Embed Size (px)
Citation preview
TWO WAY PRESTRESSED CONCRETE SLABS UNDER UNIFORM LOAD
S]MPLY SUPPORTED,;' TWO WAY PRESTRESSED
CONCRETE SLABS UNDER UNIFORM LOAD
by
Gregory John Kemp
A thesis submitted to the Facu1ty of Graduate
Studies and Research in partial fu1fi1ment of the
requirements for the degree of Master of Engineering
Department of Civil Engineering
and App1ied Mechanics
McGi11 University
Montreal, Canada
_ .. _----@ Gregory John Kemp 1972
August 1971
SIMPLY SUPPORTED, TWO WAY PRESTRESSED CONCRETE
SLABS UNDER UNIFORM LOAD
Departmeht of Civil Engineering and Applied Mechanics
Gregory J. Kemp
ABSTRACT
- i -
M. Eng. August 1971
The object of this investigation was to study the strength and
behaviour characteristics of simply supported, two way post-tensioned,
square slabs subjected to a uniformly distributed load. The uniform load
was applied by a large flexible air filled bag bearing against a stiff
reaction frame.
Three slabs were tested to destruction. The second slab was a
duplicate of the first slab and used for verification purposes. The third
slab had a lower volume of steel. than its predecessors but aIl other dimen
sions were identical.
The slabs behaved elastically whilst the section remained uncracked.
Due to the fact that the slab warped slightly due to prestressing and creep
and also the fact that the corners were free to deflect no classical elastic ar analysis cou Id be used. The slabs were analysed by the finite element
method up to failure and the results were in very close agreement with those
found experimentally.
Yield line theory predicted loads of approximately 50% of the
actual failure loads. This discrepancy was a result of the yield line
theory ~eglecting the large amount of tensile membrane action which occurred.
- il. -
DALLES EN BETON SUR APPUIS SIMPLES, SOUMISES A
UNE CHARGE UNIFORMEMENT REPARTIE, ET PRECONTRAINTES
DANS LES DEUX DIRECTIONS
Département de Génie Civil et de Mécanique Appliquée
Grégoire J. Kemp
RESUME
M.Eng. AoUt 1971
L'objet du présent travail est l'étude de la résistance et du
comportement de dalles carrées, simplement appuyées, précontraintes ultér
ieurement et soumises à une charge uniformément répartie. La charge fut
appliquée à l'aide d'un réservoir. flexible d'air comprimé., calé entre la
dalle et un cadre de réaction rigide.
Trois dalles furent essayées jusqu~ à la rupture: la deuxième
était identique à la première, et fut essayée aux fins de vérification;
la troisième avait un volume d'acier inférieur, touts les autres dimensions
restant égales d'ailleurs.
Les dalles se comportèrent é1astiquement tantqueHes sections
transversales ne furent pas fissurées. En raison du fai1 que les dalles
voilèrent 1égèrment à cause de la précontrainte et du fluage, et que les
coins étaient libres de se déformer, aucune analyse classique ne put ~tre
utilisée dans le domaine élastique. Les dalles furent étudiées ana1ytique-
·ment finis, jusqu'à la rupture; les résultats concordent de près avec
ceux des expériences.
La théorie des lignes de rupture a prédit des charges de rupture
égales à 50% des charges réelles. Cette différence est expliquée par le
fait qu'un effet de membrane sous tension important est ignoré par cette
théorie.
- iii -
ACKNOWLEDGEMENTS
The author gratefully acknowledges the assistance provided by
the following people and organizations:
The author's Research Director, Professor J.O. McCutcheon,
Chairman of the Department of Civil Engineering and Applied Mechanics,
for a Research Assistantship and guidance through aIl aspects of this study.
Laurent Labonté and other graduate students for advice on the
instrumentation and assistance during various stages of the experiment.
Ali Youssef for the finite element program used herein.
The technical staff in the Strength of Materials Laboratory
for manual assistance with the experiments.
The lecturers under whom the author has studied, for assistance
in the understanding of structures and materials.
B.F. Goodrich Pty. Ltd. for fabrication of the loading bag.
The Steel Company of Canada for donating aIl the prestressing
steel used.
Miss Wendy Grier for the excellent job in typing this thesis.
TABLE OF CONTENTS
ABSTRACT.
RESUME ••
ACKNOWLEDGMENTS .
TABLE OF CONTENTS
LIST OF FIGURES .
LIST OF PLATES. .
NOTATION. • • .
1. INTRODUCTION. • •
2.
1.1 . General .•••••••.
1.2 Object and Scope ...•••
THEORETICAL CONSIDERATIONS .••
2.1 General .•••.
2.2 Limit Analysis ..... .
2.3 Yield Criteria for Slabs .
2.4
2.5
2.6
Unbonded Prestressing Tendons.
Membrane Action. •
Review of Previous Work on Prestressed Concrete Slabs.
3 . EKPERIMENTAL SET UP .
3.1 General •.•.
4.
3.2 Description of Slabs .•••.
3.3 Control Specimens ..
3.4 Formwork and Casting .
3.5 Concrete. • . • • •
3.6
3.7
3.8
Prestressing • .
Slab Testing Frame .
Instrumentation. .
TEST PROCEDURE AND DISCUSSION OF BEHAVIOUR.
4.1 General ••••••.
4.2 Slab 1
4.3 Slab 2
- iv -
Page No.
i
ii
iii
iv
vi
viii
ix
1
1
1
3
3
3
4
7
8
9
10
10
10
11
13
13
19
22
28
36
36
36
43
5.
TABLE OF CONTENTS (Cont'd)
4.4 Slab 3 . . . . . . . . . . . . . . . . . . . . . ANALYS!S OF RESULTS
5.1 Effective Prestressing at Time of Testing .••
5.2 Yield Line Theory Analysis ••••...•.
5.3 Elastic Analysis . . •. . ...
5.4 Cracking Load ..•.••.
5.5 Finite Element Analysis.
5.6 Strains and Discussion of Resulte .•
- v -
Page No.
46
50
50
50
52
56
59
74
6. CONCLUSIONS. • • . . . . . . • . . . . . . . • . . . . • . . 81
LIST OF FIGURES
Fig. No. Title
2.1 Assumed Stress Distribution in Concrete
2.2 Square Yield Criterion. Isotropie Reinforcement
2.3 Yield Criterion for Orthotropic Bottom Steel and Isotropie Top Steel
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
4.1
4.2
4.3
5.1
5.2
5.3
Steel Layouts for Slabs 1 and 2
Steel Layout for Slab 3
Slab 1. Stress-Strain Curve of Concrete
Slab 2. Stress-Strain Curve of Concrete
Slab 3. Stress-Strain Curve of Concrete
Stress-Strain Curve for 0.276 Diameter Wire
Test Frame Details
Schematic Representation of Loading Apparatus
Slab 1, Deflection Gauge Layout
Slab 1, Strain Gauge Layout
Slab 2, Deflection Gauge Layout
Slab 3, Deflection Gauge Layout
Slab 3, Strain Gauge Layout
Slab 1. Deflection Profile
Schematic Representation of Crack Patterns of Slab 1
Schematic Representation of Crack Patterns of Slab 3
Slab 1, Load v. Central Deflection, lst Loading Cycle
Slab 1, Load v. Principal Strain at Centre. lst Loading Cycle
Slab 3, Load v. Principal Strain at Centre. lst Load Cycle
- vi -
Page No.
6
6
12
12
15
16
17
20
25
26
30
31
32
33
34
37
39
48
54
57
58
- vii -
LIST OF FIGURES (Cont'd)
Fig. No. Title Page No.
5.4 Slab 1, Load v. Central Deflection. 2nd Loading 60 Cycle
5.5 Slab 2, Load v. Central Deflection. lst and 2nd 61 Loading Cycle
5.6 Slab 3, Load v. Central Deflection. lst and 2nd 62 Loading Cycle
5.7 Load v. Strain in Wires in Slab 1. 2nd and 3rd Load .;. ;64 " '.
Cycles
5.8 Load v. Strain in Wires in Slab 3. 2nd and 3rd 65 Loading Cycle
5.9 Load v. Strain in Wires in Slab 3. 2nd and 3rd 66 Loading Cycle
5.10 Slab 1, Deflections parallel to, and 2 inches fram, 67 the Edge Supports
5.11 Load v. Central Deflection of Slabs 1 and 2. 68 3rd Loading Cycle
5.12 Slab 1, Load v. Deflection along Diagonal 69
5.13 Slab 1, Deflections along Slab Centre Line 70
5.14 Slab 3, Load v. Central Deflection. 3rd Loading 71 Cycle
5.15 Slab 3, Deflection along Diagonal 72
5.16 Slab 3, Deflection along Centre Line 73
5.17 Slab 1. Strains along Diagonal. lst Load Cycle 76
5.18 Slab 1. Strains along Diagonal. 2nd Load Cycle 77
5.19 Slab 1. Strains along Diagonal. 3rd Load Cycle 78
5.20 Slab 3. Strains along Diagonal. lst and 2nd 79 Loading Cycles
Ï't!. 5.21 Slab 3. Strains along Diagonal. 3rd Loading Cycle 80 J..
Plate No.
l
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI
XVII
XVIII
XIX
XX
LIST OF PLATES
Title
Jacking Operation
Jack, Pump and Manometer
Bearing Plates and Anchorages
Test Frame and Bag
Slab 1 and Frame, Prior to Placing Edge Supports
Slab 1 before Testing
Slab 3 at a Load of 10 in. Hg.
Slab 1, after Failure
Tension Face Slab 1 after Failure
Corner of Slab 1 after Failure (Top Face)
Bottom Face of Slab 1 after Failure
Crushing of Compression Face of Slab 1
Slab 2 after Failure (Top Face)
Slab 2 after Failure (Top Face)
Slab 2 Crushed Concrete Removed
Compression Face of Slab 2
Detail of Crushed Area on Compression Face of Slab 2
Slab 3 after Failure (Top Face)
Slab 3 after Failure (Top Face)
Slab 3 after Failure (Bottom Face)
- viii -
Page No.
24
24
24
27
27
35
35
41
41
41
42
42
44
44
44
45
45
49
49
49
E
fI C
fI P
fI S
fI t
l
k
L
Mmax
~, My, ~,
p
p
NOTATIONS
= area of prestressing or reinforcing steel
= width of section
compressive force in concrete
= plate stiffness per unit length
- ix -
= effective depth (to the centroid of the tensile reinforcement)
= Youngls Modulus of a material
= compressive strength of concrete
= compressive stress in comcrete due to prestress only, after aIl losses, at the extreme fibre of a section at which tension stresses are caused by applied loads
= ultimate strength of prestressing steel
effective prestress in reinforcement after losses
stress in prestressing steel at ultimate load
= tensile modulus of rupture of concrete
yield stress of mild steel
= mercury
= Second Moment of Area of a Section
= ratio of depth of compression zone at ultimate to effective depth
= a factor which multiplied by f~ gives the assumed ultimate stress at the outer fibre on the compression face
= span length of beam used for modulus of rupture test
= clear span length of slab
= moment capacity of section
= moment causing first cracking of slab
= maximum moment in slab
Mt = moment in the x, y, n, and t directions
= total point load
= steel percentage As/b d
= cracking load of slab per unit area
t
T
y
z
}l
- x -
= cracking 10ad reading of slab per unit area a110wing for weight of slab on the air bag
= u1timate 10ad on slab per unit area
= total slab thickness
= tension force in steel
= distance from outer fibre to neutral axis of a section
= section modu1us
= capacity reduction factor (see AC! 318-63-1504)
= transverse def1ectibn
= curvatures in the x and y directions
= Poisson's ratio
INTRODUCTION
1.1 General
Prestressed concrete slabs are becoming increasingly common in civil
engineering structures. They are used in bridges, lift slab building floors,
pressure vessels and generally in areas with large spans and large loads.
