. 7.80 34,7 0.1587 0.1164 729. 6.86 30.5 0.1395 0.1024 Tl.O 316. 1:444 941. 8.85 39.4 0~1801 0.1247 815. 7.67 34.1 0.1560 0.1080 '5.6 336. 1~538 1067~ 10.04 44.7 0~2042 0.1328 A73. 8.21 36.5 0.1671 0.1086 BO.O 356. 1'.628 1290. 12.14 54.0 0:2469 0.1517 981. 9.23 41.1 0.1878 0.1154 84.0 374. 1'.709 1622. 15.26 67.9 0.3105 001817 1124. 10.57 47'.0 0.2151 0.i259 8B.O 391. 1~790 1878. 17.67 78.6 0~3595 0.2008 1365. 12.84 57.1 0.2613 0.1459 QZ.O 409. 1.872 21!.l6. 20.28 90.2 0.4127 0.2205 1847. 17.3B 77.3 0.3535 0.1889
FleURE 4.4 TOTAL FORCE IN TRANSVERSF. STEEL (CI BLOCK 11-160 1< , J ,
1
fi ~L_ , , 1 ~I ' 1 1
: 1
1= bp/be:, a O,455-t:ES8 ,
1 1 , Zf,/P =1
1 , 1
--0- / 1
Zs/fc.uAp --~- 1 r- 'f--
>1 1 1 1
: 1 1
1 1 1
1 .. --1 tt /f 1
" ,
1 , / t , , fi
0,2
" /' li , ,.<J / ,
",r Q/ / /1 /'
1 ~/ , " /0 V 1 ,,; xl y' --
0,1 " ~ ,cr
1 "",,,, ~ ~ r'L L..2-~ 1
1 ~ .0--~ ~ r--'
r--c 1 0-
",:'
i ~ ~ P -& 1
, ,'"
)J'
'" 1;1' Lr '" r T ~'Q ~ 1
,. \2..' ~- - -- -- ---
),,-.!ii -~.- 1 1---1-- -1 --1---- -,
-.. - ... i 1
,,,--' o 0,5 1,0 1,5 2,0
104
TABLE 4.6 TOTAL FORCE IN TRANsveRSE STEEL COI
WIDE FACE bp Ib<- = o.'me NARROW FACi: bp II::!,. ... O.68e p p P e". Zs 7~ -k :b Ew. Z5 ~~ -k :1:..
kif' !.N , -ç.ç P. \c.ip I<.N ~Ap P .. Idp ~u.Ap P
11.0 49. 0:055 O. 0.00 0.0 0:0000 0.0000 o. 0.00 O~O 0.0000 0.0000 44.5 198. 0'.222 38. 0.98 4.4 0.0049 0.0221 42. 1.09 4.8 0.0054 0.0244 67.0 298. 0.334 1l2. 2.12 9.4 0.0106 0.0317 77. 1.99 13.9 0.0099 0.0297 89.0 396. 0~443 254. 6.57 29.7. 0.0327 0.0738 102. 2.64 11.7 0.0131 0.0297
111.5 496. 0~556 318. 8.23 36.6 0.0410 0.0738 127. 3.29 14:6 0.0164 0.OZ95 1:14.0, 596. 0:668 45Z. 11.69 52.0 . 0.0583 0.0873 168. 4.35 19.3 0.OZ17 0.0324 156.0 694. 0'.777 553. 14.31 63.h 0.0713 0.0917 178. 4.61 ZO.5 0.OZ29 0.02'15 l'7B.O 792. 0.687 672. 17.39 77.3 0.0866 0.0977 217. 5.61 Z5.0 O.OZAO 0.0315 205,0 912. 1.021 800. 20.70 92.1 0.1031 0.1010 275. 7.11 31.6 0.0355 0.0347 235.0 1045. 1: 171 982. 25.41 i13.0 0:1266 0.1081 465. 12.03 53.5 0.0599 0.0512 264.0 1174. 1 :315 1158. 29.96 133.3 0.1493 0.1135 592. 15.32 68.1 0.0763 0.0580 281.0 1250. 1:400 1306. 33.79 150.3 0.1664 0.1202 675. 17.46 77.7 0.0870 0.0621 306.0 1361. 1:525 1417: 36.66 i63.1 0.1827 001198 A07 .. 20.88 92.9 0.1040 0.0682 329.0 1463. 1'.639 1530. 39.56 i76.; 0:1972 0.IZ03 916. 23.75 105.6 0.1183 0.0722 351.0 1561. 1:749 1706. 44.19 196.t> 0:2202 0.1259 1106. 28.61 127.3 0.1426 0.0815 360.5 1604. 1:796 1780. 46.05 204.8 0.2295 0.1277 1780. 46.05 204.8 0.2295 0.1277
FIGURE 4.4 TDTAL FDRCE IN TRANSVERSE STEEL (DI 8LOCK 1-375
! 1 i +-1---
.-bp/bc. .. 0.4=8 10.Gôe 1
1 1
Zs/P =1 --0-- . Zs/fcuAp --Q-- -
J __ ._ , ,. , ,.
0.2 ....
" , --, , 1 ,. ,
" , - , , ,. 1
" ..! , , , ,
," , ..0- --,
0-- ,\/
1 ~~ ~ ~ ,. ,. Q
1..---' i-:?" , ;0
l-P ,. ~ ~ ;0
0.1
Ir 0 ,
17 kî' ,x' ,. ,. ~ , ,. ,. !.---
J ,,;<'
~ f..--p-
;0
,~ -- ---1- A~ - ?-IVI 1," _ n. =sr.. --~ 1--- -- .1--- ... --
1 ." _17- --17-
--- .oP.-"r-9 --
o 0·5 1·0 1·5 2·0
0,2
0,1
105
TOTAL FORCE IN TRAN~VFRSE STEEL (El BLDCK 1':'375~1
"'IOE FACE bp lèt = 0.456 NARROW FACë bp lit '" ~,68e p p P t", Zs 7~ .~ .b é.n Z$ Zs ~ b "-ip I<N ~uAp po klp I<.N ~Ap P fA \(.Ip kN ~ ... Ap P
11.0 49. 0:051 5. 0.13 0.6 0:OÔ06 0.0118 8. 0.21 0~9 0.0010 n.nlfl8 45.0 200. 0:207 47'. 1.22 5.4 0.0056 0.0270 35. 0.91 4.0 0.1)042 n.02n1 68.0 302. 0:312 78. 2.02 9.n 0.0093 0.0297 78. 2.02 9~0 0.0093 n.n297 90.0 400. 0:413 108. 2.79 12.4 0.0128 0.0310 134. 3.47 15.4 0.0159 0.0365
112.0 49B. 0:'14 169. 4.37 19.4 0~0201 0.0390 156. 4.04 18.0 0.01S5 0.03l'10 193.0 '92. 0~611 367. 9.50 42.2 0.0436 0.0714 185. 4.79 21.3 0.027.0 0.0360 154.0 6B5. 0:707 455. 11.77 52.4 0.0541 0.0764 iUl. 5.46 24.3 0.0251 0.030;4 U8.0 792. 0'.818 542. 14.02 62.4 0.0644 0.0788 240. 6.21 27.6 0.02Q5 0.03',9 205.0 912~ 0'.942 657. 17.00 75.6 0'.0781 0.0829 275. 7.11 3l.6 0.0327 0.0347 235.0 1045. 1:079 787. 20.36 90." 0.0935 0.0866 31B. 8.23 36.6 0.0378 0.0350 266.0 IlB3. 1:222 ' 940. 24.32 10B.2 0.1117 0.0914 370. 9.57 4Z.6 0.0440 0.0360 281.0 1250. 1.Z91 10RO. 27.94 124.3 0.1283 0.0994 435. 11.25 50.1 0.0517 0.0401 306.0 1361. 1:406 1244. 32.1B 143.? 0.1478 0.1052 542. 14.02 62.4 0.0644 0.0458 3'1.0 1561. 1 :612 1794. 46.41 206.5 0.Z132 0.1322 783; 20.26 90'.1 0.0931 0.0577 ~fl2.0 1610. 1'.663 1840. 47.60 211.7 0·.Z187 0.1315 140B. 36.43 16Z'.0 0.1673 0.1006
FIGURE 4.4 TOTAL FORCE IN TRANSVERSF STEEL ( El BLOCK 1-37!1N
1 !
1 bp/bc c 0.458 0.6B8
Zs/P --<>-- -n-
ZsF,uAp _ .. x-- --\7- -
" .; ~ .;
, ...., ,.-
,.'" - v ~ i?-c
IP-p-- ,.~
,.' ~
) ' .' ,.,t ,.~
.,~
.or ---=--b-.:: Vi 1 Q--
_.-- .; ~---~ --
a...- 1 1 'f"-;--.L----w---r--o 0.5 '.0
1 1 --l 1
·/1
j-, 1 1 J 1
1
1 J 1
.-il ,
L 11 . / : l
~" .,' 1 ,
/1 0--'''-
~ ~ 1
1 1
r--!- -- '-J
'·5
_.