Very little research has been carried out on prestressed concrete slabs and to
date most slabs are designed assuming that they behave elastically. Prestressed
concrete beams have received considerable research attention and are designed
by both elastic and ultimate load methods, but the behaviour of their slab
counterparts, particularly near failure, remains almost unknown. This is inspite
of the vast knowledge accumulated on reinforced'concrete slabs.
The fact that two way prestressed concrete slabs are far more difficult
to model mathematically than prestressed concrete beams or reinforced concrete
slabs is readily seen. Prestressed concrete beams can be considered two dimen
sional and thus it is only necessary to determine moments, strains, stresses,
and crack patterns in one plane. The behaviour of reinforced concrete slabs
can be fairly easily modelled because of its almost perf~ctly elasto-plastic
moment-curvature characteristic. This has led to yield line theory and mathe
matical programming techniques.
1.2 Object and Scope
The purpose of the investigation was to .determine the behaviour of
simply supported two way prestressed concrete slab from zero load to failure.
A uniform load was chosen for several reasons, namely:
1) it is the type of load for which slabs are designed using code
specifications.
2) it is a load which causes the least complications in actual
loading and has no punching or shear consequences.
3) it lends itself weIl to analysis.
In addition to these foregoing factors, uniformly loaded slabs are becoming
increasingly more used by investigators of slab behaviour and hence it lends
- 2 -
itself to comparison with former research. This investigation carries on from
that of Scordelis, Pister, and Lin (1).
The simply supported boundary condition was chosen mainly because
of its fundamental nature. Corners were left free to deflect and consequently
no classical elastic solution exists. The most convenient analysis of plates
with irregular boundary conditions would seem to be by the method of finite
elements. This method was employed herein.
Thus the investigation was set up in order to answer the following
questions:
1) Does the slab behave elastically up to the load causing cracking?
2) Can the cracking load be predicted by elastic theory, using the flexural
tensile strength as determined from plain concrete specimens?
3) How does the behaviour compare in the elastic region with a simply
supported slab in which the corners are held to the supports?
4) How weIl does the yield line theory predict the failure load of prestressed
concrete slabs?
5) What is the physical behaviour of such slabs through the plastic range and
at the ultimate load? In particular, (~ what is the crack pattern?
(b) what is the magnitude of corner
deflections?
(c) does membrane action occur?
(d) is the failure sudden or graduaI?
6) How does the experimental work compare with the finite element analysis up
to failure?
The first slab tested was square, post-tensioned with unbonded tendons.
A duplicate test was made to verify results of the first slab. In addition a
third slab was tested being identical to the first test except for an increase
in the spacing of the prestressing steel. Emphasis is put on a full understand
ing of the behaviour of simply supported prestressed slabs in general and not
on the effects of different parameters.
- 3 -
2. THEORETICAL CONSIDERATIONS
2.1 General
Prestressed concrete structures are generally designed by assuming
that they behave elastically. Prestressed concrete slabs are designed similarly,
however, there is often difficulty in determining the elastic moments due to
complex boundary conditions. Nevertheless these problems can be overcome for
design purposes by using suitable approximations and assumptions. Examples are
the load balancing method and the use of code design specifications. The
ultimate strength of statically determinate beams can be determined and is
already adopted in most codes. Indeterminate structures are a more difficult
problem since their load capacity is not just a property of the cross-section
but also depends on their rotation ability and the geometry of the structure.
The same factors will also effect prestressed concrete slabs.
For reinforced concrete slabs this problem has been overcome by
assuming that the material behaves as a r~gid plastic material. There has been
a large amount of work done on reinforced concrete slabs and many of the
theories can be said to yield a satisfactory estimate of the ultimate load.
The overall procedure is called Limit Analysis and assumes a rigid-plastic
material.
2.2 Limit Analysis
The most essential requirement is a material with a constant yield
point at levels of strain many times that of first yield. Mild steel satisfies
this requirement over a large range. Thus steel plates give a good response
to plastic theory. High tensile strength steel such as that used for prestress
ing is not compatible with the assumption underlying limit analysis. The
theory also assumes a perfectly plastic moment curvature relationship. Thus
for any points on the slab at less than the plastic moment there is no curva
ture i.e. sections away from the fracture lines remain perfectly plane. Also,
once a section has reached its plastic moment it will remain at this level of
moment even though the rotation increases until failure.
There is in practice no perfectly rigid plastic material. Reinforced
concrete does not even behave perfectly elasto-plastically because of the concrete
section not being particularly plastic. Whilst by equilibrium considerations
the constant tension forces in the steel at yield must equal the compressive
force in the concrete the distance between the centroids of these two forces
- 4 -
increases very slowly with rotation thus giving a moment curvature diagram with
an ever increasing moment.
Using rigid plastic assumptions, it is not possible to obtain the
complete load-deflection characteristics of a slab but only the collapse load.
Limit analysis draws attention to the fact that even exact collapse loads are
not always possible and may only provide an upper or lower limit called
"bounds".
These upper and lower bound analyses are weIl documented (2), (3),
(4) and it will suffice to mention only bri~fly their salient points. To
determine an upper bound on the collapse load it is assumed that the slab
behaves as a mechanism and rotates either about a fracture line at yield
(called a yield line) or a boundary. The equilibrium of the surface between
these hinge lines is examined thus equating the slab load and the plastic
moment. This leads to the collapse load for that particular mechanism. In
the terminology of Limit Analysis this is called a "Kinematically Admissible
Velocity System." The theory does have limitations in that if the wrong
failure mechanism is chosen, too high a collapse load will be predicted. Also
the theory does not provide any information as to the distribution of moments
in the non-yielding segments of the slab.
On the other hand, lower bound solutions require the determination
of the entire moment field over the slab. If a pattern of bending and twisting
moments can be found such that the equilibrium conditions are satisfied and
the yield cr~terion is nowhere exceeded then this constitutes a "Statically
Admissible Stress Field" and is a lower bound. A lower bound solution can
never predict a load greater than the true ultimate load.
2.3 Yield Criteria For Slabs
The plastic moment of a section is usually determined by conventional
ultimate load analysis. Both for reinforced and prestressed concrete slabs the
theory assumes that steel is bonded to concrete, that failure is primarily
flexural and the load considered is a short time static load and not a result
of impact, fatique or long term loading. Most slabs will have a small amount
of steel, and failure will proceed by plastic deformation of steel until
rotation of the section reduces the effective concrete depth so that final
collapse is caused by crushing of the concrete. Rarely will steel reinforce
ment rupture. At failure it is usual to assume a rectangular stress distribution
in the concrete. Many elaborate compressive stress distributions have been
- 5 -
postu1ated by researchers but they a11 give an u1timate moment of approximate1y
the same magnitude. In reinforced concrete slabs the force in the steel at
yie1d ASfy is equated to the compressive force k1f~kbd to obtain the effective
depth kd. Refer to Fig. 2.1.
Thus
Locating C at the midd1e of kd yie1ds
M = A f (d - ~) s y ~
= A f d (1 -s y
k1 is usua11y taken to be equa1 to 0.85
where
0.59 pf) f' c
A, p = ~ , the steel percentage.
bd
• • • • • (1)
· • • . • (2)
• • • . • (3)
• • • • • (4)
• (5)
U1timate prestressed concrete design uses the same equation except
that it assumes the steel is stressed to its u1timate strength at the rupture
of the beam. Thus f~ replaces fy where f~ is the u1timate strength of the
prestressing steel (5). However, if the steel percentage is great enough so
that at fai1ure it is not near its u1timate it must be rep1aced by fsu' the
stress in the steel at 'u1t'imate load. There are trial and error methods of
computing fsu for statica11y determinate structures (5).
It is usua11y assumed that the yie1d moment in one direction is
independent of the yte1d moment at right angles or in fact if the moment is
at yie1d or not. This has given rise to square'yie1d criterion as shown in
Fig. 2.2 where the slab is reinforced in the 'x' and 'y' directions. The
slab yie1ds when the bending moment in the 'x' direction MX reaches the plastic moment ± M, providing that the reinforcement is the same top and
bottom. The same app1ies in the 'y' direction providing the reinforcement is
the same as in the 'x' direction. A change in reinforcement in one direction
or at the top will cause a change in the yie1d criteria but it will be
I __ -;;"~ C
d
~T
Fig.2.1 Assumed Stress Distribution in Concrete
+y
My t M
-M -x
-M - y
f~ A
M
C
permissable strains
+x
- 6 -
Fig.2.2 Square Yield Criterion. Isotropic Reinforcement top and bottom.
m'
-~ M x
-ID
Fig.2.3 Yield Criterion for Orthotropic.Bottom Steel and Isotropic Top Steel
- 7 -
still rectangular ego assume negative moment is reduced ta m and 'y' moment
reduced ta m' for positive moments. The yield criteria is as shown in
Fig. 2.3.
In Fig. 2.2 the point A is the condition when the moments along
mutually perpendicular axes reach the yield moment. Such is the case at the
centre of a square slab symmetrically loaded. Point C would refer to the
corner of the same slab providing negative reinforcement is provided. Point
A infers no twist and uniform moment in aIl directions whilst point C implies
twist equal ta the yield moment itself.
Square yield criterion also implies that it applies ta a yield line
not in the direction of the reinforcement. This is justified by assuming
across any fracture line not in line with the reinforcement axes that both
sets of bars must yield. Thus Mx = M and My = M, then for axes 'n'and 't'
likewise Mt = M.
This really only applies to point A where there is a state of aIl
round uniform moment. At other places on a yield line the moment across it is
at maximum but the moment along it will change. Thus Mn = M but Mt(M.
Since Mn is a maximum it is thus a principal moment and consequently so is
Mt' This implies both Mx and My are both less than M. Thus it would seem
that if reinforcement across a diagonal yield line is forced to yield then
the plastic moment is likely to exceed M and thus square yield criterion
would seem to be conservative. Tests carried out by the Building Research
Station in England confirmed this facto A yield moment at 450
to the
reinforcement axes gave up to 16% greater strength than one in the direction
of the reinforcement for slabs spanning one way.
Square yield criteria do not take into consideration the increase
in concrete strength under biaxial compression. The research on this factor
to date (6) would indicate that the uniform biaxial stress is 10 to 80%
greater than the uniaxial strength. Under biaxial stress Young's Modulus
is found to increase slightly and poisson's ratio to decrease.
2.4: Unbonded Prestressing Tenoons
The fore-mentioned considerations aIl assume that the reinforcing
steel is bonded to the concrete. When the tendons are unbonded the stress
- 8 -
in the steel can be distributed a10ng its 1ength. For this reason the steel
stress at the most critica1 section for bending moment is 1ess in an unbonded
section than a bonded section.
As an unbonded beam is 10aded there is 1ittle increase in steel
stress and cracking is not distributed very much. As a resu1t, for the same
moment an unbonded beam has a 1arger rotation at the critica1 section than a
bonded beam. Consequent1y the unbonded beam fai1s at the 10wer moment.