--
1-
.-
1
---, 1
_ .. ....J
~§ l ""'l . .-J
1
. __ J
-1 ~-
_.1 --
-t 1 -1 i
--j '-'-j
_J
T-I 1
--i
-lJ ---
~ --1 _ .. -1' --·-1
.... .1 '-r-'- __ J 2.0 PI fcu/'p
106
TnTAL FORCE IN TRANSVERSE STEEL (F' BLOCK Il ~375
WICE FACE 'op 'kt. = 0.456 NARROW FACt: 'op lit = O,éë6
P P P r..n Zs 7!t ~ :b. ~ Zs Zs ~ b. kip \(N ~\,\Ap po Idp It.N ~Ap P
'"' Idp 1<.\01 ~u..Ap P
11.0 49. 0:054 5'. 0.26 1.? 0'.0013 0.0235 7. 0~36 1.6 0.0018 n.03i!9 4,..5 198. 0.219 i7. 0.R8 3.9 0.00lt3 0.0198 44. 2.28 11).1 1).0112 n.0512 67.0 298. 0'.330 45. 2.33 10.4 0.0115 0.0348 64. 3.31 14.7 0,0163 0.0494 89.0 396. 0.438 54. 2.79 12,4 0.0138 0.0314 80. 4.14 lA.4 0.0204 0,0465
111.5 496. 0'.549 219. 11,33 50,4 0~0558 0.1016 120. 6.21 27.6 0,0306 n.0557 134.0 596. 0'.660 314. 16.25 72,3 O.OAOO 0.1213 151. 7.81 34.8 0.031\5 0.05A3 156.0 694. 0·.76A 369. 19.09 84,9 0.0940 0.1224 180. 9.31 41.4 0.04';8 0.05'l7 110.0 792, 0.876 445. 23.03 102,4 0'.1133 0.1294 221. 11.44 50.9 0.0563 0.0642 205.0 912, 1:009 539. 27,89 124,i 0.1373 0.1360 2b8. 13.87 61.7 0.0(1)3 0.0676 235.0 1045. 1'.157 623. 32.24 143,4 0.1587 0.1372 333. 17.23 76.6 0.0848 0.0733 264.0 1174. 1'.299 703. 36.38 161,8 0.1791 0.1378 413. 21.37 95.1 0.10~2 0.0809 281.0 1250. 1..3!13 811. 41.96 186.7 0.2066 0'.1493 457, 23.65 105.2 0.11"4 0.0842 306.0 1361. 1~5('16 892. 46.16 205,3 0~2272 0.1508 538. 27.84 123.8 0,1370 0.0910 329.0 1463. 1~619 1192. 61.68 ?74,3 0.3036 0.1875 696. 36.01 160.2 0.1773 0.1095 351.0 1561. 1.728 1309. 67.73 301, :3 0.3334 0.1930 758,' 39.22 174.5 0019'31 0.1117 373.0 1659. 1.836 1400. 72,44 322.2 0.3566 0,1942 828. 42.84 190.6 0.2109 0.1149 396.0 1761. 1~949 1457. 75,39 335,3 0~3711 0.1904 937. 48.48 215.7 0.2387 0.1224 400.0 1779. 1:969 i780. 92.10 409,7 0.4534 0.2303 1268. 65.61 291'. B 0.3230 0.1640
FIGURE 4.4 TOTAL FORCE IN TRANSVERSF. STEEL ( FI BLOCK 11-375
l' ,,'
Z,;Tw.Ap 1'/ 1
13 Zs/P
q,
1 ! 0.3 - III'
bp/br, .. 0.458 1 o,eô8 ,
J Zs/P =1=' 1
~ , Zs/fc.uAp
, 1 il 1
1-
/~ ---
1 / 1-/
/ ;1 1 1
K 1
0.2 -1- --1 " Vol
1 '" p--;--" " /;~ '1 '" Q
,,"" --2 v,' ~ , - 1
~ ~ ~
'" ~I 1.-J'r'" --0" , ,,'" ri ;", "'v ..u--
~ " " 1,-.... r
0.1 1 , ,
V 1
/''1. '" 1--' , .»" .--a , ,," ~ f-o- --
1 --1 / --0- -;;:' rr: "Q
-~ 0/ .sr'''' .... _-- -, j ~ l '-
-0--1,0'" L .- _ ......
- .--];1--
Q .. 'tI. .. /
1 -"-'-
' .. :'::. ---" 0 0,5 ',0 1,5 2,0 PI fcuAp
107
slope displaying the effect of progressive cracking and the graduaI
transfer of aIl tensile forces onto the rein forcement. The tran-
sition between the initial linear portion of the curve and the
portion of increasing slope took place at loads Pif A between 0.4 cu p
and 0.6. The curves representing the wide and narrow faces, which
had been nearly superimposed up to the transition point, became clearly
separated at this point. The transition is more evident in the
z IP vs. P curves, in which a sharp discontinuity was often observed. s
This occurrence was generally much more pronounced in the wide face,
where higher tensile stresses are created as a result of wider com-
pressive stress trajectories. It can be supposed that significant
internaI cracking took place at this load, although the presence of
externally visible cracks was not detected until the load reached
approximately half the ultimate load, i.e. Pif A between 0.8 and cu p
0.9.
The Z IP curves reflect the graduaI progression of cracking s
after the transition, which resulted in the small and nearly constant
positive slope of these curves over their major portion.
A second transition occurred when diagonal cracks associated
with the spalling mode of failure took form, at a load of approximately
0.85 P • u
These cracks were immediately apparent on outer surfaces,
and their progression down the face of the block was more rapid than
that of the midside cracks. In the Z IP curves, the second trans
sition marks the passage from the straight line of small slope to a
curve of increasing slope; a step-type transition was observed
only in block 1I-375. The second transition occurred at loads
pIf A between 1.3 and 1.5. cu p
108
Since the unreinforced blocks
failed as a consequence of spalling at a mean PIf A ratio of 1.397, cu p
it seems that the provision of lateral reinforcements of types land
II had no influence on the onset of the failure mechanism. This
can be attributed to the position of the reinforcement, which was
placed as closely as possible to the tendon duct in order to be in
the region of highest tensile stresses. (It must be emphasized
that the block model tests were meant to simulate the anchorage
zone in a buttress, and that the rein forcement therefore did not
conform to practical design considerations for prestressed con crete
.beam end blocks.) The increase in ultimate capacity with the
presence of rein forcement was a consequence of the transfer of tensile
forces from the skewed plane of the concrete wedge, onto the re-
in forcing bars along faces perpendicular to those in which spalling
took place. (Fig. 4.5). Because the bearing plate was square in
shape, the wedges were of. similar form and spalling occurred on the
four faces simultaneouslYi the sudden increase of strain up to the
yield point in bars close to the bearing plate along both the long
and narrow faces at loads approacfling P is thus explained. The u
discrepancy between the strain distribution observed on concrete
surfaces and that found in the reinforcing bars (Fig. 4.3) can be
attributed to the effect on the bars closest to the bearing plate of
the tensile stress~s which lead to spalling. The increase in
ultimate strength in going from detail l to detail II was a result
of the closer spacing of the bars, rather than the greater number of
barsi since the bar spacing was decreased from 4 to 3 inches
109
TABLE 4.7 TnTAL TfNSJlE FORCE; THEORETJCAL PREDICTIONS
Z/P
10 DE NARRml FACE FACE
bp/bc 0.456 0.686
GUYO~ 0.127 0.072 MAGNEL 0.138 0.094 MORSCH 0.146 0.091 LEONHARDT 0.163 0.094 8LEJCH (; SEIVERS 0.196 0.099 ZIELINSKI & ROHE 0.276 0.176
-: ~.
~~'\.~ ~,,"J
--:L- 0'--/ 0 ,7-0 --~--~- ::.-X-..L __ ~ ~--~ / 0 0 \ /
Figure 4.5 Failure mechanism
110
(112 to 76 mm), the increase in strength was not proportional to
the doubling of the number of bars.
A comparison of the theoretical values of the ratio Z/P
according to the various mathematical and empirical solutions, and
the experimentally observed Z /P ratios, will allow a distinction s
to be made between the conservative and non-conservative solutions.
(Z represents the resultant of tensile stresses. across the central
plane integrated over the area of the block, and Z is the resultant s .
of forces in the reinforcing bars.) Table 4.7 summarizes the pre-
dicted Z/P values according to the various theories, after a table
(3) by Zielinski and Rowe • In Fig. 4.6, these results were indi-
cated in the form of a scale, against which aIl experimental values
of Z /p from Fig. 4.4 are presented for the various block models at s
aIl loading stages, along both the wide and narrow faces. A
linear envelope was drawn, and the loads Pif A equal to 1.0 and cu p
1.5 were marked out, as they represent respectively the maximum
initial prestressing force and the load at the onset of spalling.
It must be pointed out that this envelope is not conservative, because
higher Z /P ratios could be obtained with the use of an anchorage zone s
reinforcement providing more steel than detail II. (Note that
detail II provides bars to a distance of approximately b /2 from the c
bearing plate, whereas most authors report significant.tensile stresses
to a distance b , c
therefore, as an increase in Z occurs when detail s
II is substituted for detail l, a further increase in Z would result s
from the use of a still greater number of bars.) Keeping in mind
that the required steel force could thus be higher than the results
Zs p
3·0
2·0
1·0
1 1 1 1 __ ! 1 1 1 1 1 1 ; 1 Til 1 1 WI~E ~ 1 j 1 1 1 1 1 ~ ACE b,J",=0A58 1
D 1- 160 1
_ ~ 1-160N 0 1-375 ~=·t=j=t±t-1I-t-+-+=+l=tt±-• 1-375N ~ '- 1 • 11-160 'Il-ô75 1 1 -~ l 1 l 1 1 _..1_ 1 -~- -~ ~ 1 • ~ ~ i - 1 • ~~II ~ l ' 1 • < • ~ . i Il! . ' ,1 1
jt-"l \--1- 1 _ : i ! 1 -.\---I ! _ • • >---=9 .~ • i. " - . 1.· ~ ~ 1 • o'
• 0 ·iO
""
.. _.. ____ 1--] 0.0 • • o--~ . ' . -,_ . 0 E';;-"-'--Io- · ' ID ID ID ID D - • 1-
-v-"Q .... Q--fl- __ ,_' .,_ 0 ' D U D - T---~ 0 0 0 la ilJ ~r\-I-lr-' +-4--1-ID "," ID T --
~ ."_ ID " 11." ID --.-- - - • . " --1 D J J . . J J JI' -
1 J -J--.J-_I 1 J 1-_...1 ---J-_~_l 1
7.IELlNSl<1 2. ROWE.
SlEICI-l Z. SIEVERS
LEONHARl>i
..... o?SCI-I
MAC.NEL
GUYON
o 0.5 1.0 1·5 2·0 PJfcuAp
Figure 4.6 Total force in transverse steel, all models (a) Wide face
1-' 1-' 1-'
b p
3.0
'2.0 1
1·0
! i 1
NARROW l=ACE b/bc =0.688
o I-160 o 1-375 13 I-1bON (;) 1-375 Q Il-160 x 11-375
..
i 1
1 1
1
1
I 1
l---: --f..-- Q -f--- Q Q
k-' ,~ Q Q 1"-
l x
---~xQ 1 1 x 1 >C
1)(' 1 )( I!I [ G G
i- m . 10 0
!'" !:> I!I ~'l:o ~o Ild e o ~ i:l~ 1- 0 aj QJ
e o 0
0 :
0
~ 0 - tJ 1
1 1 1
i 1
o 0·5 1-0
1
1
1
'1:
p Q
X -;;-~ Q li: -- Q G IJ
X n
~ - 0
f:-0
G 0
:J 0 ~ 0 -
0
~ °t (;)
1 1
1·5
Q
x
~
~
1 1
--1
'--
1 1
1 ! 1
1
1 L_._
1 i 1
! -! --
h-! 1 ,
1
L~ONH~R\)'t t- ~ 1.I~)./.1. '"1\
1 I-1 1
1 ! t---1 1 1 1
'EI.'~!)l<.I ROWE
.lEICIIl. 'EVE~
Ml5R~~
GU'{ON
2-0 P/fcuAp
Figure 4.6 Total force in transverse steel, all models (h) Narrow face
.....
..... toJ
113
fa11ing within the enve1ope, it is seen that on1y the solution of
zie1inski and Rowe gives a conservative estimate of the required
tensi1e capacity.
4.6 DISCUSSION OF END BLOCK DESIGN
The ana1ysis of b10ck mode1s loaded through a bearing
plate giving b lb (bearing plate size to b10ck width) ratios of p c
0.458 and 0.688, has 1ed to the fo11owing observations:
(1) The fai1ure mechanism was one in which the spa11ing of
concrete wedges from the edge of the bearing plate outwards
1ed to fai1ure in aIl specimens, with or without reinforce-
ment and/or midside crack initiators.
(2) The load at which tensile stresses were sufficient to induce
spalling was apparently the same in reinforced and unreinforced
spe'cimens; this load could be predicted by the equation:
3 P f A .;'r--A-/""':"A-
cu p cp p
where f is the concrete compressive strength, cu
A is the net area of the face of the end block, cp
A is the net area of the bearing plate. p
In the present case, this formula leads to a predicted
capacity of 1.391 fA; cu p
tests of unreinforced blocks gave a mean
load of 1.397 fA, while reinforced blocks disp1ayed spalling cracks cu p
and a significant increase in tensile stresses at loads between 1.3
and 1.5 fA. cu P
114
(3) The provision of reinforcement of Type l (Fig. 3.3)
increased the ultimate capacity to 1.795 fA, while the use cu p
of Type II, which doubles the area of reinforcement, increased
it to 1.935; the increase in strength is not proportional
tothe increase in steel area, due to the predominance of the
spalling mode of failure over tensile rupture along the mid-
side lines. The presence of rein forcement decreased the
variability of test results, and increased the ductility at
failure.
(4) The absence of crack initiators caused a decrease in the
strength of models of Type l to 1.650 fA; this effect may cu p
be due to the earlier propagation of spalling cracks when
points of stress relief are not provided.
(5) The total tensile force in the concrete at maximum initial pre-
stressing forces could only be predicted conservatively from
the experimental data of Zielinski and Rowe(3) , all other
theories having given underestimates or insufficiently conser-
vative estimates of the total tensile force in the reinforcing
bars at loads which did not cause the onset of spalling.
(6) An ultimate strength design of end blocks is feasible, which
would require a smaller amount of lateral reinforcement than
design techniques based on the total tensile force concept;
for bp/bc ratios in the range 0.46 to 0.69, detail l is seen
to provide sufficient lateral restraint to satisfy a load
factor of 1.5 and an undercapacity factor of 0.9 on the ulti-
mate strength of the anchorage. However, if the presence
Ils
and width of cracks is a significant concern for
serviceability, the design procedure of Zielinski and Rowe(3)
is recommended, since both designs l and II (which are non
conservative by this standard) led to visible cracking and
sorne reinforcing bar stresses in excess of the allowable
stress in lateral ties, at the maxim~ initial prestressing
force.
(7) The distribution of reinforcing bars in the anchorage zone
should have a concentration of bars near the plate, where
spalling tendencies affect both the strength and service
ability of the anchorage, rather than near the theoretical
point of peak tensile stresses, where generally lower bar
stresses were recorded.
4.7 DISCUSSION OF SIMILITUDE
Good similitude was obtained in aIl aspects of behavior,
including the mode of failure, the ultimate strength, and the strains
and forces in the rein forcement.
The non-dimensional ultimate strengths of block models
(Table 4.1) show remarkable agreement between the models of scales
1.60 and 3.75; ultimate strength data was not available for the
prototypes. The mode of failure of blocks of both scales is illus
trated in Fig. 4.1.
116
The randomness of the strain data and of the total tensile
force data (Figs. 4.3 and 4.6) was noticed among models of the same
scale as weIl as those of different scales, and can therefore not be
attributed to a size effect. The three-dimensional distribution
of compression trajectories and of paths of crack propagation seems
to be the cause of this randomness. Comparing, for example, data
points for models 11-375 and 11-160 in Fig. 4.6· (a and b), it is
seen that bars along the wide face are more heavily strained for
model 11-375 than for 11-160, while the converse is true along the
narrow face; such behavior is clearly the result of the randomness
of cracking, and not of the similitude of the models. Similar
observations could be made regarding specimens reinforced with detail 1.
It can, therefore, be assumed with confidence that an
investigation of anchorage zone problems conducted at a scale 1/6,
making use of the scaling procedure and mate ri aIs employed in the
~~ model test, would reproduce exactly the mode of failure and
ultimate strength of the prototype structure, and would give a satis-
factory assessment of strains and total tensile forces. The use of
structural modeling in the foregoing investigation of the behavior·of
the anchorage zones in buttresses is seen to be feasible and reliable.
CHAPTER 5
BUTTRESS MODEL TEST
5.1 TEST PROCEDURE, INSTRUMENTATION AND TEST DATA
Anchorages in the buttress model were subjected to bearing
pressures due to tendon loads equal to and greater than the design"
prestressing force in the prototype structure. The simultaneous
application of vertical pressure and of a temperature gradient was
also imposed upon the specimen. Strain readings were taken at
24 points on lateral reinforcing bars.
The effeètive prototype prestressing force was taken as
1460 kips (6.50 MN), and represented by Pd. AlI tendons in the
model were first loaded to 1.14 Pd; the uppermost four tendons
were later taken incrementally to loads up to 2.00 Pd.
Tendons in each of the five groups were taken to this load
in a uniform sequence C, A, B, D, as shown in Fig. 3.5. Strain gages
were read initially, and after each cable had been shirnrned. Midplane
tendons were stressed first, fol1owed by groups 1 and 2. Uppermost
tendon lA was then drawn incrementally to loads up to twice the maxi-
mum design anchorage force. At each load increment, the strain
gages were read and the exposed concrete surfaces were scrutinized
for the appearance of cracks. Tendon 2A was then loaded in an
identical manner, after lA had been locked off at twice its design
load. Tensioning on the face without crack initiators proceeded
similarly in the order: Group 4, Group 3, Tendon 3A, Tendon 4A.
117
118
The buttress was left under full load for seven days,
at the end of which strain readings were taken, and the thermal
loading was initiated. Boiling water at 2l0F to 2l2F (99C to
100C) was pumped through the embedded piping at a rate of approxi-
mately 15 pounds per hour (6.80 kg/hr) for a period of twelve hours,
after which the dis charge attained a fairly constant tempe rature of
95F (35C). Straih readings were taken throughout the warming-up
period; those reported herein were taken three hours after the
steady state had been attained. For safety reasons the boiler
was shut down for a period of eight hours at night. On resumption
of the test the following morning, the effluent temperature had de-
creased to 88F (31C). Steady state was re-established in four
hours, at an effluent temperature of approximately 90F (33C), the
lowering being due to a reduced ambient laboratory temperaturei
final strain readings were taken after six hours of application of
this gradient.
Strain gages were mounted on lateral rein forcing bars in
the anchorage zones lB, lC, 2A, 2C. The strain readings are re-
ported in Table 5.1 for the loading conditions of interest. The
strain gage locations are indicated in the sketches which accompany
the tables. (Fig. 5.1)
The concrete had attained a compressive strength f = 4~50 ksi cu
(3.10 MPa), and a tensile strength fct = 380 psi (262 kPa) at the
beginning of testing. Both of these strengtl1 evaluations are the
average of test results from seven 6 x 12 inch (152 x 305 mm)
cylinders.
x {. ~ ~ \
9 ra c~ ®\ ® ® 0
-- -~ - -1 t"' \ ® 7 8 ® • J 1 t \ ® ~ â ® .. \ 1 \ \ ® ® • "'0 1 0 \ \ ® 1 i 2 ® 0
1 \ 1 0 o 0
0 :.:~.;. :::":: .. :.;:::.:.
x- Ia \ \ 21A
o 1 'O5®1
o 1 1 0 ~ 3
o 1 1" ® 1
a
.. :.:~.;.
Figure 5.1 Strain gage locations in buttress model
1
! o ® 51 .• ® 4 • ® 3 a®2 o ®
1 l i o 0
..: •..... ;.;.:::.:
21B 21C
Section x-x
I~" 1
~~ ~Ç(
Section X-X
o 0
" 0 a 0
" . 0
" 0 a a
::":':.':'~::'.'.:
1 0 o 1
~ o 1
~ o ,
!
1
1
21D
0
0
.. 0 .. ..
::.':.:
10
la
"0 ..
: :.', ~ .