Un1ike a bonded beam the moment-curvature re1ationship (11) of an
unbonded member cannot beana1ytica11y predicted with any assurance since
there is no way of accurate1y knowing the stress in the steel at any particu1ar
10ad. This is further comp1icated by the fact that the stress will vary a10ng
the steel 1ength as a resu1t ·of frictional forces with the concrete. This
is significant at high deformations.
Before the member has cracked the steel stress remains constant
(5) but after cracking it increases at an increasing rate but the steel
stress at fai1ure is a1ways 1ess than maximum strength (10). The tendon stress
at fai1ure de pends on the steel percentage in the section and tests (5), (10)
have shown the increase in stress varies from 20 to 80 k.s.i. This is higher
than the ACI Building Code (ACI 318 - 63) value of 15 k.s.i. Even if it were
possible to estimate the variation of the steel stress between cracking 10ad
and u1timate 10ad the strain distribution over the section wou1d be unknown
(because the steel strain is not equa1 to the strain in the adjacent concrete)
and hence curvature cou1d not be ca1cu1ated.
2.5 Membrane Action
Most slab analyses are based upon sma11 def1ection theory and break
down once the def1ection gradients become large. These large def1ections
cause an increase in stabi1ity and the 10ad at plastic fai1ure can increase
indefinite1y as a resu1t of the slab deforming into a tensi1e membrane. On
the other hand there may occur compressive membrane forces if the slab is
prevented from deforming 1atera11y as in the case of mu1ti-pane1 slabs or
slabs encastered by stiff beams. Compressive membrane action causes sma11
def1ections but is high1y uns table. Membrane action is another factor which
makes yie1d 1ine theory safer than if the materia1 behave~ as it is assumed.
To date most analyses of tensi1e membrane action (2), (7), (12) in concrete
slabs are approximate and re1y upon the steel yie1ding at constant stress and
as such are unsuitab1e for prestressing stee1s. Compression membrane action
has also been investigated (8) and (9), however, this work is of no concern
- 9 -
for the slabs tested herein. It has been found that the relative increase
in strength over the yiêld 1ine solution is greatest for low steel percent
ages wh en the membrane action is tensile (13).
2.6 Review Of Previous Work On Prestressed Concrete Slabs
Most of the published literature on prestressed concrete slabs
has been on methods of design or construction techniques and behaviour.
A1most aIl reports on actual slabs have been non-destructive and no analyses
of ultimate behaviour have been pub1ished. As a result most of the literature
is of little use as background material for this work.
The most significant work carried out has been that by Scordelis,
Pister, and Lin (1) where a 15 feet square, unbonded concrete plate was
tested to destruction under a uniform load. The slab was supported at each
of the four corners on rollers. The stressing tendons were straight with
a centroid at the mid-depth of the slab. In many aspects this test was
similar to those herin, the major difference being the boundary conditions,
which resulted in quite different behaviour.
The conclusions from this test were:
~) Elastic theory gives deflections and strains in very close agreement to
the measured values for loads below the onset of first cracking. The
modu1us of elasticity was estimated from cy1inder tests.
b) The cracking load can be closely predicted by e1astic theory using the
flexural strength of concrete as determined by p1ainconcrete beam
specimens in pure uni-directional bending.
c) The stress in the wires at ultimate fai1ure of the slab was 84% of the
ultimate strength. The effective prestress in the wires before testing
was 51% of the u1timate strength.
d) Ultimate rupture of the slab occurred gradually and at this stage the
slab deflected continuously with no further increase in load.
e) Failure occurred as a result of crushing of the concrete along the
transverse centre line.
f) Large residua1 deflections remained in the slab after the removal of load.
g) If the ultimate stress in the wires can be determined or accurately
estimated the yield line theory gives good results. It should be noted,
however, that in this case the slab failed in a similar manner to a
simply supported beam. This was then a determinate system and 1ittle
membrane action could occur.
3 • EXPERIMENTAL SETUP
3.1 General
The decision variables in a prestressed concrete slab are:
i) The type of loading.
ii) The boundary restraints.
iii) The size of slab.
iv) The method of prestressing.
v) The size and spacing of prestressing wires.
vi) The instrumentation required.
- 10 -
It was decided to test the slabs under a uniform load using air
pressure contained within a light flexible bag. The edges were stmply supported
with the corners free to deflect. The size of the slab was determined from
the following considerations:
i) The mintmum size of prestressing steel and the associated
anchorages and jacks available.
ii) The pressure which cou Id be developed within the air bag.
iii) The size of the loading frame which could conveniently fabri
cated in the laboratory.
The slabs were post-tensioned as is usual for slabs in actual construction
projects. This method is simple and requires no heavy reaction bed as does
pre-tensioning.
3.2 Description of Slabs
AlI slabs were 76 in. by 76 in. and two inches thick. The distance
between the supports was 74 in. in both directions. AlI were post-tensioned
using 0.276 in. diameter wires placed with their centroid at the mid-depth
of the slab. The small thickness of the slab made using draped cables inad
visable for the following reasons:
i) It would be difficult to position them accurately since the
wires used were quite stiff over this small span.
ii) The deflections and warping due to prestressing would have
been large with such a shallow slab. Tensile stresses wou Id result from
this type of prestressing and cracking could occur.
- 11 -
iii) Draped cab les wou1d have increased the u1timate load which was
not wanted since the u1timate load per square foot with straight tendons
was many times higher than that which slabs are expected to resist in practice.
A1so the reaction frame may have deformed excessive1y if the u1timate load
was significant1y increased.
Slab 1 had the tendons spaced
steel percentage of 0.75% and the value
by ACI 318-63.
at 8 inch centres. This gave a
pfsu was 1ess than 0.3 as required -f-'-
c
The steel running east-west was p1aced at the mid-depth and the
steel running north-south was p1aced a1ternative1y above and be10w the layer
at the mid-depth. This a110wed nine wires to be p1aced in each direction.
The wires in the north-south direction were symmetrica1 but one more wire
was p1aced on the tension side of the neutra1 axis than the compression side.
Refer to Fig. 3.1.
Slab 2 was identica1 to Slab 1 and was used for verification
purposes.
Slab 3 had the wires spaced at 12 inch centres and this resu1ted
in six wires in each direction. In the north-south direction a11 wires
were at the mid-depth of the concrete section. In the east-west,'direction
they were p1aced a1ternative1y above and be10w those in the north-south
direction but for symmetry four were p1aced on the tension side and two on
the compression side. Refer to Fig. 3.2.
3.3 Control Specimens
Each slab with their Eespective control specimens required nine
cubic feet of concrete. This necessitated three'batches in the three cubic
feet capacity open pan type concrete mixer. From each batch four 6 in. x 3 in.
and two 12 in. x 6 in. cylinders were cast. The 6 in. x 3 in. cylinders were
chosen for u1timate strength determination at the time of prestressing and
testing. This size was considered to be most suitab1e for the slab thickness
under consideration. The 12 in. x 6 in. cylinders were used for obtaining
the stress-strain characteristics at the time of prestressing and testing.
In addition, a 14 in. x 2 in. x 2 in. beam was cast for each batbh
for obtaining the te~sile modu1us of rupture of the concrete. The beam was
simp1y supported over a 12 in. span and loaded at its third points.
00
.u _Ill
00
76"
8 at 8" = 64"
- 12 -
•
•
•
• •
• • •
--{ t- 2" ~.
r-~------~------~------~~----~~. tension face ~ __ • __ -:.~ __ • __ ~. ____ • __ ~.~ __ • __ ~.~ __ • __ ~I under load
Fig.3.1 Steel 1ayout for Slabs 1 and 2
76"
5 at 12" = 60"
00
o \0
Il
: N .-1
.u III
11"\
00
• •
Fig.3.2 Steel layout for Slab 3.
• • • •
•
•
•
•
•
• - tension face under load
-4 t- 2"
- 13 -
Thus for each slab, there were twe1ve 6 in. x 3 in. and six 12 in.
x 6 in. cy1inders cast, as we11 as the three sma11 beams.
3.4 Fo~ork and Casting
The slabs were cast on a specia11y treated 3/4 inch p1ywood.
The p1ywood was braced underneath in both directions with 4 in. x 4 in.
timber to prevent warping of the slabs during casting. The side formwork
consisted of 2 in. x 2 in. x 1/4 in. ste~l angle sections bolted to the
formwork at 10 inch centres. The fo~ork was 1evel1ed by p1acing chocks
at appropriate positions under the 4 in. x 4 in. timber bracing. By means
of a surveyors 1eve1 and rod the maximum variation in the leve1 of the form
work prior to casting was kept to within 0.006 ft. The stiff timber bracing
and the steel side forms prevented the p1ywood base from def1ecting significant1y
under the weight of the fresh concrete.
The method of preventing bond between the steel and concrete was
by greasing the tendons prior to p1acing them. The wires were then p1aced
in the formwork with sufficient 1ength protruding from the side forms to
permit jacking. The concrete was then cast over the greased wires. This
method of preventing bond required no expensive ducting. Ducting wou1d
a1so take up a large amount of slab depth and wou1d cause the wires running
in the direction of a1ternating effective depths to be further from the
centroid. This is an advantage of using smooth wires compared to mu1ti
wire strand which wou1d mechanica11y bond to the concrete un1ess ducting was
provided.
The concrete was cast for the full depth of the slab and vibrated
with a 2 inch diameter penci1 type vibrator. Afte~ the who1e slab area
had been adequate1y vibrated the surface was 1eve11ed by using a long 6 in. x 3 in.
timber section with a very straight edge. The surface was fina11y finished-
off with a hand trowe1 to give a very smooth surface. The 12 in. x 6 in.
control cy1inders from each bat ch were hand rodded in three layers whilst
the smaller cylinders and the beams were vibrated on a vibrating table.
3.5 Concrete
Type III high ear1y strength Portland Cement manufactured by the
Canada Cement Co. was used for aIl slabs. The coarse aggregate consisted of
a type of crushed limestone and had a maximum size of 1/4 in. The fine
- 14 -
aggregate consisted of crushed silica sand. Two sizes were used in equal
quantities, these corresponded to the A.S.T.M. sieve sizes no. 24 and no. 40.
The mix proportions for·each slab were the same. The ratio of
cement: coarse aggregate; fine aggregate was 1: 1.61: 1.65. The water
cement ratio was 0.50. The aggregate was thoroughly dry prior to mixing.
A representative stress-strain curve is shown for each slab in
Figs. 3.3, 3.4, 3.5. It was found that the envelope of the stress-strain
curves at the time of testing encompassed the stress-stnain curve at the time
of prestressing. Thus it was assumed that there was no difference in the
modulus of elasticity between the time at prestressing and the time at testing.
The strains on the 12 in. x 6 in. control cylinders were measured
using 20 mm. electrical resistance strain gauges. Two gauges were connected
in series in both vertical and horizontal directions on opposite sides of the
cylinder. The horizontal strains allowed the Poisson's ratio to be determined.
Poisson's ratio varied considerably for each cylinder.as has been reported
by many previous investigators. AlI tests indicated a constant ~ over the
first 80% of the ultimate load and increasingsharply at failure. This
constant value of p was always very close to 0.20. Table 3a lists the
measured physical properties of the concrete for each slab. For aIl slabs
the slump was 1.5 inches and the initial tangent modulus of elasticity was
5.0 x 106 p.s.i.