1
1-' 1-' 1.0
TABle 5,1 STRAIN REAOINGS~ BUTTRESS MOOEl CA)
LOAOlNG CONOITImlS 102 183
INITIAI. READINGS 004 019
MIOPLANE TENSION" TENDON OC -04 016 OA,e -06 014 OA~B~C -04 020 Oh,B~C.rO 000 019
SURFACE 2 TENSION~ TENDON 2C -25 -24
1
2A,C 000 - 017 2A, [b C -49 010 2A,B~C,O -64 010
SURFACE 1 TENSIDN~ TENDON lC 039 034 lA,C 058 060 1A,B~e 177 116 lA,B~C~D _ 179 125
TENDON 2AI INCREHENTAl TENSION, 1,25P 188 124 2,00P ~84 129
TENDON lAI INCREMENTAl TENSION, 1.14P 188 108 1.25P IS8 114 1.38P 189 116 1.S0P 190 118 l.sap 190 118 1.75p 190 110 1.85p 190 128 2.00P i 192 177
LOADING ON OPPOSITE ~UTTRESS, 1 SURFACE 4 TE~SIUN, TENDONS 4A~B~CID 214 187
SURFACE 3 TENSIDN~ TENQUNS 3A,B,C~D 248 190 1 TENDON 3A, INCREMENTAL TENSION, 1.25P 2'IQ 184
2.00P 245 185 TENDON 4A, INC~EMENTAl TENSIDNI 1.25P 244 184
2.00P 250 190
7TH DAY UNDER SUSTAI~ED FUll TENSION 275 195 FULL TENSIC'N & THERIJ,Al GRADIENT 289 190
8TH DAY THER~AL GRADIENT REST ART 294 190 THERMAL GRADIENT STEADY 300 200
GAGES IN ANCHORAGE ZONES lB AND lC
184 lBS 1B6 lC 1 1 C 7- lC3 1e4
-14 -14 -14 000 010 004 -14-
-17 -10 .. 19 -06 on'. 000 020 -17 000 -10 -04 005 005 004 004 014 017 014 -09 028 -20 000 004 007 015 -10 020 -30
004 004 004 015 -16 03 /, 006 -15 -04 -08 -09 -24 029 004 -10 -004 005 -20 -47 030 004 -10 004 006 -35 -Sb 034 005
070 019 038 160 134 168 O'JB 079 049 039 166 109 160 074 144 087 078 240 120 19~ 105 154 089 090 254 2:)5 205 167
164 094 095 264 237 218 135 154 092 098 260 234 227 136
15a 094 100 264 230 214 139 167 096 .095 264 231 215 1'39 160 098 099 264 NO 218 1'.5 164 100 104 265 2',0 230 000 154 100 104 259 -220 215 107 155 107 100 260 226 205 114 157 115 COO 260 228 214 118 167 125 107 267 238 217 108
177 125 106 287 237 230 115 180 124 104 326 274 240 124 178 120 104 314 260 230 124 174 118 094 -324 3(,4 234 106 180 117 096 324 270 234 124 176 119 096 330 274 239 114
248 149 119 430 3'.0 317 179 268 135 115 447 349 330 l'JO 290 144 116 460 347 354 190 290 139 120 484 368 374 198
UNE
01
02 03 04 05
06 07 OB 09
10 11 12 13
1'+ 15
1~ 17 la 19 20 21 22 23
24 25 26 27 28 29
30 31 32 33
~ I\J o
TABLE '.1 STRAIN ReADINGS, BUTTRESS MaDEL (B l
~OAOiNG CONDITIO~S 2A1 2A2
INITIAL REAOI:'1GS 000 -14
MIDPLANE TENSION, TENDON OC 000 -10 OA,C -0-' 000 Oll" B, C -10 004 OA, B, C, 0 -07 -04
SURFACE 2 TENSION, TENDON 2C -14 010 2A .. C 110 125 2A .. B,C 118 167 2A,B,C,D ua i70
SURFACE 1 TENSION, TENDON 1C 120 179 1h .. C 095 159 1A .. B,C 084 150 1A,B,C,D 090 140
TENDON 2A .. INCREMENTAL TENSION .. 1.14P 120 080 1.25P 158 06'. 1.3SP 200 084 1.50P 266 080 l.sap 274 069
! 1.75P 3':10 060 - 1.85p . 32'. 050
2.00P 479 369
TENDON lA, INCREI:ENTAL TENSION, 1.25P 479 . 367 2.00P
LOADING ON OPPOSITE r-UTTRESS, 4Q5 370
SURfACE 4 TENSIJN, TE~DONS 4A,B,C,D 4S4 390 SURFACE 3 TE~StONJ TE~DONS 3A,S,C,O 489 407 TENDON 3A, lNCREHENTAL TENSIONJ 1.25P 464 379
2.00P 464 390 1 TEN"ON 4', INC'EME~T'l TENSION. 1.25P 460 390
2.00P 464 385
17TH OAY UNDER MAINTAINED FULL TENSION 428 394 1 FULL TEr~SI[lN f.. THER~II\L GRADIENT 405 38S (TH DAY THERI~AL GRüOIENT RESTART 404 374
THERMAL GRADIENT STEADY 389 387
GAGES IN ANCHORAGE ZONES 2A AND 2C
2A3 2M 2A5 2A6 2A7 2AB 2A9 2eI
-06 014 006 -15 -07 -06 -15 -15
-07 014 005 -15 000 000 -14 ... 17 -10 004 004 -20 000 005 -10 -26 -oa 019 010 -07 024 020 008 -20 -10 010 010 -OB 007 020 0'.9 000
-09 028 000 005 020 030 034 1114 110 139 086 Q94 074 119 0'.0 110 119 230 110 169 095 184 065 17 /• 120 224 108 174 094 190 059 179
120 248 108 178 104 197 105 1"0 105 230 097 160 108 184 144 }64 099 228 0'J4 158 105 190 114 156 094 229 097 156 097 184 134 140
046 180 0(,6 150 080 174 110 ]40 055 195 077 159 064 190 17.4 150 074 204 084 16'. 094 194 130 1313 100 204 085 170 090 200 134 160 104 214 090 156 096 194 134 14:1 110 195 090 15'. 100 1 S'. 130 145 124 180 100 154 104 194 148 138 310 34Q 178 214 174 250 174 141
315 350 170 214 174 246 190 138 329 341 167 215 176 245 199 137
334 355 166 219 178 257 190 150 360 376 174 226 184 264 11)0 164 330 360 166 218 180 246 l'JO ]64 340 356 160 214 179 256 190 184 339 350 160 214 174 255 188 168 334 354 160 214 184 254 190 164
358 380 164 216 190 327 3B9 210 364 390 16S 218 180 336 430 ?l8 350 389 157 220 194 36 ft 534 21B 360 386 168 205 196 344 559 220
2e2 2e3 2C4
-05 035 -06
000 035 1)04 -04 010 006 005 000 014 -06 000 C07
117 095 100 128 115 100 208 165 181 216 195 178
214 1~5 194 189 154 194 194 125 210 188 110 214
190 000 214 190 0(10 218 190 000 214 188 000 27.4 190 000 214 190 OCO 7.30 190 000 216 190 000 222
190 000 230 180 000 2',6
204 000 234 216 noo 239 205 oeo 214 209 000 Z18 7.07 000 226 214 000 217
240 000 234 258 000 220 237 000 234 237 000. 220
2es
-07
-04 004 'no 030
c90 094 144 160
170 1b9 169 )70 .
1641 174
t7f! 1110 178 IRa 1 S'. 185
1 RI •. 204
Z04 215 . 205 208 7.09 224
COO 000 000 oon
UNE
51
52 .53 54 55
56 57 !i8 59
60 nl "2 63
64 65 66 67 68 69 70 71
77. 73
74 75 76 17 78 79
110 131 82 113
1-' /IJ 1-'
122
5.2 GENERAL BEHAVIOR
The formation of cracks on exposed concrete surfaces would
be, in the present structure, the prime cause and indication of loss
of serviceability of the proposed anchorage zone details. It is,
therefore, quite significant that none of the tendon anchorage zones
showed any externally observable cracking at loads ranging from 1.14
to 2.00 times the maximum design working load irrespective of the
bearing plate form, of the amount and distribution of end zone steel,
and of the presence of crack initiators.
One portion of the buttress surface did suffer sorne
accidentaI damage (Fig. 5.2) when the jack seat was inadvertently
misaligned, and bore on the outer concrete cover. This occurred
after aIl tendons of groups 1 and 2 had been drawn, and tendon lA was
being reloaded incrementally. Spalling began suddenly at a load
just over Pd' and was confined to a layer of about 3/8 inch (1 cm)
thickness; none of the rein forcing bars were exposed due to the
loss of this cover. On noticing the damage, loading was stopped
immediately, the system was realigned, and incremental loading was
resumed as scheduled to a final force valve of 2 Pd. Despite the
partial loss of cover, no cracking or other signs of further distress
were noticeable; Fig. 5.2, which was taken after the top tendon had
been locked off at double load, shows no visible crack~ng.
The top anchorage zones could not be taken to the onset of
failure, as had been originally planned, due to the unforseen reserve
of strength in the buttress. From considerations of dimensions
and scaling, the largest usable tendon consisted of seven 7-mm wires,
123
Figure 5.2 Superficia1 damage in buttress mode1
124
and could develop a specified ultimate force of 100 kips (445 kN).
By stressing the tendons to 0.81 of their capacity, a bearing load
of 2 Pd could be generated, which was hoped to be sufficient to bring
about sorne structural damage. Although the desired cracking and
ultimate loads could not be established experimentally it was decided
not to load the tendon beyond 0.8 f , on the grounds that high su
elongations beyond yield and excessive stresses would cause difficul-
ties in jacking and possibly jeopardize the safety of the test.
5.3 DISCUSSION OF STRAIN DATA
5.3.1 General observations
All but one of the instrumented lateral rein forcing bars
attained a stress of 20 ksi (13.8 MPa) under any loading condition
as can be irrferred from the strain values all being well within the
equivalent strain limit of 680 ~.
In the following sections, individual loading conditions
will be deduced from the combined cases by taking differences between
pairs of lines in Table 5.1.
5.3.2 Effect of middle plane tendons
The middle plane tendons were loaded to the same tensile
force as the tendons in the anchorage zones under study. In addition
to simulating the effect of the continuous circumferential tendons by-
passing the buttress under consideration, they provided a vertical
component of force by virtue of their inclination, which amounted to a
total of 664 kips (2.95 MN) in a prototype structure.
TABLE 5.2 EFFECT OF MIDDLE PLANE TENDONS, SUSTAINED LOADING, AND THERMAL.GRADIENT
LOADING FROM TENDON FULL THERHAL GRADIENT ANCHORAGE BAR TENSION ZONE ~FJ • OC OA,C OA,8"C OA/û,C/D DAY 7 DAY l DAY 2 STEADV
lB 162 -OB ~10 -08 000 025 014 Dl'> 025 153 -03 -05 001 000 005 -05 -OS 005 154 -03 -03 028 024 on 020 042 042 135 004 000 OZ8 008 030 -14 -os -10 1B6 -05 OO't 031 C21 023 _\)f, -03 001
le lC 1 -06 -04 014 015 100 017 030 054 lC2 -06 -05 -19 -20 06b 009 001 028 1e3 -04 001 024 016 082 013 037 057 1C4 034 028 -06 -16 065 011 011 019
TABLE 5.1 LFlES 02GOl 03&Ol 04&Ol 05&01 30&29 31&30 32&30 33&30
IZA 2Al 000 -07 -10' -07 -36 -23 -24 -39 2A2 00'1 014 01B 010 009 -06 -20 -07 7.A3 -01 -04 -02 -04 024 006 -oa 002 2A4 000 -la 005 -04 026 010 009 006 ZA5 -01 -02 004 004 004 004 -07 004 2Ab 000 -05 008 007 002 fl02 004 -11 2A? 007 007 031 014 006 -10 004 006 ZAS 006 ou 026 026 067 009 037 017 2A9 001 005 023 064 189 041 145 170
2e 2e1 -02 -11 -05 015 046 018 OOS 010 2C2 005 001 010 -01 -2b 018 -03 -03
1 2C3 ClOO -7.5 -35 -35 - - - -
1
2C4 Ole 012 020 013 017 -l" 000 -14 2e5 00) 011 037 037 373 143 253 302
ITABLE ~.1 Lt~ES 52&51 53(.51 54&51 55&51 80&79 81&80 82&ôO B3&80 -- -- .- -_._------ ---- ---
i
!
1-' l\J V1
126
The effect of middle plane loading on the lateral re-
infor~ing bars was small as shown in Table 5.2. Bars farthest
from the bearing plate, i.e. bars placed in the body of the buttress
as thermal and shrinkage reinforcement, underwent the greatest strain-
ing, viz. 64 p in gage 2A9, 37 p in gage 2C5. Other bars experienced
smaller strains, of the order of ± 30 p, the sign and magnitude depend
ing on the position of the bar relative to that of the tendon being
loaded.
5.3.3 Effect of sustained loading and thermal gradient
Table 5.2 lists the increases in strain after seven days of
sustained loading (including the top-most tendons being maintained at
a load of 2 Pd)' and at three stages in the application of the thermal
gradient.