The tensile strength f~ was ob~ained by testing the small concrete
beams in an Instron testing machine under a symmetrical two point load. The
total load P was measured to the nearest pound weight. The associated
moment in the shear free middle zone is thus 1/2 P x 1/3i = P ~ /6 The b d2
section modulus Z = ---6--- and thus the stress in the outer fibre is ~~ ~
For the size of the beams tested herein the stress in the outer fibre equalled
3/2 P p.s.i. The above calculations assume that the beam behaves linearly
elastic until rupture. The author placed strain gauges on the top and bottom
faces of. the beams during testing and could detect no non-linearities.
AlI slabs and control specimens were cured under moist burlap for
seven days.
· ..-f
· II)
· p..
Ul Ul
~ .E-c
Ul
- 15 -
finite e1ement ana1ysis
400 800 1200 1600 2000 2400
STRAIN x 106
Fig.3.3 S1ab 1. Stress-Strain Curve of Concrete
- 16 -
7000
6000
"'
5000
"" . fi) . "p.. 4000
11) 11)
~ ~ 11)
3000
2000
1000
STRAIN x 106
Fig.3.4 Slab 2. Stress-Strain Curve of Concrete
- 17 -
7000
6000
5000
~ . fil . 0..
CI) 4000 CI)
~ ~ CI)
3000
2000
1000
400 800 1200 1600 2000 2400
STRAIN x 106
Fig.3.5 Slab 3. Stress-Strain Curve for Concrete
- 18 -
Table 3a
CONCRETE PROPERTIES
Age at Age at Mean Std. Dev. Post Testing fI c f' c ft:. Tensioning (days)
Batch 1 6560 526 910 Slab 1 Batch 2 6800 317 945
Batch 3 6550 646 960 Average , 6640 481 940 41 63
Batch 1 6920 438 720 Slab 2 Batch 2 6730 282 760
Batch 3 6130 228 690 Average 6490 461 720 7 9
Batch 1 7148 734 870 Slab 3 Batch 2 7074 873 770
Batch 3 7682 350 790 Average 7301 684 830 10 11
- 19 -
3.6 Prestressing
The smallest commercially available plain wire prestressing system
was chosen. This was the Freyssinet mono-wire system using 0.276 inch (7 mm.)
diameter wires anchored by cone anchorages. The mono~wire jack allowed the
wire to pass through the centre of the piston. Each wire was stressed by
the jack and then anchored off by hammering in the radial friction type wedges.
See Plates I, II, III.
The mild steel bearing plates were 4 in. x 2 in. x 1/4 in. In order
to reduce the tensile strains under the bearing plates two 1/8 in. diameter
mild steel deformed bars were placed behind the plates. These were 1-1/2 inches
vertically apart and ran along the whole length of each edge. This also pre
vented spalling of the concrete edges due to handling and yet because of its
size and position would have no effect on the load-deflection characteristics
of the slab.
The 0.276 inch diameter wire had a guaranteed ultimate strength of
235 k.s.i. ie. 14.1 kips and a proportional limit of 196 k.s.i. ie. Il.8 kips.
The 0.2% offset occurred at 224 k.s.i. ie. 13.4 kips. The manufacturers
figures were verified by testing three 15 inch long specimens in the Instron
testing machine. Two strain gauges were placed on each specimen and the
stress-strain curve found to correspond to the one given by the manufacturer.
The ultimate strength of each specimen was always ab ove the guaranteed value
and the average value of these tensile tests was 245 k.s.i. ie. 14.7 kips.
The final elongation was 5%. The stress-strain curve is shown in Fig. 3.6.
The manometer for the jack was calibrated especially for this size
of jack and had a scale graduated in 200 pound increments. In order to cal
culate the coefficient of friction in the jack a wire was tested in a 22 ft.
long pre-tensioning bed. Four strain gauges were placed on the wire and the
strains were recorded for each one kip increment of load as shown on the
manometer. The coefficient of jack friction was found to be 3% which was
in exact agreement with that quoted by the manufacturer.
The complete prestressing operation consisted of the following
stages:
a) The wires were weIl greased and placed in the formwork together with the
bearing plates and the mild steel reinforcement at the edges. The wires
were supported on specially made concrete blocks. The concrete was poured
and allowed to cure before removing from the formwork.
- 20 -
U.T.S.--' 240
200
. ~ . 160 11)
~
tJl tJl ~ ~
120 E-t tJl
80
40
0.002 0.004 0.006
STRAIN in. / in.
Fig.3.6 Stress-Strain Curve for 0.276 Diameter Wire.
- 21 -
b) Prior to post tensioning a11 strain gauges on both the steel and concrete
were read. For the wires in Slabs 1 and 2 which did not have strain
gauges attached the wires were pu11ed in order to obtain the residua1
frictiona1 resistance. The bond broke at an average 10ad of 1200 pounds
and the residua1 friction had an average value of 600 pounds. There was
1itt1e variation for the 26 wires tested.
c) The protruding wires and bearing plates were c1eaned and the anchorages
p1aced and secured at the non-jacking end. The anchorages were 100se1y
p1aced at the jacking end and the jack p1aced over the wire and anchorage
so that it reacted against the bearing plate. The jack was 10aded to
1400 pounds to draw the anchorage at the non-jacking end against its bear
ing plate. The 10ad was removed to 600 pounds and the protruding piston
1ength measured. This gave a value eO' The jack was hand pumped unti1
the manometer read the desired 10ad and the piston 1ength was again
measured, el' The wedge in the anchorage at the jacking end was hammered
in and the 10ad in the jack removed to 600 pounds and the piston 1ength
again measured e2' By taking these piston measurements the wire e10ngation
cou1d be measured and used to verify the manometer readin~ and the strain
gauge readings.
Elongation before transfer = el - eO
Elongation after transfer e2 - eO
The 10ss in e10ngation of the wires at transfer, ie. 10cking-off
10sses, amounted to 0.10 inches which for the 1ength of wire in the slab
amounted to an average 10ss of 2.2 kips ie. 36.6 k.s.i. The effective pre
stress force in the wires at the time of transfer cou1d thus be determined
by knowing the 10ss at transfer. These resu1ts were a1so in close agreement
with the straingauge resu1ts.
As a resu1t of the large 10sses at transfer it was necessary to
stress the wires initia11y to the proportiona1 1tmit in order to obtain final
effective stresses of the same magnitude as those found in practice. The
wires were stressed beginning near the corners of the slab and progressing
in towards the centre. The first wires were over stressed slight1y to a110w
for further compression of the concrete as subsequent wires were stressed. The
amount of over stressing was proportional1y reduced unti1 the final wire was
stressed. The wires were a1so over stressed to compensate for the amount
- 22 -
of residua1 friction. This amount of over stressing being ha1f the residua1
friction ie. 0.3 kips. Prior to transfer the force in the wire at the jack
was 12.1 kips and thus the force in the wire after transfer was 9.6 kips
ie. 67% of the U.T.S. This resu1ted in a concrete stress of 600 p.s.i. for
Slabs 1 and 2 ans 400 p.s.i. in Slab 3. The associated strain in the concrete
was 120 x 10- 6 in. lin. for Slabs 1 and 2 and 80 in. lin. for Slab 3. The
associated 10ss in prestress force in the first wire stressed due to subsequent
stressing was 0.21 kips for Slabs 1 and 2. Thus the first wires were over
stressed by 0.2 kips. This is on1y 1.6% of the total app1ied 10ad. Slab 3
was a1so over stressed by this amount.
The first wires stressed will be further stressed by subsequent
stressing of wires perpendicu1ar to the wires under consideration as a resu1t
of the Poisson effect. Thus for Slabs 1 and 2 where the strain in the concrete
is 120 x 10-6 the strain in the perpendicu1ar direction is p ttmes this -6 amount. For poisson's ratio of 0.2 this amounts to a strain of 24 x 10
and hence the first wires wou1d be further stressed by an amount of 0.04 kips.
This represents 0.3% of the desired 10ad and was thus ignored. A1so it wou1d
have been beyond the range of the manometer sensitivity.
Upon complet ion of the stressing operation the wires protruding
beyond their anchorages were mechanica11y cut off and a11 strain gauges were
read. The slab was then 1ifted into the testing frame and positioned for
testing.
3.7 Slab Testing Frame
The slabs were 10aded by air pressure acting against a steel frame
as shown in Fig. 3.7 The air was encased in a 1ight flexible rubber bag
and was oversize so as to a110w complete contact with the slab without in
ducing membrane stresses in the bag itse1f. Thus theoretica11y the bag shou1d
not need to take any pressure but mere1y act as an air entrainer. The bag
was fabricated from B.F. Goodrich Storitainer materia1 which is a ny10n
reinforced rubber. The materia1 was 2 mm. thick, possessed a high tensi1e
strength and was very flexible. On the under side of the bag was an in1et
for both the air and the hose to the manometer. A polyethylene sheet was
p1aced between the bag and the slab during testing to a110w the bag to slide
easi1y over the concrete surface as it was inf1ated and a1so as protection
for the bag.
- 23 -
The testing frame had to be very stiff in comparison to the slab
in order to ensure the idea1 boundary conditions assumed for the analyses.
The frame was more than adequate in this regard since the channel section and
the box section acted composite1y. Thus differentia1 def1ection a10ng the
supports was neg1igib1e. The base of the reaction frame consisted of
6 ft. x 2 in. x 8.2 lb. channe1s at 10 inch centres in both directions
we1ded to a 3/16 in. thick base plate. At the corners of the frame the
channel section and the box section were bo1ted to their counterparts in
the perpendicu1ar direction. This prevented differentia1 def1ections between
the two edges at the corner.
The slab was set up for testing in the fo110wing way:
a) Prior to p1acing the slab the channel sections were removed and 1/8 in.
amber gum rubber strips p1aced on top of the box section in order to
cushion the slab as it was positioned.
b) The slab was then 1ifted in and positioned.
c) The channel sections were rep1aced and 3/8 in. clearance 1eft between
the slab and the 1 in. diameter support bar. A 1/8 in. thick strip of
neoprene rubber was inserted between the slab and the support. The
remaining gap was required to a110w for the lifting of the corners.
d) The channel section was then accurate1y 1eve11ed, with the use of a
surveyors 1eve1 and rod by adjusting the nuts ab ove and be10w the channel.
This ensures that the slab bears against supports which border one plane.
The channel section was then tight1y bo1ted.
e) In order to prevent the wire or anchorages from shooting out from the
frame in the event of a wire breaking during testing a 1/8 in. thick steel
plate was bo1ted to the f1ange of the channel section as shown in Plate XIV.
The pressure on the slab was measured using a U tube manometer with
mercury. The mercury 1eve1 cou1d be read to 0.02 in. accurate1y. This
represents a pressure of 1.4 1b./sq. ft. The air was supp1ied by bott1ed
compressed air. The inf10w and outf10w of the air was contro11ed by a standard
y valve regu1ator. Refer to Fig. 3.8.
Plates IV and V show the reaction frame prior to p1acing the
channel supports.
24 -
Plate I. Jacking Operation
Plate II. Jack, Pump, and Manometer
Plate III Bearing Plates and Anchorages
- 25 -
~" thick web stiffener
slab
25 lb.
PLAN
" ç--~" thick web s; iffener
12" x 3" x 25 lb. Channel -protective i1 r1 plate E~~=.~==d3.~~~~~~====~ ____ - ~'1" dia. bar
3/4" dia. bo1ts 1: neonrene strio at 10" centres rr-l 9" ~.. .~ ~ ~ ,
'"~ G '. :., '. t
r .... ' j 1_
1 rubber bag
7'-0" x 10" x 5" box ___ ... r3/4" p1ywood base
JI)-~t----------------~ L---1 . L-.J 8' -0" x 6" x' 2" x 8.2 lb. Channel at
~'~'~'~~~~~~~~~~~~'~".==.=="="='='=='="='="=~'''~='.~.'.~.~~~~~ r- .r 'II, '11 10" centre
L 8' -0" X 8' -0" X 3/16" plate
SECTION A-A and.B-B
. Fig.3.7 Test Frame Details
Fig.3.8
slab .'q ...... ., ........ ~ -'" '<1 ". 0.0$ 'v • ·'0
air filled bag
Mercury manometer
regul.ators
Schematic Representation of Loading Apparatus
valve
compressed air
frame
N' 0\
- 27 -
Plate IV. Test Frame and Bag
Plate V. Slab 1 and Frame prior ta Placing Edge Supports
- 28 -
3.8 Instrumentation
A symmetrical square slab possesses the property that each octant is
identical. However, for the slabs tested, it was impossible to have the pre
stressing steel at the same height in both directions and thus they can only be
considered to have identical quadrants. Hence only one quadrant of the slab
was elaborately instrumentated. A check for symmetry was made by taking some
deflections and strains in other quadrants.
The deflections of aIl slabs were measured using 0.001 in. dial
indicators mounted on a steel frame supported by the channel sections. This
gave the deflections of the slab relative to the supports. The deflections
were measured at positions shown in Figs. 3.9, 3.11, 3.12 for Slabs 1, 2, and 3
respectively. In addition it was desired to know the differential deflection
of the supports and so dial gauges were placed along one edge of one channel
section for Slab 1. The deflections were measured relative to the floor but
only the difference in deflections along the edge was of importance. The
positions of these dial gauges is also shown in Fig. 3.9.
Electrical resistance strain gauges were used in Slabs 1 and 3
on both the steel and concrete. The gauges placed on the prestressing wires
were of special significance since to the author's knowledge straingauges
have notbeenplaced on post tensioning wires which use no ducting. The concrete
will tend to scrape the concrete off the wires in such a system because of
there being only a very thin layer of grease between the steel and the concrete.
The problem was overcome by first placing aIl gauges at the non-jacking end
of the wires where relative movement between steel and concrete is a minimum.
Small 5 mm. long gauges were used. The gauges were glued to the surface, water
proofed and then their wires bent back at their middle. The strain gauges were
then placed between sheets of teflon with their ends left protruding. These
ends were soldered to the electrical lead wires. The whole lot was taped with
a plastic backed tape and coated with a special sponge rubber compound. The
whole lot was then re-taped. Thus any movement of up to 3/8 in. between the
steel and concrete'could occur without failure of the gauge. The principle
being that the strain gauge wires were free to slide between the teflon sheets.
In Slab 1 only four out of the ten wires behaved satisfactorily but this was
- 29 -
improved for Slab 3 where nine out of the twe1ve gauges were satisfactory.
The most 1ike1y causes of trouble are:
i) Damage due to casting and vibrating.
ii) Fau1ty soldering of the 1ead wires.
iii) Breaking of the gauge wire because of stressing.
iv) Short circuiting the strain gauge' wires which move during stressing.
By taking due care a11 of these prob1ems can be overcome.
The strain gauges p1aced on the concrete surfaces were a11 20 mm.
long. The strain gauge 1ayouts for Slabs 1 and 3 are shown in Figs. 3.10
and 3.13. The gauges which performed satisfactori1y on the wires are a1so shown.
Plates VI and VII show the test frame with the dia1 gauge 1ayouts for
Slabs 1 and 3.
- 30 -
channel support - - -,
...... --support
1
• -..
Fig.3.9 Slab 1, Def1ection Gauge Layout
."
w S i ~N
E
- 31 -
+-r +-1 . ..
IIIII~~ Prestressing wires ---- ---- ----
LI -1-1-+ +-1 ,I-t LI~ lB%"
~
t-I-trl-i
at 5~"
Fig.3.10a Slab 1, Strain Gauges on Tension Face and Prestressing Wires
18~"
, ..
1"
S--+j--E
Fig.3.10b Slab 1, Strain Gauges on Compression Face
N
- 32 -
edge of slab
101--- support
18~"
1"
2"
Fig.3.11 Slab 2, Def1ection Gauge Layout
74"
- - - --,
18~" 9],..11 9],.." 9],..11 7~" 1~------O--~----O~O~b~O--
9~"! '\ 18\"
1 1 1 ~-------
9~ .. 1 ~ o 0./.;
9~" 1 ~ 6. a
--~~~!- ~~'~ --
Fig.3.12 Slab 3, Def1ection Gauge Layout
- 33 -
2"
1"
1 1 l , 1 1-'-1-'·-1
, 1-1 ~I-I-I 1-1-1-1-1-1--1-1-1-1-1 1
- 34 -
Fig.3~13a Slab 3, Strain Gauges on Tension Face and Prestressing Wires -
1"
Fig.3.13b Slab 3, Strain Gauges on· Compression Face
- 35 -
Plate VI Slab 1 before Testing
Plate VII Slab 3 at a Load of 10 in. Hg.
- 36 -
4. TEST PROCEDURE AND DISCUSSION OF BEHAVIOUR
4.1 General
The density of the concrete was found by weighing the control cylinders
and it was estÜRated that the unit weight of the concrete with the stressing
steel was 145 pounds per cubic foot. This corresponds to 0.34 inches of mercury
(24 p.s.f.) for a two inch thick slab. Thus a load reading of 0.34 on the mercury
manometer meant that the slab was floating freely on the air-filled bag. A
load of 0.68 inches of mercury was equivalent to the slab being loaded by its
own weight. This condition was used as datum for aIl measurements of deflections
and strain. A pressure of 1 in. of mercury corresponds to 70.7 p.s.f or
346 kg./sq.m.
The test program consisted of three distinct loading cycles. In
each cycle the load was incrementally brought up to a certain level and then
completely removed incrementally. The first cycle tested the slab before
cracking; the second cycle checked the first cycle and the slab was then brought
pa st the cracking load; cycle three checked the second cycle and brought the
slab to failure.
4.2 Slab 1
Prior to testing Slab 1 it was noticed that some warping had occured.
This was thought to be a result of the eccentric prestressing in one direction
~ the fact that creep had taken place in the 22 days between post tensioning
and testing. The slab profile was determined by means of a surveyor's level
and rod. It found that a fIat plane could be drawn between four corners.
Fig. 4.1 shows the deflection of the slab relative to this plane.
The first load cycle took the load up to 6 inches of mercury (425 p.s.f.)
in half inch (35.4 p.s.f~ increments and unloaded in 1 inch (70.7 p.s.f.)
increments. This upper level was below the cracking load as calculated in the
next chapter. No cracking was observed at this load level nor was any indicated
by the strain gauges on the tension face nor from the load-deflection curves.
The second cycle took the load to a mercury level of 12 inches (850 p.s.f.
in one inch (70.7 p.s.f.) increments. Cracking was visually observed at 10 inches
of mercury (707 p.s.f.) but indicated by the strain and dial gauges at 8 inches
(565 p.s.f.).
The third loading cycle loaded the slab to failure in one inch of
mercury increments. Strain and dial gauges were last read at a load level
0.02 - 37 -
0.01 ~North ~~ _____________________________________ ~ ________ ~edge
0.00
0.03
0.02
0.01
0.00
~ ______________________________________ ~L/4
0.04
0.03
0.02
0.01
o.oo-----------------------------------------J L/2
0.03
0.02
0.01
o.oo--------------------------------------------------J 3L/4
0.01 ~ / South
0.00 '--__________________ ~_..::::~ edge ~--------------
J "'
J~
W 6.16ft .
~1Il"
6.16 ft.
N
'ft 6.16 ft. '~
S
L/4
L/2
3L/4
Fig.4.1 a) Slab 1. Def1ection profile b\ Key for Def1ection Profile
E
- 3'8 -
of 23 inches (1625 p.s.f.) and failure occurred Just prior to a load of 24 inches
of mercury (1700 p.s.f.) being reached. At high load levels much creep occurred
and thus once the load level had been reached gauges had to be read quickly
since the pressure would drop off due to an increase in deflections. The
deflections at these high loads were very large and it was quite obvious that
the slab was being subjected to tensile membrane action.
The visual crack pattern started at a load of 10 inches of mercury
(707 p.s.f.). The cracks originated near the centre and roughly followed the
diagonal but usually fanned out at the corners. With subsequent increases in
load the cracks along the diagonal widened and many new ones formed. Thus the
slab had a wide zone of cracking emanating near the centre and going to the
corners. The cracks fanned out to the edges rather than pass exactly through
the corner.
At loads around 18 in. of Hg. (1270 p.s.f.) deflections became large
and cracks parallel to the boundary supports formed about 16 ins. frorn the centre.
The deflected shape appeared conical and 'the cracks opened to over 1/4 inch
in parts. The largest crack"-- -; widths occurred for those cracks parallel to the
bounda'ries and these were caused as a result of tensile membrane action. The
slab exhibited a jig-saw puzzle type of cracked pattern with distinct differences
in the level of concrete sections adjacent to a crack. In other words sorne
parts of the concrete looked as though they were being punched out from the
adjoining concrete sections. Figs. 4.2a to 4.2c give the sequence of crack
patterns.
The final rupture was explosive in nature and occurred through com
pression failure of the concrete. The failure zone was not Just a narrow line
but rather a yield region consisting of a wide area of crushed concrete.
The yield zone occurred along the diagonals and along a central crack paraI leI
to the boundary as shown in Fig. 4.2d. The square yield pattern at the interior
of the slab covered a finite width but was centred over the second prestressing
tendon from the centre. That is, the yield line was approximateay 16 ins.
from the centre and thus the square yield pattern had a side length of approxi
'o. mately 32 ins.
This square type of yield pattern has been obtained in most investiga
tions where tensile membrane action is present. Taylor et al (13) found this
central square yield pattern in aIl of their tests on simply supported, square
reinforced concrete slabs. They also noted that near failure the deflections
- 39 -
(a) (b)
Load of 12 in. Hg. Load of 17 in.~' Hg.
( c) (d)
Load of 20 in. Hg. Rupture Lines after Failure
r ' Fig.4.2 Schematic Representation of Crack Patterns of Slab 1.
- 40 -
were large and tensile membrane action was present. Final failure also occurred
by compression of the concrete along the diagonals even though it did not occur
explosively.
Plates VIII to XII show both faces of the slab after failure.
After the test the following points were observed about the destroyed
slab.
a) The explosive type rupture caused myriads of new cracks to open. Along·
the failure lines there were so many cracks that much of the concrete in
this region fell away from the slab upon lifting out of the test frame.
b) AlI corners were cracked at right angles to their diagonals as seen in
Plate X.
c) Across the failure lines the prestressing wires had been permanently bent.