At anchorage surface 1, where details of types III and IIIa
were used, increases in strain not exceeding 100 p and 60 p were in
duced by the sustained loading and the thermal loading, respectively.
Bars lB2, lB4, ICI, lC3, which act in the space between tendons lB
and lC (c.f. Fig. 5.1 and Table 5.1), were most heavily strained,
while bars between ID and lC underwent comparable, although slightly
smaller s trains. The similarity of readings between gages ICI and
lC3, and between gages lC2 and lC4, can be noted. L?teral bars be-
tween tendons lA and lB, instrumented with gages lB3 and lBS, were
almost unaffected by the sustained and thermal loads, despite the in-
tentional overloading of tendon lA; thus the individual bearing
plate may have a beneficial effect in creep and thermal considerations.
127
At anchorage surface 2, the range of strains was somewhat
less than at surface l, due to the use of the larger steel percentage
of detail 1. With the exception of gages 2A9 and 2C5, most strain
readings were less than ± 40 ~, and in many cases were negligible.
Considerable increases in gages 2A9 and 2C5, which stand farthest
from the bearing surface, may point to stress redistributions in the
massive buttress under differential creep or thermal dilation.
These results emphasize the necessity of reinforcing for inelastic
and thermal effects, and may warrant further study. The hypothesis
of creep in the gages seems improbable, in view of (i) the previous
readings, (ii) the coincident locations of the gages, (iii) the
slowing of the rate of increase with the attainment of a steady state
thermal gradient (later readings did not differ from those reported
in the last lines of Tables 5.la & b), and (i v) the fact that com
parable increases took place in the 7-day creep period and in the
2-day heat application periode
5.3.4. Effect of direct loading on anchorage zone
Strain measurements taken in each instrumented zone, while
the tendon bearing thereon was loaded, are reported in Table 5.3.
By virtue of the arrangement of gage locations, the effects of con
tinuous and distinct bearing plates, light and heavy lateral rein force
ments, and the sequence of load application can be studied.
Comparison of results for tendons IC and 2C shows the
difference in the efficiency of details IlIa and l, respectively, at
working loads. Both designs are strained considerably below the
TABLE 5.3 EFFECT OF· PRESTRESSING FORCE APPLIED DIRECTLV ON ANC~ORAGE ZONE
STRAINS IN ANCHO~AGE ZONE lB UNDER TENSIONING OF TENDON lB (iAôLE 5,1 LI~ES 12&11)
IB2 133 lB'. 185 lB6 119 056 065 038 039
1
ISTRAINS IN ANCHQPAGE ZONE le UNDER TENSIONING OF TENDON lC .(TA3LE 5.l LI~ES 10&09) 1 1 le 1 lCZ lC3 1e4 1 195 200 134 093
1
ISTRAINS IN ANCHORAGE ZONE 2C UNDER TENSIO~ING OF TENDON 2C (TA3lE 5.1 LINES ~5&55)
2e 1 2C2 2.C3 2C4 2C5 114 123 095 093 0(:0
,STRAiNS IN ANCHORAGE ZONE 2A UNOER TENSIONING OF TENDON 2A
1 LO.dOS UNES 2A1 2A? 2A3 2A'. 2A5 2A6 2A7 2AB 1 57&56 124 115 119 111 086 089 054 089 ,1.14? !l.l4P 64&63 03n -60 -48 -49 -31 -06 -17. -10 1.25P 65&!J4 038 004 009 015 011 009 004 016
t 1.;S?· 66&64 080 004 028 024 018 014 014 020 Il•
501'
67&64 14" ono 054 024 019 020 010 026 1.58!' 68&64 15
', -11 058 03', 07.4 006 016 020
!.ï!>P 69&64 180 -20 064 OlS 024 004 020 010 Il.B5P 70&64 204 -30 078 01)8 034 004 024 020 2.0QP 71&64 359 7.49 2~4 240 088 064 094 076
~~-_ .. _~--
2A9
006 -24 014 020 024 024 020 038 064
TABLE 5.4
ANCHORAGE ZONE
lB
lC
2A
2C
EFFECT OF PRESTRESSING ~ORCE ON AryJACr-NT ANCHORAGE ZONES
aAR LOADING APPLIED AT ANCHORAGE NO.
lA le ID 133 026 024 009 l!~5 030 015 002 1'12 01C) 103 OOZ Id4 O')') OêO 010 186 001 032 012
1 lA lB Hl lC 1 OOb 074 014 lC3 -OB 031 ooa lC2 -25 011 085 1C4 -24 031 062
2n 2C 2D 2A1 OOR. -07 000 2A3 009 oot 001 2A5 024 -10 -02 2A7 021 013 . -01 2A9 025 -15 -06 2A2 042 Ol't 003 2A4 09l OIS -06 2A6 075 013 005 2A8 065 010 -C6
2:\ 2B 20 . 2Cl 004 064 005 2e2 011 080 ooa 2C3 020 050 030 2C4 000 081 -03 2CS 004 050 016
1
! 1
1
1
1 1
1-' t\) ())
129
a110wable working strain 1eve1 of 680 ~, being of the order of 120 ~
for detail land 200 ~ for detail IlIa. In either case, the
tranverse bars nearest the bearing plate show the 1argest strain, and
since both details employ the large No. 10 (32 mm) trim bar as addi
tional lateral reinforcement, its contribution to splitting resistance
can be assumed to be quite significant. Thus the reduction of the
number of the bars from detail l to detail III increases the efficiency
of force transfer to these bars. Since bar stresses at working load
are still less than one-third of the allowable value, it seems that
a further reduction in steel percentage, such as the use of No. 4
(13 mm) bars in detail III in the sarne location as the No. 8 (25 mm)
bars in detail IlIa, may be perrnissible.
Because tendon lB was loaded after adjacent tendons lA and
lC had been tensioned, a reduction occurred in the magnitude of ten
sile stresses and in the extent of the zone of stress gradients in
anchorage zone lB. It is important to notice that the stressing
of tendons above and below the anchorage zone not only lowers the
tranverse stresses in the vertical direction by lirniting the width of
the spreading compression trajectories, but also reduces the horizon
tal tranverse stresses as a consequence of the triaxial nature of the
stress distribution. ~le maximum strain in the first tranverse bar
of anchorage lB was 120 ~ or approximately 60 percent of that which
occurred when identical anchorage lC was loaded in isolation.
The first application of loading to tendon 2A gave strains
closely similar to those obtained in the tensioning of tendon 2C.
Since the anchorage zones 2A and 2C were detailed identically,it appears
130
that the width of the bearing plate and its continuity have no
significant effect on the generation of tensile forces in the
anchorage zone reinforcement.
When load was first applied to tendon 2A, only anchorage
2C had been previously loaded, and its location was sufficiently re-
mote for anchorage 2A to be considered in isolation. In the second
loading to 1.14 Pd' aIl adjacent anchorages were under working bearing
pressures as tendon 2A was unloaded and reloaded. The same reduction
in lateral stresses which was observed in the case of anchorage lB
occurred here; the second entry in Table 5.3 gives the reduction
quantitatively, as calculated from the difference between the new strain
values at 1.14 Pd and the immediate preceding loading condition.
IncrementaI prestressing of tendon 2A did not have a signi-
ficant effect on gages other than 2Al and 2A3, until the load level
1.85 Pd was attained. In the increment from 1.85 Pd to 2.00 Pd'
strains in the first and the second tie each increased by more than
200 ~ to values ranging between 240 and 359 ~, while strain in the
more distant bars increased proportionately, but remained below 100 ~.
The confining effect of adjacent pressure zones was again present,
as the higher strains recorded in the odd-numbered top-side gages
indicate. Despite this sudden increase in steel strains, which
probably resulted from accelerated internaI cracking not visible on
outside surfaces or at the root of crack propagators, the anchorages
were far from failure at a load of 2 Pd. Moreover, the stress in the
lateral rein forcement at this load did not exceed Il ksi (7.59 MPa),
approximately, which is weIl within allowable working stresses.
131
5.3.5 Effect of loading on adjacent an ch orage zones
Table 5.4 shows the strains which result in the four
instrumented anchorage zones, when each of the adjacent anchorage
zones undergoes loading. Results were obtained from differences
between the appropriate pairs of lines ranging between 9 and 13,
and between 55 and 59, in Table 5.1.
It can be readily observed that the distance between ten
don centerlines defines the width of the zone over which rein forcing
bars are significantly influenced by the bearing load. Bars
situated beyond this zone show sorne strains, but of a negligible
magnitude (less than 20 ~). For example, bars between tendons lB
and lC are not appreciably strained by loads on tendons lA and 10.
Similarly, odd-numbered bars 2Al and 2A9, which lie further away
from tendon lB than even-numbered 2A2 to 2A8, show strains not greater
than 24 ~ under load at lB, while the latter undergo strains up to
91 ~ in the same condition; the effect of loading in anchorage 2C
on these same bars is less than 18 ~, and that of anchorage 20 is less
than 6 ~. Numerous other examples could be drawn from this table.
As observed in the previous section, the sequence of
tensioning has an effect on the strains in the horizontal ties. It
must be noted that tendon lC is loaded in isolation, while lB and 10
are tensioned after their neighbours lA and lC. It is seen that
strains in IB2, IB4, IB6 under loading of lC are higher than those
in ICI and lC3 due to loading of lB, or those in lC2 and lC4 due to
loading of 10. Therefore, the reduction of lateral splitting
stresses in the horizontal direction, which accompanies the narrowing
132
of the spread of compression trajectories in the vertical direction,
is again noticeable.
The continuity of the bearing plate did not appear to be
of importance. Under the action of tendon 2B, gages 2Cl to 2C5
generally reacted in the same way as even-numbered gages 2A2 to 2AB.
The most important observation to be drawn from these re
sults'is that aIl bars in the immediate vicinity of a tendon undergo
straining due to the stress distribution in bearing. The contri-
bution of bars in the anchorage zones of adjacent tendons can be
significant. Strain readings in the vicinity of tendon lC, for
example, are: 195, 200, 134, 93 ~ for bars ICI to lC4, and 103, BO,
32 ~ for bars lB2, lB4, lB6, respectively. Thus strains in the
neighbouring bars are of the order of half those measured in the
anchorage zone. To illustrate the case of incremental loading
beyond the design load, strains in bars lB3 and lBS increased by 26
and 30 ~, respectively, when tendon lA was loaded to 1.14 Pd' and
exhibited a further increase of 69 and 31 ~, respectively, and the
tendon load was increased to 2 Pd.
It could be argued that this effect is inconsequential,
since superposition of the lateral forces from the several tendons
will eventually produce the same strain in aIl reinforcing bars.
This, however, is a conservative reasoning, because the reduction of
the anchorage zone width in any tensioning sequence will always reduce
the lateral strains induced by subsequent tensioning.