Thus the sudden rupture caused yielding of the wires in this area.
d) The width of the failure lines on the compression face was also wide.
e) The fact that the concrete could be removed along the failure lines permitted
the slab thickness to be measured in these areas. It was found that the
slab, in a11 places measured, was 2 ins. to within :!" 1/64 in. It was ob
served, however, that the wires in some areas had floated up during casting
giving an increase in the effective depth of the slab. As far as cou Id be
determined this increase in effective depth was no more than 1/16 in.
f) The thin layer of grease on the wires had not deteriorated the physical
appearance of the surrounding concrete.
g) After the concrete had been removed from around the strain gauges it was
found that aIl gauges were in good condition, none having been scraped from
the tensioning wire. The fact that sorne gauges could not be used during
the test seemed to be due to
i) Faulty soldering of the lead wires.
ii) Lead wires broken during casting.
iii) Strain gauge wites shorting during stressing.
h) Whilst the interior of the slab suffered appreciable permanent deformation
the edgesremained fairly straight.
i) The central square shape crack pattern did not form simultaneously. Each
side formed at a slightly different load. The actual rupture line occurred
on the crack:,;; Une which was last to forme
j) The anchorages on the prestressing wires were no longer flush against their
respective bearing plates indicating that there was no load in the wires
- 41 -
Plate VIII Slab 1 after Fai1ure
Plate IX Tension Face Slab 1 after Failure
Plate X Corner of Slab 1 after Failure (Top Face)
- 42 -
Plate XI Bottom Face of Slab 1 after Failure
Plate XII Crushing of Compression Face of Slab 1
- ~, -
after failure.
k) None of the wires broke during testing.
4.3 Slab 2
Slab 2 was manufactured to be the same as Slab 1 for verification
purposes. The slab was tested after 9 days compared with 63 days for Slab 1.
Both the compressive and tensile strength of Slab 2 concrete was less than that
in Slab 1, as shown in Table 3a. It was considered that verification could
be provided by a comparison of the load-deflection curves and by comparison of
ultimate load. Consequently Slab 2 was only instrumented with 17 dial gauges.
The loading cycles and increments were exactly the same as in Slab 1.
The behaviour of Slab 2 was almost identical to Slab 1. Visual cracks
commenced at the same load of 10 ins. Hg. (707 p.s.f.). With increase in load
a band of cracks formed approxima~ely parallel with the diagonals. Further
load resulted in large deflections and the formation of cracks parallel to the
boundary. This square pattern formed in the same place as Slab 1 and at a load
of 17 ins. Hg. (1202 p.s.f.) compared with 18 ins. Hg. (1272 p.s.f.) in Slab 1.
The square shaped crack pattern was even more obvious than in Slab 1 as seen
in Plates XIII to XVII.
Prior to cracking it was quite common for cracks to cross over other
cracks indicating a re-distribution in the direction of the principal stresses.
Towards failure the cracks were again very wide (ie. over 1/4 in.), especially
the cracks parallel to the boundary.
Failure occurred explosively and under a constant load of 22 ins. Hg.
(1560 p.s.f.). Slab 1 failed at less than 24 ins. Hg., estimated at 23.7 ins. Hg.
(1675·p.s.f.\., Thus Slab 1 and 2 varied in ultimate load by 7%. The discrepancy
was thought to be mainly due to a small difference in the effective depth.
Assuming at failure that the ultimate capacity is proportional'to the square
of the effective depth then a difference in effective depths of 3·1/2% or
1/30 in. would cause this discrepancy. This shows the extreme care which must
be taken in placing steel in such shallow sections. Another important reason
for the difference in ultimate capacities was the difference in concrete strengths.
After failure the same observations were made as in the previous section
except that two of the edges no longer remained straight but bowed in at the,
middle. This was a result of the large central deflection. The slab had large
permanent deflections but when the slab was pushed back into a fIat plane the
edges also straightened out considerably.
- 44 -
Plate XIII Slab 2 after Failure (Top Face)
Plate XIV Slab 2 after Failure (Top Face)
Plate XV Slab 2 Crushed Concrete Removed
· ,... "!I§; ,la. ·4 q ,:. '
- 45 -
Plate XVI Compression Face of Slab 2
Plate XVII Detail of Cru shed Area on Compression Face of Slab 2
- 46 -
4.4 Slab 3
The first cycle of load went to 5 in. Hg. (354 p.s.f.) in 1/2 in. Hg.
increments and decreased to the dead load (01.68 ins. Hg.) in 1/2 in. Hg. increments.
No cracking occurred during this cycle.
The second cycle brought the load to 10 ins. Hg. (707 p.s.f.) in 1 in.
-Hg. (70.7 p.s.f.' increments and was removed in the same increments. The first
visual cracking occurred at a load of 8 ins. Hg. (565 p.s.f.). Cracking along
the diagonals did not originate from the centre of the slab but rather at the
corners of a 12 inch square central crack pattern as shown in Fig. 4.3a. This
square crack pattern formed between loads of 8 and 10 ins. Hg.
The third cycle loaded the slab in 1 in. Hg. increments until the slab
failed explosively at a load just less than 20 ins. Hg. (1414 p.s.f.). The slab
behaviour was similar to the previous ones. At a load of 15 ins. Hg. (1060 p.s.f.)
two sides of a square formed over the second wires from the centre ie. 18 - 20 ins.
from the centre. The other two sides of the square crack pattern did not forro
clearly as seen in Plate XVIII. The failure lines formed along parts of the
diagonals and on the two cracks parallel to the boundaries and 18 inches from the
centre of the slab as seen in Fig. 4.3c and Plates XVIII to XX.
In comparison to the previous slabs this slab did not crack so exten
sively on the tension side but the cracks were wider, especially the ones on the
central 12 inch square which measured over 3/8 in. in width. On the compression
face there were many more cracks than in Slabs 1 and 2. Large segments of the
concrete had completely crushed along the failure lines and fell away from the
slab as it was lifted from the test frame as seen in Plate XX. After failure
of the slab similar observations were made as for the previous slabs. The most
important being the following:
a) The slab thickness was 2 inches at aIl the places that could be measured in
the interior of the slab.
b) The distance from the compression surface to the centroid of the steel was
often 1/16 in. to 1/10 in. greater than intended. This was thought to be due
to the steel floating upwards slightly because of vibrating during the pour
ing of the slab.
c) Failure lines went directly to the corners. AlI corners had cracked perpendi
cularly to the diagonals.
d) The edges had bowed in at the middle noticeably due to the large vertical
deflections.
- 47; -
e) Wires were permanently bent and most had no load in them after failure.
f) All strain gauges were in good condition after the test.
- 48 -
Fig.4.3a slàb 3 Load of 10 in. Hg.
Fig.4.3b Slab 3 Load of 18 in. Hg~
Fig.4.3c Slab 3 Rupture Lines after Fai1ure
- 49 -
Plate XVIII Slab 3 after Failure (Top Face)
Plate XIX Slab 3 after Failure (Top Face)
Plate XX Slab 3 after Failure (Bottom Face)
- 50 -
5. ANALYSIS OF RESULTS
5.1 Effective Prestressing at Time of Testing
The strain gauges on the wires in Slabs 1 and 3 showed that the wires
were initial1y stressed to an average strain of 5,300, ie. a wire force of
9,200 lbs. A110wing for an average frictiona1 force of 300 lbs. the average
force in the wire wou1d be 9,500 lb. This compares extreme1y we11 with the value
of 9,600 lb. ca1cu1ated from wire e1ongations in section 3.6.
In the time period between prestressing and testing, losses in the
effective prestress wou1d occur as a resu1t of relaxation in the wires and
shrinkage and creep in the concrete. These losses were estimated from documented
tests concerned with these phenomena.
For the stress-re1ieved wire used, Magura et al (14) measured stress
relaxations of 10% of the initial prestress for wire of this age and 1eve1 of
prestress for wires in Slab 1. This amounts to 950 lbs. in the wires. The'
creep of concrete under the prestressing load plus the sma11 amount of shrinkage
between prestressing and testing was estimated, using the work of Neville (15),
to total 200 micro strains or 350 lbs. in the wires. Thus the total effective
force in the wires of Slab 1 were 9500 - 950 - 350 = 8200 lbs.
Simi1ar1y the effective prestresses in Slabs 2 and 3 at the time of
testing were estimated to be 8,500 1b.and 8,300 1b.respective1y. The.uniform
concrete compression stress in Slabs 1, 2 and 3 at time of testing was thus
512 p.s.i., 532 p.s.i. and 346 p.s.i. respective1y.
5.2 Yie1d Line Theory Ana1ysis
Yie1d 1ine theory is on1y usefu1 in predicting the fai1ure load.
A1so it assumes steel yie1ds at a known constant value. Prestressing wires do
not yie1d at a constant load and thus at fai1ure of a member the stress in the
wires will be a function of the properties of the particu1armember. Thus if
the u1timate moment capacity ofl a member is ca1cu1ated from the universa11y
known equation
Mult.= ~ts fsud (1 - 0.59 Asfsu) 1b.in.fin. f~ b d
the value of fsu is not known with any certainty, un1ess experimenta11y measured.
- SI -
The A.C.I. code (ACI 318-63) specifies a value of f = f + 15,000 p.s.i. su se
The measured increases in stress ranged from 78,000 p.s.i. at the centre of
Slab 3 to 18,300 p.s.i. at the edges of Slab 1. Refer Figs. 5.7, 5.8, and 5.9.
For the purposes of ana1ysi~ it wou1d not be conservative to assume
fsu = fse + 50,000 p.s.i.
The intended effective depth was 1 inch, however, due to vibrating
of the concrete during casting the effective depth varied from 1 inch to 1-1/16
inch. Thus for an effective depth of 1-1/16 inch and a value of ~ = 1 the
u1timate moment capacity of each slab was:
Slab 1: Mu1t = 1,300 lb. in. lin. width
Slab 2: Mu1t = 1,310 lb. in. lin. width
Slab 3: Mu1t = 930 lb. in. lin. width
For a square slab on simple supports, distance L apart, under uniform
load qu1t per unit area, yie1d 1ine theory gives
24Mu1t qu1t'= i2
For L = 74 in. = 6.16 ft. the u1tfmàte load for each slab is:
Slab 1: = 1,300 x 24 6.162
= 820 Ib./sq.ft.
= 11.6 inches of mercury
It shou1d be noted that even when the dead and live load are removed the manometer
will still record the dead weight of the slab because the slab has been loaded
from underneath. Therefore, with the dead load effect the slab shou1d fai1 at
a manometer reading of 11.6 + 0.34 = 12.0 inches of mercury (844 p.s.f.).
The slab actua11y fai1ed at a reading of 24 inches Hg.
Slab 2: qu1t = 830 1b./sq.ft.
= Il.7 in. of Hg.
- 52 -
Therefore with dead weight, fai1ure reading shou1d have been 12.0 in.Hg. com
pared to an actua1 fai1ure pressure of 22 in.Hg. (1560 p.s.f.).
Slab 3: qu1t = 590 1b./sq.ft.
= 8.35 in. Hg.
Therefore, with dead weight effect fai1ure reading shou1d have been 8.7 in.Hg.
rather than the actual fai1ure 10ad of 20 in.Hg.