133
5.3.6 Effect of loading on opposite side of buttress
The application of load to an anchorage zone induces stresses
throughout the buttress. This stress distribution has an effect on
reinforcing bars located on the bearing surface opposite to the point
of application of the load.
is srnall, but not insignificant.
Table 5.5 shows that this effect
BeneficiaI reductions in strains
of the order of 50 ~ were generally observed, although sorne bars
underwent strain increases of a comparable magnitude. The largest
strain differences in either case were +85 ~ in gage 2A9, and -85 ~
in gage 2C3. The only apparent significant trend in the overall
behavior is that bars located closest to the bearing plate undergo
strain reductions, while bars farther away from it display both in-
creases and decreases in strains. It seems, therefore, that load-
ing applied to the opposite side of the buttress is not detrimental
in the horizontal bars which are most heavily stressed under direct
bearing loads.
Also in Table 5.5, the effect of the individual tendons
was deduced from superposition. The results show no conclusive
trends in behavior. However, a reduction in the magnitude of
strains with the distance between the loaded anchorage zone and that
being examined, is observeable in the strains recorded at lB under
the action of 2D, and those at 2A under the action of ID.
The loading of cables on the opposite buttress, that of
bearing surfaces 3 and 4, had a limited effect on anchorage zones in
the instrumented buttress (c.f. lines 24 to 29, and 74 to 79, in
Table 5.1). This will not be discussed here, as it did not signi-
ficantly alter the recorded strain readings, and is of no relevance
in practical design and analysis.
TABLE 5.5 eFFEeT OF PRESTReSSING FORCES ON OPPOSITE SIOE OF BUTTRESS
3TRAtNS.AT ANCHORAGE SURFACE l UNDER PRESTRESSING AT SURFACE 2
LOADING FROM TENDON ANCHORAGE BAR
IZONE Na. 2C ZA,C 2A"B"C ZA/B"CJD 2A"Z.00P
lB 182 -25 - -49 -64 005 133 -43 -02 . -09 -09 004 184 004 -15 -10 -10 000 165 000 -08 000 000 003 106 -03 -15 -02 -01 008
lC le 1 000 -24 -35 -50 006 1CZ -06 -14 -37 -56 029 le3 014 009 010 014 OZ2 1C4 036 034 034 035 -31
TABLE 5.1 LIl~ES 06&05 07&05 08&05 On05 15&13
STRAINS AT ANCHUPAGE SURFACE Z UNDER PRESTRESSING AT SURFACE 1
LOADING FROM TENDON ANCHrJRAGE BAR ZONE IlO. le lA,C lA" B .. C .1AJB,'/D lA,Z.OOP
2A 2A1 ooz -Z3 -45 -Z8 -74 2A2 009 -11 -20 -30 001 2A3 000 -15 -21 -26 019 ZA4 024 006 004 005 001 2A5 000 -11 -14 -11 -11 2A6 00'+ -14 -16 -18 001 2A7 010 014 OU 003 002 2AS 007 -06 000 -06 -05 2A9 046 085 055 075 025
2e 2el 001 ~15 -23 -39 -04 2e2 . -02 -27 -22 -28 -10 2C3 -30 -41 -70 -85 -2C4 . 016 ·016 032 036 024 2C5 010 009 OOB 010 019
TABLE 5.1 Lif,jES 60&59 61&59 62&59 63&59 73&71
EFFEtT CF INDIVIOUAL TENDONS
'ZA ZB 2C 20
lB2 - - -z5 -15 lB3 041 -07 -43 000 1B4 -19 005 004 000 lB~ -OB 008 ono 000 1B6 -IZ 013 -03 .. 01
. lC 1 -Z4 -11 000 -15 lC2 -IZ -23 -nô -19 1C3 -05 001 n14 004 lC4 -02 000 '03f: ~10 1
TBI,. 07 08 e() 09 5.1 &06 &07 &05 r.OS
-EFFECT CF INDIVI~UAL TENDONS
lA lB le 10
ZA1 -25 -11 ooZ COb ZA2 -Zo -09 009 -10 ZA3 -15 -06 000 -05 214 -18 -02 024 001 2A5 -11 -03 orO 003 2A6 -18 -oz 004 -oz ZA7 -04 -03 010 -06 2A8 -13 006 GO-' -o~ 2A9 039 -30 0 /,6 020
2el -lb -Da 001 -16 2ez -25 005 -n2 -Ob 2C3 -11 -29 -30 -15 2C4 000 016 016 004 ZC5 -Dl -01 010 002
T8L 61 62 (,0 63 5.1 &60 &61 C.!l9 1:62
------_ ..
1-' w ~
135
5.4 CORRELATION BETWEEN BLOCK MODEL AND
BUTTRESS MODEL TESTS
The block models were investigated for the following
purposes: (1) to determine the behavior of unreinforced and re
inforced beam end-blocks and compare it with theoretical predictions;
(ii) to contrast the behavior of the continuous buttress with that
of thé finite prismatic end-block; (iii) to assess the similitude in
small-scale models in this type of investigation.
The behavior was described in Chapter 4, in terms of the
mode of failure, the ultimate capacity, and the force in the lateral
rein forcement at every load level. Strains in the lateral bars of
the buttress model at working loads were similar to strains at very
early load stages prior to cracking in the block model tests.
Good similitude was observed between the block models of
various scales, especially in the ultimate loads and the mode of
failure. The scatter which was observed in measurements of lateral
strain and force in rein forcement was also noticed among models of
the same scale, and can be attributed to the randomness of cracking
mechanisms', rather than to a size effect.
The block models provided a very conservative estimate of
the behavior of a buttress anchorage zone. Although the finite
limited dimensions wou Id appear, in a two-dimensional analysis, to
induce smaller tensile stresses than would occur in the buttress,
this effect is greatly compensated by the fact that the limitation
of the area over which stresses may distribute increases the stresses
themselves more significantlYi the net result is a conservative
136
simulation. The central position of the bearing plate on the
block model, and the outer facesbeing parallel to the line of
action of the prestressing force, are two more features which
render the block model more conservative.
In the block models reinforced with the light detail of
type l, the highest bar strains were in the range 500 to 700 ~ along
the narrow face, and 800 to 1100 ~ along the wide face (Table 4.4),
at a load level Pif A = 1.0, i.e. one at which the mean pressure ~p
over the bearing plate PIA equals the concrete compressive strength p
f. In a similarly reinforced buttress anchorage zone, strains cu
were of the order of 50 to 120 ~ at a corresponding load P = 1.14 Pd
(Table 5.3). These strains did not increase by more than 359 ~
when the prestressing force was increased to P = 2.0 Pd. Theb~~
models therefore gave a very conservative estimate of the strains in
the reinforcing steel. They likewise were very conservative in the
estimation of general behavior which could be inferred from them, as
visible cracking and ultimate crushing occurred at load levels of
pif A equal to approximately 0.8 and 1.7, respectively, whereas the cu p
buttress model showed no detectable cracking under even higher load
levels.
5.5 STRESS DISTRIBUTION IN BUTTRESS
The buttress test specimen demonstrated that conventional
methods for evaluating tensile forces in the end anchorages of beams,
and the common design rules for detailing the lateral rein forcement
137
against sp1itting, are excessive1y conservative in the context of
anchorage zone design for the buttresses of circu1ar structures.
Design and ana1ysis procedures for beam end zones were
evo1ved from two-dimensiona1 studies, in which the width of the
plate and the end-b1ock were assumed equa1 in the normal direction.
When this condition is not imposed, tensi1e stresses are reduced
because the, compression trajectories radiate in a three-dimensiona1
distribution which is 1ess pronounced than that encountered in the
p1anar stress trajectories. This fact was recognized by Leonhardt(27)
and Guyon (18) , but was not given practica1 consideration, because
the design method which they advocated required the provision of re-
in forcement to carry the total tensi1e force, which wou1d be near1y
the same regard1ess of the stress distribution, as a resu1t of the
integration of sma11er stresses over a 1arger area. Moreover, the
effect in beams end-zones was of far 1ess significance than it cou1d
be in a structure of continuous structural integrity such as a buttress.
The experimenta1 resu1ts confirmed the extensive reduction
of tensi1e stresses in the continuous structure. Despite the use
of structural detai1s providing as 1itt1e as one-quarter of the steel
area required from usua1 design considerations, no structural damage
cou1d be detected under normal loads of 1.14 Pd' and incrementa1
loadings up to 2 Pd' where Pd is the effective design prestressing·
force. The absence of externa11y visible cracking on faces with
and without crack propagators, and the low level of strains in the
horizontal 1ateral steel, suggest that internaI cracking did not arise
under any of the loading conditions to which the buttress was subjected.
138
Reliance on the tensile strength of. concrete to resist
lateral stresses seems realistic and feasible in view of the
experimental results. The provision of lateral rein forcement
could be kept to the minimum steel percentage required to maintain
tolerances on crack widths in such occurrences as cold joints or
unavoidable initial cracking, and to ensure an adequate load factor
on ultimate capacity in compliance with the structural reliability
philosophy incorporated in current ultimate strength design concepts.
Suggested future research endeavours into these and other aspects of
buttress anchorage zone design will be dealt with in a following
section.
5.6 SUGGESTIONS FOR FUTURE RESEARCH
5.6.1 Study of vertical anchorage rein forcement
The present investigation was devoted to finding the stress
distribution in the horizontal lateral reinforcement. It was found
that the tensioning sequence had an influence on the distribution of
the horizontal transverse stresses, even though the cables were situa
ted in a common vertical plane, and the change in shape of the com
pression trajectories would expectedly be predominant in the vertical
direction. The effect of the tensioning sequence on vertical
stresses therefore needs to be investigated, in order that the design
of the vertical anchorage zone steel be optimized with respect to a
predetermined pattern of prestressing force application.
139
The presence or absence of crack propagators in this
case might be quite significant, since the cracking induced by
their presence might be beneficial to the strength of the individual
anchorage zones, by transforming the conceptual symmetrical prisms
of the numerical analogs into physical pris ms bounded by cracked
planes, and offering a much narrower path of travel to the compression
trajectories, than an uncracked integral mass •.
The functional efficiency of the steel could then be
evaluated not only by measuring the strains in the reinforcing bars,
but also, by observing crack widths on external surfaces. The design
method of Gergely and sozen(38), wherein crack widths at working loads
form the principal design criterion, could be given a thorough con
sideration.
Various steel arrangements could be investigated, to find
an optimum reinforcing detail consistent with chosen design criteria,
based on tolerable crack widths, capacity and behavior at working load
and ultimate strength, or sorne combination of these.
The effect of the structural weight and of the longitudinal
prestressing forces would play an essential role in resisting or r~
tarding the progress of cracks, and hence their contribution should be
evaluated and taken into account. Thermal gradients would also have
a pronounced effect in the longitudinal direction. Construction
procedures should be given consideration so that factors contributing
to the anchorage capacity could be mobilized to their fullest extent.
The use of continuous or distinct bearing plates, like the
provision of crack initiators, might have a greater effect on the
140
stress distribution in the vertical steel than it had in the
horizontal bars. The flexural characteristics of continuous
bearing plates would be an important factor.
An adequate determination of the strain distribution in
vertical rein forcing bars would require the placing of strain gages
at the level of each tendon centerline, and midway between these
locations. A scale model investigation is feasible, provided that
the proportion between the length of the strain gages and the spacing
between them is sufficient to avoid a detrimental effect on the bond.
As shown in the models of scale 0.375, interference of the strain
gages with the embedment bond is not critical at that scale and with
gages of the type used. A model of a scale close to 1/3 would
therefore be appropriate for such a study. On the other hand, in
the sixth-scale model investigated in the present program, complete
instrumentation of any vertical transverse bars was out of the question.
The strain gages along with their protective coatings would have dis
turbed the bond over ~ much larger proportion of the area and hence
the stress distribution would not be representative. The investi-
gation therefore concentrated on the study of lateral reinforcement,
and the vertical rein forcement performed a perfunctory role in the
design of anchorage zone details.
A test specimen of the same nature as the buttress model is
suit able for the investigation of strains in the vertical reinforce-
ment. The four bearing surfaces would provide for the inclusion of
continuous and distinct bearing plates, with and without induced
crack initiation. Three tendons at each of the bearing surfaces
141
would suffice to assess the effect of the tensioning sequence.
The thermal system should be maintained, but the middle plane in
clined tendons should be replaced by vertical (longitudinal) tendons
placed as they would be in the containment structure.
AlI anchorage zone reinforcement details in the above
model should be of the same design, in order to achieve a clear dis
tinction between the significant parameters influencing the stress
gradients in the anchorages. Subsequent models could optionally
be fabricated to investigate various anchorage plate designs.
Once the significant parameters have been quantitatively
assessed, optimization of the vertical steel could be attempted on
block models, and finally verified on another buttress specimen
where a single instrumented anchorage zone'would confirm the adequacy.
5.6.2 Further study of horizontal anchorage reinforcement
Lighter steel reinforcements could be investigated, with
the aim of optimizing the lateral reinforcement with respect to the
list of criteria formulated earlier for the vertical reinforcements.
The effect of bearing plate continuity, tensioning sequence, circum
fer~ntial and longitudinal prestressing, and thermal effects being
already known, a "two-dimensional" treatment would be sufficient,
wherein a single "slice" of the buttress containing one plane of
tendons would constitute the test specimen.
Gergely and Sozen's method cou Id be used to determine the
efficiency of crack arresting reinforcement, by measurement of crack
widths and/or relative rotations at the centerline of the tendon,
and at the corner where the buttress joins the containment wall.
142
5.6.3 Study of unreinforced anchorage zones
The continuity of the buttress in the vertical direction
and its integrity with the structural wall result in a three-
dimensional distribution of longitudinal and lateral stresses of
such a conservative nature that the resistance of plain concrete
may be taken into account in design considerations.
Tests to destruction of specimens rep~oducing the essential
features of the buttress geometry should be conducted to determine
the ultimate strength and the mode of failure of the unreinforced
anchorage zone. The resistance to crushing, splitting and spalling
at working loads could be investigated with regard to bearing plate
continuity, presence of crack initiators, thermal and inelastic
effects, vertical and horizontal prestressing, and other parameters
of influence.
The purpose of research into unreinforced anchorage zone
behavior does not come into conflict with that of optimizing the
lateral rein forcement , since tl1e lateral steel is provided to arrest
any unforeseen cracks, and as such it is indispensable. Knowledge
of the resistance to cracking and failure of the unreinforced con
crete would be use fuI towards establishing the degree of structural
reliability (the load factor) inherent in the calculations of the
bearing and the splitting capacities. The possibili~y of erecting
and maintaining a truly crack-free structure is also of significant
concern, particularly when environmental safety is at stake, as is
the case with the construction of nuclear reactor containments.
143
5.6.4 Finite element analyses
The finite element method has already proven its value
as a tool of numerical analysis in structural and continuum mechanics.
The study of the anchorage zone problem as outlined herein has been
successfully attempted by design offices, using a two-dimensional
treatment in the horizontal and vertical planes. (51) . Stress distri-
butions in the con crete were obtained and used in the same way as the
results of classical mathematical or photoelastic analyses of anchorage
zones: reinforcement was provided to resist any tensile forces.
Even in such a simplified form, the analysis"is of considerable value,
as it is not time consuming and is unlimited in its treatment of un-
usual geometrical shapes. It is often simpler to use than other
numerical methods (due to the availability of numerous reliable
packaged programs), and considerably cheaper and faster than experi-
mental investigations.
. (54) The three-dimensional treatment of Yettram and Robb~ns,
which has already yielded significant new data on anchorage zone
stress distributions for structural units having eccentric and
muitiple ~chorages(55) and abrupt changes in geometry, (56) has
particular potential for the solution of the buttress anchorage zone
problem.
This method would be used most advantageously in conjunction
with failure criteria for plain concrete under combined triaxial
stresses in tension and compression. Failure envelopes for concrete
under biaxial compression and tension combinations, in the presence of
a prestressing force applied in the third coordinate direction through
144
a bearing plate or an internal anchorage, were determined by Taylor~58)
in the process of the study of anchorage bearing stresses in pre-
stressed concrete pressure vessels. The criteria for the onset
of rapid crack propagation under tensile stresses, in addition to
those for failure by crushing, would warrant further study, in view
f th b t , b 'l' k' d (4) d 1 (58) f h' h o e 0 serva 10n y Z1e 1ns 1 an Rowe an Tay or 0 19 er
tensile strengths for concrete confined in a compressive stress field.
Thus this method could be used not only to predict stress conditions
with great accuracy, but also to recognize structural damage and
failure, and determine serviceable load levels as well.
Since the introduction of reinforcing elements into the
fini te element formulation can be achieved, this method is also
. (37) compatible with the approach of Lenschow and Sozen. By setting
limits on the tolerable widths of cracks (pre-existing discontinuities
imposed in the geometry), the optimum reinforcing detail could be
designed to insure the structural integrity, safety and appearance,
at unavoidable or unpredictable cold joints or other such discontin-
uities in the concrete mass.
The finite element method can also deal with the presence
of crack initiators, the continuity, flexural stiffness, form and
position of the bearing plates, the sequence of tensioning, the im-
position of strains due to creep, shrinkage, self-weight and thermal
gradients, and any other particularities which can be expressed in
the geometrical, physical or loading data.
By successive trials and improvements in the design of the
buttress and of the anchorage devices and reinforcement~ a structural
145
detail making optimum use of the strengths of the concrete and
rein forcing steel could be achieved. A thorough finite element
analysis constitutes a feasible alternative to the experimental
design and analysis procedure outlined earlier. Direct model
investigations of a much narrower scope could then serve as a veri
fication of this numerical solution.
Studies are presently under way at McGill and elsewhere
to develop a finite element method capable of simulating all be-
havioral aspects of reinforced concrete. Wh en , in the near
future, such characteristics as material non-linearity, steel
concrete interfacial force transfer, failure criteria, constitutive
relationships, crack propagation and the three-dimensional nature
of the problem are fully integrated into the formulation, a complete
analysis and design method capable of reproducing the anchorage zone
behavior up to and including failure, will be available.
CHAPTER 6
CONCLUSIONS
Good similitude of behavior was observed in tests to
failure of micro-concrete models of scales 1/6 and 3/8 of rectangular
prismatic blocks simulating the anchorage zone of one tendon. Each
specimen consisted of a square bearing plate and a tendon duct of a
diameter equal to half the plate width, embedded in a concrete block
whose rentangular face gave ratios of plate width to block width of
0.458 and 0.688, and having a length equal to the wider dimension.
The mechanisms of crack propagation and failure were identical in
models of both scales, and excellent similitude was obtained for
the ultimate strengths. The strains in the individual reinforcing
bars and the total force in the rein forcement showed adequate simili-
tude; sorne scatter of data was observed, which could be attributed
to the mode of crack propagation rather than to a size effect.
Spalling was the cause of failure in aIl reinforced and
unreinforced blocks. The load at which spalling cracks developed
could be accurately predicted by the equation:
where f is the concrete strength, and A and A are the net areas ~ p ~
of the bearing plate and of the face of the end block, respectively.
Unreinforced blocks failed at this load, whereas the presence of re-
inforcement increased the ultimate bearing capacity by an amount
146
147
which depended on the area of steel provided in the region directly
behind the bearing plate. An ultimate strength design of end blocks,
based on the provision of sufficient reinforcement to provide a
suitable factor of safety on the bearing capacity of the anchorage,
would require a smaller steel area than that required by the classical
theories which underlie current design methods. However, if cracking
is a signif~cant criterion of serviceability, the design method of
Zielinski and Rowe(3} , which req~ires the use of more rein forcement
than the classical solutions, would be more appropriate, and not un-
duly conservative.
Load testing of a 1/6-scale micro-concrete model of two
buttresses, reproducing the geometrical configuration of the buttress
and anchorage devices, and containing sixteen tendons bearing on con-
tinuous and individual bearing plates, showed that stress conditions
in the anchorage zones were less severe than could be inferred from
design methods applicable to bearn end blocks. No cracking occurred
at any point, irrespective of the presence or absence of crack initiators,
in spite of anchorage zones being reinforced with considerably less
steel(than theoretically required, and being subjected to tendon loads
up te twice the maximum initial design prestressing force, applied
simultaneously with thermal gradients and additional longitudinal and
vertical prestressing. AIso, the strains measured on lateral re-
inforcing bars were aIl below the allowable working stress level.
since the instrumented bars were in the region where spalling tensile
stresses predorninate, and because the buttress is less vulnerable to
spalling than the block specimen due to the angle of the bearing force
148
and the presence of additional rein forcement placed for this
purpose, it can be conservatively assumed that factors of safety
on the onset of cracking and failure are at least 2 and 3, respectively.
Strains were not measured in the vertical reinforcing bars because·
such instrumentation was not feasible at this scalei however, the
absence of cracks on faces with and without crack initiators at bearing
loads equal to twice the design prestressing force allows for the
generalisation of this last conclusion to cracking in both the horizontal
and vertical planes.
Suggestions for future research include the study of the
tensile stress distribution in the vertical bars y the optimization of
the rein forcement in light of the low stresses observed, and the appli
cation of three-dimensional finite element techniques accounting for
failure criteria and constitutive relationships under combined triaxial
stress states.