Thus, it is seen that the conventiona1 yie1d 1ine theory (an upper
bound) predicted 10ads which were around 50% of the actua1 fai1ure 10ad. The
difference being due to three important factors:
1. Yie1d 1ine theory assumes no membrane action.
2. Yie1d 1ine theory assumes a perfect1y rigid-p1astic steel.
3. Yie1d 1ine theory assumes bonded reinforcement.
5.3 E1astic Ana1ysis
An object of the investigation was to determine whether the slabs
remained e1astic prior to cracking and if so whether a c1assica1 theory cou1d
be app1ied. No c1assica1 theory exists for slabs which a110w corner def1ections.
However, if the corners are he1d to the supports so that the def1ection a10ng
the supports is a1ways zero a c1assica1 ana1ysis exists.
The c1assica1 ana1ysis of a square slab, simp1y supported under uniform
10ad has been treated by Jaeger (16) and many other authors. It is not intended
that this work be a rigorous treatment of such an ana1ysis and thus on1y basic
equation sha11 be produced.
An e1ement of the slab:under a uniform 10ad q satisfies the Laplace
operator in two dimensions;
v 4w = q/D • . . . . 5. 1
where W = transverse def1ections
D = Et3/12(1 - p2) = plate stiffness per unit 1ength.
A simp1y supported plate has zero def1ection and zero curvature at the supports.
A uniform 10ad q can he put into a Fourrier series form for the purpose of
ana1ysis.
- 53 -
. . . . . 5.2
Substituting into equation 5.1 for the centre of the plate where x = L/2,
y= L/2 gives
. . • • . 5.3
The summation was computer ca1cu1ated for m and n up to the 15th order and was
found to be equa1 to 0.244.
For t = 2
• W- =
Therefore, central def1ection W- = 169 L4 x 0.244
DTT6
in. ; E = 5 x 106 p.s.!.; }l
D = 3.47 x 106 lb. 2/. . .1n. 1n.
0.0352 q inches, where q is
0.0174 q inches, \' where q is
'-
in
in
0.00406 ~ . . . •. 5.4 D
0.2; L 74 in. ;
1b./in. 2 5.5 . . . . . inches of mercury 5.6
The re1ationship between central def1ection and load is shown in F'ig. 5.1
together with the measured ;Load-def1ection curve for Slab 1 (load cycle 1)
and a1so the finite e1ement curve for Slab 1.
The large discrepancy is attributed main1y to the fact that the slab
was not initia11y f1at. As a resu1t of this warping the slab was actua11y sup-
ported on four corner points and thus had a far sma11er stiffness unti1 the edges
came into contact with the supports. From the edge def1ection characteristics
and a1so from a finite e1ement program of a slab supported on four corner points
it was seen that the edges of the slab came fu11y into contact with the supports
at a live load of 2 in.Hg. (141 p.s.f.). Consequent1yonce the edges of the slab
came fu11y into contact with the supports the stiffness increased as seen by
the measured load-def1ection curve in Fig. 5.1. Another major reason for the
discrepancy being the fact that corners were free in the tests and this a1so
gave a lower stiffness than the e1astic mode1.
6.0 r-
5.01--
~ 4.0 " = ..-1
1 ~ 3 3.0
/ 2.0t-- .
)C
I /
2
,,~ ~lastic theory
/ /~ x
./ -/''/ ~
/_./' ./ Je . . . . . Finite Element Resu1ts
CENTRAL DEFLECTION in.
Fig.5.1 Slab 1, Load v. Central Def1ection, lst Loading Cycle VI ~
- 55 -
The greatest bending moments occur at the centre of the slab and
by c1assica1 e1astic theory are re1ated to the curvatures at the centre by
) . 5.7
Substitution into equations 5.1 and 5.2 for the centre of the ~z..w ~2.w
S 1ab where ~ = ~ yie 1ds 0;)(:. "'~
m+n-2 ...lo.(m==--2 _+:.......!::p-:....:n:...2-::-) ...:Io(_-..=.1)=-_-=-2 __ . • • . 5.8
m n ( m2 + n2 ) 2
For p = 0.2 the summation to the 15th order gives
Mx = My = 16q L 2 x O. 269 rr 4
= 0.0442 q L 2
243 q ,
119 q
1b.in./in. for L = 74 in.; q in p.s.i.
1b.in./in. if q is in inches of Hg .•. 5.9
The curvature is re1ated to the moments by
Therefore, curvature at the centre = 28.8 x 10-6 q • • • • • 5.10
where q is in inches of mercury.
The curvature in any direction is equa1 to the strain at the sur
face in that direction divided by the depth to the neutra1 axis from the
surface. For the slabs tested this depth, equa1 to ha1f the slab thickness,
was 1 inch.
Therefore principal strain at centre = d\l."", x t
where q is in inches of mercury.
d',X," 2
-6 = 28.8 x 10 q
- 56 -
••••• 5.11
This re1ationship is plotted in Figs. 5.2 and 5.3 together with the measured
principal strains on both the tension and compression faces of Slabs land
3 respectively. The discrepancy is attributed to the same reasons as that
for the discrepancy in the load-deflection curves.
The ab ove elastic relationships are applicable to aIl the slabs
tested. The small amount of steel placed at the centroid has negligible
influence on the moment of inertia of the section and thus aIl slabs had
the same stiffness prior to cracking.
'5.4 Cracking Load
The cracking load will vary for each of the slabs since it is
dependent upon the level of prestress in the concrete and the tensile strength
of the concrete. The, slab is assumed to crack when the tensile stress in
the concrete exceeds the modu1us of rupture.
For a level of prestress of fp and a modulus of rupture of ft
the bending moment causing a stress equàl to the modulus of rupture is
given by
Mc = (fp + f~) l y
. • • . • 5.12
The load causing the cracking moment is obtained from equation 5.9.
lb. in. lin. where qc is in inches of Hg.
cracking load qc = ~(_fLP~+~_f~t~)~I_ 119 y
inches of Hg.
This results in the following cracking moments for Slabs 1, 2 and 3.
6.0
5.0
bO 4.0 = s:: r-..-l
~ 3.0 ~
1
1 2.0
1.0
/ 1
/
4'
)t
. elastic theory ~;'
,
/
/
40
/ " tension face
;'
I .1
././
~
80 120
YY· compression face
)1;
" . • . . • Finite Element Results at Point P
200
STRAIN x 106
280
Fig.5.2 Slab 1, Load v. ~rincipal Strain at Centre lst Load Cycle
VI .....
6.0
5.0
. 1 = 4.0 ....
elast:l.c theory tens:l.on face Il
~ o 1./ /..- compression face ~
~
STRAIN x 106
~ • • • • • F:l.nite Elemen~ Results
VI 00
Fig.5.3. Slab 3, Load v. Principal Strains at Centre, lst Load Cycle
- 59 -
Slab 1: qc = 8.1 inches of Hg. (572 p.s.f.)
a110wing for dead load effect the slab shou1d crack when the
mercury 1eve1 read q~ = 8.5 inches.
Slab 2: qc = 7.0 inches Hg. (495 p.s.f.)
.'. with dead load effect q~ = 7.4 inches Hg.
Slab 3: q = 6.6 inches Hg. c (467 p.s.f.)
.'. with dead load effect q~ = 7.0 inches Hg.
The st~ain gauges on the tensi~n face of Slab 1 indicated crack
ing commenced between load 1eve1s of 7 and 8 inches of Hg. The strain gauges
on the tension face of Slab 3 indicated cracking commenced at a load of
6 inches of mercury .(425 p.s.f.).
Cracking can a1so be considered to commence at the start of the
non-1inear section of the load def1ection curves. From Figs. 5.4, 5.5,
and 5.6 this is seen to be at loads of 8; 8~nd 6 inches of mercury for ::'
Slabs 1, 2 and 3 respective1y.
5.5 Finite Element Ana1ysis
This experimenta1 work was ana1ysed by A. Youssef using the
finite e1ement method. The program deve10ped by Youssef (17) made the
fo11owing assumptions.
1. The slab behaves incrementa11y e1astic. Thus for each incre
ment of load the slab hehaves e1astica11y. At the end of each load the
amount of cracking is determined and a new slab stiffness, based on the
effective depth, found.
2. The stress-strain curve of concrete is bi-1inear as shown
in Fig. 3.3.
3. The force in the wires remains constant for the first and
second loading cycles. This was found from the strain gauges on the wires
to be approximate1y va1id. Refer Figs. 5.7, 5.8, 5.9. At higher loads,
when membrane forces became apparent, the forces in the wires were increased
in the finite e1ement program to the experimenta1ly measured values.
4. The slab was considered to be supported on four corner sup
ports up to a live load of 2 in. Hg. (141 p.s.f.) and afterwards simp1y
12.0
10.0
00 8.0 ::t:
~
~ 6.0
4.0
2.0
". / 3rd load. cycle
)C . .. Finite Element Results
DEFLECTION in.
Fig.5.4 Slab 1, Load v. Central Def1ection, 2nd Loading Cycle
0\ o
12.0
10.0 ~
. bO ::t: 8 0 s:: ~
~ 3 . 6 0
40
2.0
Fig.S.S Slab 2. Load v. Central Deflection. lst. and 2nd. Loading Cycle.
.. - X
0.2 0.4
/-
/"
~~ 3rd loading cycle ,/
Je. . Finite Element. Results
0.6 0.8 1.0 1.2 1.4
CENTRAL DEFLECTIONS in.
Q\ ....
10.0
8.0
. bD ::t: . s:: 6.0 ~
.~ 0 ,..::
4.0
2.0
1
/ /
/ /
;1"
/
,., /
~ 3rd. Loading Cyc1~
Jt . . . Finite Element Results .
CENTRAL DEFLECTION in.
Fig.5.6 Slab 3, Load v. Central Deflection. lst. and 2nd. Loading Cycle
0\ N
- 63 -
supported a10ng a11 four edges. The corners were a110wed to def1ect.
By examining the def1ections a10ng the boundary it was possible to estimate
the point at which the e1ement was no longer in contact with the edge
because of corner lift. Refer Fig. 5.10. This point was estimated to be
8 inches from the corner. Thus in the ana1ysis, after a load 'of 2 in.Hg.
(141 p.s.f.) the slab was supported a10ng a11 of the edges except for a
free hanging 1ength of 8 inches from the corner.
5. Fai1ure of the slab occurred when a11 e1ements a10ng the
diagonal had comp1ete1y crushed on the compression side and cracked on
the tension face.
The resu1ts of the finite e1ement program have beengiven previ
ous1y in Figs. 5.1, 5.2, 5.3, 5.4, 5.5, and 5.6. A11 the fo11owing load
def1ection curves show the finite e1ement resu1ts where applicable.
The finite e1ement program yie1ded a fai1ure load of 22 in. Hg.
(1550 p.s.f.) for Slabs 1 and 2. This is in very close agreement with the
experimenta1 values of 24 and 22 inches of mercury. The load def1ection
curves were a1so very close at a11 load 1eve1s inc1uding the u1timate.
Refer Figs. 5.11, 5.12, 5.13. The biggest discrepency however, did occur
near the u1timate. This was a resu1t of the slab actua11y increasing in
stiffness b~cause of extra st?bi1ity arising from membrane action. The
finite e1ement program did not direct1y account for the membrane action
and thus gave central def1ections 15% 1arger for Slab'l and 6% 1arger
than the central def1ections of Slab 2 at incipient fai1ure. At load 1eve1s
be10w 19 in. Hg. (1340 p.s.f.) the def1ections were extreme1y close to the
experimenta1 values at a11 points in the slab as shown in Figs. 5.12 and
5.13.