Abbreviations:
ACI
ASTM
AS CE
C&CA
CE & PWR
CPCI
CSA
IABSE
ICE (CPCPV)
JPCEA
MCR
PCI
SCS
WCPC
149
LIST OF REFERENCES
American Concrete Ins'i::.itute
American Society for Testing and Materials
American Society of Civil Engineers
Cernent and Concrete Association, London
Civil Engineering and Public Works Review
Canadian Prestressed Concrete Institute
Canadian Standards Association
International Association for Bridge and Structural Engineering
Institution of Civil Engineers, London (Conference on Prestressed Concrete Pressure Vessels, 1968)
Japanese Prestressed Concrete Engineering Association
Magazine of Concrete Research
Prestressed Con crete Institute
Structural Con crete Series, Departrnent of Civil Engineering and
Applied Mechanics McGill University
World Conference on Prestressed Concrete,· San Francisco, 1957
1. PCI
150
Prestressed Concrete Building Code Requirements
2. ACI Building Code Requirements Committee 318 for Reinforced Con crete
3. Zie1inski, J. An Investigation on the Stress Rowe, R.E. Distribution in the Anchorage
Zones of Post-Tensioned Con crete Members
4. Zielinski, J. Rowe, R.E.
5. Zielinski, Rowe, R.E.
6. Rowe, R.E.
7. Rowe, R. E.
8. CSA
9. Iyengar, K.T.S.R.
10. PCI
11.
12. Morsch, E.
13. Magne1, G.
J.
The Stress Distribution Associated with Groups of Anchorages in PostTensioned Concrete Members
Design of Anchorage Zones of Post-Tensioned Prestressed Concrete Elements in the Light of Experi-mental Investigations
End B10ck Stresses in Post-Tensioned Concrete Beams
Local and Bearing Stress~s
Prestressed Con crete
Two-Dimensiona1 Theories of Anchorage Zone Stresses in PostTensioned Prestressed Beams
List of Se1ected References on Anchorage Zone Stresses
Bib1iography on Prestressed Nuc1ear Vesse1s
Uber die Berechnung der Ge1enkquader
Design of the Ends of Prestressed Concrete Beams
ACI Standard 318-63
C&CA Research Report 9, London, Sept. 1960
C&CA Research Report 13, London, Oct. 1962
Archiwum Inzynierii Ladowej 9(1) 1963
Structural Engineer 41(2) Feb. 1963 (54-68) Discussion: 41(12) Dec. 1963 (411-415)
Prestressed Concrete Deve10pment Group, London, 1965
CSA Standard A-135-1962
ACI Journal 59(10) Oct. 1962 (1443-1466)
PCI publication, Aug. 1964
Oak Ridge National Laboratory
Beton und Eisen (No.12) 1924 (156-161)
Concrete & Const. Eng. 44(5) May 1949 (141-148)
14. Magne 1 , G.
15. Bortsch, R.
16. Bortsch, R.
17. Guyon, Y.
18. Guyon, Y.
19. Sievers, H.
20. Sievers, H.
21. B1eich, F.
22. Tesar, M.
23. Sargious, M.
151
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Die Spannungen in Wa1zge1enkquadern
Wa1zge1enke und Ste1zen1ager aus Eisembeton
Contraintes dans les pièces soumises à des forces appliqUées sur leurs bases, au voisinage de ces bases
"Prestressed Concrete" (First Ed.)
Die Berechnung von Auf1agerbanken und Auf1agerquadern von Brückenpfei1ern
Über den Spannungszustand im Bereich der Ankerp1atten von Spanngliedern vorgespannter Stah1betonkonstruktionen
Der gerade stab mit Rechteckquerschnitt aIs ebenes prob1em
Détermination expérimentale des tensions dans les extrémités des pièces prismatiques munies d'une semi-articulation
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Concrete Pub1. Ltd. London, 1950
Beton und Eisen 35(4) Feb. 1935 (61-66)
Beton und Eisen 37(19) Oct. 1938 (315-318); 37(20) Oct. 1938 (328-332)
lASSE (Vol. Il) 1951 (165-226)
Contractors Record and Municipal Engineering, London, 1953
Der Bauingenieur 27(6) June 1952 (209-213)
Der Bauingenieur 31(4) Apr. 1956 (134-135)
Der Bauingenieur (No.9) 1923 (255-259); (No.10) 1923 (304-307)
lASSE (Vol. 1) 1932 (497-526)
Dr.Eng. Thesis, u. of Stuttgart 1960
24. Christodou1ides, A Two-Dimensiona1 Investigation Structural Engineer
25.
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Christodou1ides, Three-Dimensiona1 Investiga-S.P. tion of the Stresses in the End
Anchorage B10cks of a Prestressed Con crete Gantry Bearn.
Structural Engineer 35(9) Sept. 1957 (349-356)
26. Ban, S. Nuguruma, H. Ogaki, Z.
27. Leonhardt, F.
28. Ramaswamy, G.S.
Goel, H.
29. McHenry, D.
30. Spieth, H.P.
31. Dowrick, D.J.
32. Okada, K. Fujii, M. et al.
33. Gardner, A.T.
34. Eimer, C. Doroskiewicz,
R. Michalski, B. Szule, J.
152
Anchorage Zone Stress Distributions in Post-Tensioned Concrete Members
"Prestressed Concrete" Design and Construction (Second Ed.)
Stresses in End Blocks of Prestressed Beams by Lattice Analogy
Lattice Analogy in Concrete Design
The Behaviour of Concrete under High Local Pressure
Anchorage Zone Reinforcement for Post-Tensioned Concrete
Sorne Considerations on Stress Concentration in the Notched Anchorage Zone of PostTensioned Con crete Members
Strength and Behaviour of Spiral Prestressed Con cre te Cylinders Under Eccentric Loading
Anchorage Zone in Prestressed Concrete Elements in the Light of Photoelastic Investigations on Reinforced Models
35. Rydzewski,J.R. Short End Blocks for PreWhitbread,F.J. stressed Beams
36 . Som, P. K. Ghosh, K.
Anchor Zone Stresses in Prestressed Con crete Beams
Proc. WCPC, San Francisco, 1957 (16.1-16.14)
W. Ernst & Sohn Berlin, 1964
Proc. WCPC (23.1-23.4)
ACI Journal (Vol. 20) 1948 (129-138)
Beton und Stahlbetonbau (Vol. 56) Nov. 1961 (257-263)
CE & PWR 59(698) Sept. 1964 (p.ll0l)
JPCEA Journal 6(6) Dec. 1964 (32-39)
M.Sc. Thesis, U. Colorado, 1965
Archiwum Inzynierii Ladowej (Vol. 12) 1966 (p.149)
Structural Engineer 41(2) Feb. 1963 (41-53) Discussion: 41(12) Dec. 1963 (411-415)
ASCE Proceedings 90(ST4) Aug. 1964 (p.49) Discussion: 91(ST2) Apr. 1965 (171-186) 91(ST5) Oct. 1965 (351-353)
153
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stressed Beams
38. Gergely, P. Sozen, M.A.
39. Gergely, P. Sozen, M.A. Siess, C.P.
40. ASTM
41. ASTM
42. ASTM
43. ASTM
44. ASTM
45. ACI Committee 318
46. ACI Committee 104
47. Syamal, P.K.
48. Hsu, C-T.
49. Tsui, H. Mirza, M.S.
Design of Anchorage Zone Reinforcement in Prestressed Concrete Beams
The Effect of Reinforcement on Anchorage Zone Cracks in Prestressea Con crete Members
Method of Test for Compressive Strength of Molded Concrete Cylinders
Concrete Test Specimens, Making and Curing in the Laboratory
Method of Test for Splitting Tensile Strength of Cylindrical Con crete Specimens
Method of Test for Verification of Testing Machines '
Metric practice Guide
Proposed Revision of ACI-318-63 Building'Code Requirements for Reinforced Con crete
Proposed Standard: Preparation of Notation for Con crete
Direct Models in Combined Stress Investigations
Investigation of Bond in Reinforced Concrete Models
Model Microconcrete Mixes
50. Labonté, L.R.S.Size Effects in Microconcrete Mirza, S. Cylinders
ACI Journal 62(11) Nov. 1965 (1421-1439)
PCI Journal 12(2) Apr. 1967 (63-75)
Structural Res. Series, No. 271, 1963 U. Illinois (Urbana)
ASTM Specification C39-66
ASTM Specification C192-69
ASTM Specification C496-69
ASTM Specification E4-64
ASTM Specification E380-70
ACI Journal 67(2) Feb. 1970 (77-186)
ACI Journal 67(8) Aug. 1970 (573-581)
M.Eng. Thesis, SCS No. 17, McGill U., 1969
M.Eng. Thesis, SCS No. 14, McGill U., 1969
SCS No. 23, McGill U., 1969
SCS (in prep.) McGill U., 1971
51. Ku1ka, F. Wahl, H.W.
52. Hanson, N.W.
53. Huang, T.
54. Yettram, A.L. Rabbins, K.
55. Yettram, A.L. Robbins, K.
56. Yettram, A.L. Robbins, K.
57. Ho11and, J.A.
58. Taylor, S.J.
59. Caza1y, L. Huggins, M.W.
60. Labontê,L.R.S. Mirza, M.S. Sabnis, G.M.
61. Mirza, M.S.
62. Wahl, H.W. Kosiba, R.J.
154
American Practices in the Design of Prestressed Concrete Containment Structures
Sei smic Resistance of Con crete Frames Witl1 Grade 60 Reinforcement
stresses in End B10cks of a Post-Tensioned Prestressed Bearn
Anchorage Zone Stresses in Axia11y Post-Tensioned Membe~s of Uniform Rectangu1ar Section
Anchorage Zone stresses in Post-Tensioned Uniform Members, with Eccentric and Multiple Anchorages
Anchorage stresses in Post-Tensioned I-Section Members with End B10cks
Dynamic Relaxation Applied to Local Effects
Anchorage Bearing Stresses
"CPCI Handbook"
Similitude Ana1ysis of Bearing Effect in Post-Tensioned Concrete Construction
PCI Journal 13(3) June 1968 (40-61)
ASCE proceedings 97(ST6) June 1971 (1685-1700)
ACI Journal 61(5) May 1964 (589-602)
MCR 21(67) June 1969 (103-112)
MCR 22 (73) Dec. 1970 (209-218)
MCR 23(74) March 1971 (37-42)
ICE (CPCPV) Group H, Paper 51 (587-595)
ICE (CPCPV) Group H, Paper 49 (563-576)
CPCI 1964
SCS No. 71-7 McGi1l U., Oct. 1971
Direct Models in McGil1 Structural SCS No. 25, Con crete Investigations (1964 to McGill U., Nov. 1969 date)
Design and Construction Aspects of Large Prestressed Con cre te (PWR) Containment Structures
ACI Journal 66 (5) May 1969 (400-412)