For Slab 3 the finite e1ement program a1soshowed excellent agree
ment with the measure~ load-def1ection curves for a1l points in ,the slab,
up to a load of 16 in. Hg. (1130 p.s.f.), as shown in Figs. 5.14, 5.15
and 5.16. The predicted fai1ure load was 19 in. Hg. (1340 p.s.f.) compared
to the actua1 value of 20 in. Hg. (1414 p.s.f.) and was again in excellent
agreement. At loads above 16 in. Hg. (1130 p.s.f.) the finite e1ement
solution predicted def1ections 1arger than the measured values for the
same reasons as it did for Slabs 1 and 2. Thus the predicted fai1ure 10ad
was accurate to 5% and the final def1ections accurate to 23%. For loads
be10w 16 in. Hg. the method gave def1ections which were genera11y within
6% of the measured values.
24.0
20.0
. 16.0 bO ::r:
= ..-1
@ 12.0 . ...:.
8.0
4.0
Position of Wire in Slab
3
INCREMENTAI. STRAINS x 106
Fig.5.7 Load v. Strain in Wires in Slab 1. 2nd and 3rd I,oading Cycles
2
1
4 1
--1,.-- ...
0\ ~
24 0
20.0
. 16.0 tlO
::r: .
c:: o,.f
~ 0
12:0 ~
8.0
4.0
Fig.5.8 Load v. Strain in Wires in Slab 3. 2nd and 3rd Lo~ding Cycle
INCREMENTAL STRAINS x 106
~osition of Wire in Slab
1 2 3 4
5 6
7 8 9- 10 Il 12
76"
C'\ \J1
Fig.5.9 Load v. strain in Wires in slab 3. 2nd and 3rd Loading Cycle.
20.0
16.0
. bD
~ 12 .. 0 1 . / / / /. ~ ~osition of Wire in slab s:: ....
~.8.0[ A ~ fiJ//1~ 1 2 3 4 5 (;
4.01- IJ/ 1 Il . \1 / J , / \1 / • • 7 8 9 10 Il 12
400 800 1200 1600 2000 2400 2800 3200 3600 4000
INCREMENTAL ST RAINS x 106
4400
0\ 0\
0.4
1
0.3 ~ = -..1
CIl 0.2 z 0 t-I E-4 r-e,)
~ rz.. r:J 0.1 I=l
-0.1
~ .12 in. Hg.
~ •
6 in. Hg.
// ~ 3 in. Hg.
~
ct.
DISTANCE FROM CORNER in.
Fig.S.10 Slab 1, Def1ections parallel to, and 2 inches from, the Edge Supports. 0\
"
24.0
20.0
16.0 . bD
== = opof
~ 12.0
< 8
8.0 1-
4.0
Fig.5.ll Load v. Central Deflections of Slabs 1 and 2. 3rd Loading Cycle.
-
~~ • • • Slab 1 • • • Slab 2
U" ". • Finite Element Results
0.4 0.8 1.2 1.6 2.0 2:4 2.8 3.2
DEFLECT ION in.
0\ 0)
3.0
2.S
!i 2.0
z S ~ u ~ 1.5 ~ ~
1.0
0.5
Fig.S.12 Slab 1, Load v. Deflection along Diagonal.
6 in. Hg.
centre of slab DISTANCE FROM CENTRE in.
1( ... Finite Element Results
0\ \0
.>
Fig.5.l3 Slab l, Deflections along Slab Centre Line.
3.0 ~
)t ••• Finite Element Results 'T 20 in. HIZ. --............
= ....
~ .~2.0
~ ~
12 in. Hg. 1.0,"-
8 in. Hg. 'C' ~~"'- ec!ge
If Ji' ~
"'" 5 10 15 20 25 30 35
..... DISTANCE FROM CENTRE in. 0
centre
16.0
. tIO
=t: .
oS A
~ ...:1 8.0
Fig.5.l4 5lab 3. Load v. Central Deflection. 3rd Load Cycle.
x·
'-
". • • Finite Element Results "-
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6
CE~RAL DEFLECTION in. "'-1 ....
"
.s ~ H E-I tJ
~
3.5 a..--__
3.0
2.5
~ ~ -1.5
centre
Fig.s.ls Slab 3. Deflection along Diagonal
8 in. Hg._
6 in. Hg.
DISTANCE FROM CENTRE in.
)(. • • Finite Element Results
....a N
4.0
3.0
c:: ~
al Z 0 H E-I 2.0 CJ
~ ~ ~ Q
1.0
centre
Fig.5.16 Slab 3. Def1ection a10ng Centre Line
in. Hg.
in. Hg.
5 10 15 20
~. • . Finite Element Results
25 30 35
DISTANCE FROM CENTRE in. " w
- 74 -
_ 5.6 Strains and Discussion of Resu1ts
The strain gauge resu1ts are shown in Figs. 5.17 to 5.21. Strain
gauge resu1ts are not a1ways representationa1 and this fact often resu1ts
in erratic strain curves. The gauges were on1y 20 mm. in 1ength and thus
cou1d give unrepresentationa1 resu1ts, dependent upon crack locations.
A1so cracking cou1d cause the epoxy precoat materia1 to spa11 of thus giving
erroneous resu1ts. These factors account for measured strain distributions
which are impossible to exp1ain. Neverthe1ess definite trends are dis
cernib1e.
At loads be10w the cracking load strains were greatest at the
centre and 1east at the corners. Before cracking occurs the strains a10ng
the diagonal are the same on both- the tension and compression faces.
This indicates that the strain gauges were working we11 -at this 1eve1 of
load. Once cracking occurred the strain gauges on the tension face were
of no further use since the resu1ts were direct1y inf1uenced by the loca
tion of the cracks.
At loads up to 12 in. Hg., the same trend of strain distribution
as previous1y, occurred along the diagona1s of Slab 1 but with much 1ess
regu1arity as seen in Fig. 5.18. The location of cracks obvious1y inf1u
enced the individua1 readings. At loads higher than this the strains in
the centre reduced whi1st those towards the corners increased. Towards
incipient fai1ure the strains in the centre due- to the load were a1most
zero. Thus the compressive f1exura1 stresses had been reduced by tensi1e
membrane action. Concomitant with this region of tensi1e membrane stresses
was an outer ring of compression stresses. This behaviour had been wide1y
acknow1edged for reinforced concrete slabs. The strain gauges on Slab 3
gave the maximum strains at the corners as stated by Woods (2), for reinforced
concrete slabs. The gauges on Slab 1 gave the maximum strains near fai1ure
at a position about 17 inches in from the corner. It is difficu1t to
know whether this is a true picture of the strain distribution a10ng the
fai1ure 1ines or unrepresentationa1 local readings. However, it can be
categorica11y stated that towards fai1ure the compressive stresses are
very low near the centre and reach a maximum at or near the corners on_ the
diagonal. Thus fai1ure occurred initia11y near the corners due to compres
sion fai1ure of the concrete. Once the concrete had comp1ete1y failed
in this region the load was immediate1y taken by an adjacent section on
the diagonal. This subsequent1y fai1ed, thus resu1ting in a "domino"
- 75 -
type fai1ure a10ng the diagona1s towards the centre. The cDushing did not
continue to the centre but stopped at the square crack pattern around the
centre (which had been caused by tensi1e membrane action). Fina11y the
concrete fai1ed'.in compression a10ng one or more of the sides of this
square crack pattern resu1ting in a mechanism and complete destruction of
the slab.
The magnitude of the strains of Slab 1 near fai1ure were about
10% 1arger than those recorded by the uniaxia1 compression test at u1timate.
For Slab 3 the recorded strains were 50% 1arger. This was to be expected
since the compressive tests on the concrete cy1inders cou1d not measure
strains past the u1timate load. Thus no descending portion of the stress
strain curve cou1d be found. The maximum strains recorded were of the
same order of magnitude as those connnonly pub1ished for uniaxia1 compression
ie. u1timate strains around 0.003. Thus the biaxia1 state of stress did
not significant1y affect the u1timate strain capacity of the slab. However,
the stresses at these 1eve1s of strain cou1d have been significant1y ine
creased due to the biaxia1 state of compression.
The synnnetry of the loading system was checked by p1acing def,lec
tion gauges a10ng the diagonal on both sidesoof the centre as we11 as
a10ng two sides of the centre 1ines. A11 slabs showed very good synnnetry
espeêiàlly.' âfter ·loads of 4 in. Hg. (283 p. s. f .). Prior to t,his load the
warping a10ng the edges of the slab caused def1ections to be unsynnnetrical.
llast a load of 6 in. Hg. (424 p.s.f.) the synnnetry of def1ection was within
5% and this value became sma11er with increasing load 1eve1s.
400
300
\0 L 0 .... ~
~ 200 ~ ~
100
!
centre
( Strains on Top Surface
o Strains on Bottom Surface
~ 6 in. Hg.
25 30
DISTANCE ALONG DIAGONAL in.
Fig.5.17 Slab 1. Strains a10ng Diagonal. lat Loading Cycle. -.-J 0'\
1400
1200
1000
\0 0 .-1
~
z 800 H
~ E-I tJ)
600
400
200
Fig.S.18 Slab 1. Strains along Diagonal. 2nd Load Cycle.
12 il\ Hg.
Hg.
Hg.
T\T~'l'ANr11' AT()Nr.! T\TAr.!()NAT. ;n_
corner
40' 45
...., ....,
3000
2500
\0 2000
0 ~
~
Z H
~ 1500 ~
1000
500
centr~
Fig.5.l9 Slab 1. Strains along Diagonal. 3rd Load Cycle.
5 15 20 25 30
DISTANCE ALONG DIAGONAL in.
35 40 45 50 ..... co
400
300 \D
0 ..... ~
z 200 1-1
~ E-4 CIl
100 .
centre
Fig.s.20 Slab 3. Strains Along Diagonal. lst and 2nd Loading Cyc1es~
--.......... 5.0 _in. Hg.
2.5 in. iIg. ----
5 10 15 20 25 30 35 40
DISTANCE ALONG DIAGONAL in.
45 50
corner
...... \D
\0 0 .-4
~
z H
;1 E-I CI)
Fig.5.21 Slab 3. Strains a10ng Diagonal. 3rd Loading Cycle.
3500
3000 1 /
19 in. Hg.
1 / 2500
".
2000
1500 1 // ~
15 in. Hg. r- I/ / 1000
500 E ~ 12 in Hg.
~> 5 10 15 20 25 30 35 40
centre DISTANCE ALONG DIAGONAL in.
45 50 00 o
- 81 -
6. CONCLUSIONS
1. Before cracking occurs the slabs can be considered to behave
e1astica11y. In other words they will have the same 10ad def1ection curves
for subsequent 10ading cycles. The agreement will not be perfect due to
the presence of micro-cracking as we11 as to time effects.
2. The ana1ytica1 e1astic mode1 of a square slab under uniform
10ad was not va1id because for the slabs tested the edges were not straight
and thus the boundary conditions were not compatible. A1so, the theory
assumes,; that the corners are he1d to the support whereas in the experiment
the corners were free to def1ect.
3. The cracking 10ad as determined by the e1astic theory of a
square simp1y supported slab with corners he1d gave a reasonab1e estimate
of the cracking 10ad. The maximum discrepancy was for Slab 3 where the
theory predicted a value of 14% greater than that inèicated by the strain
'gauges as we11 as, that observed from the 10ad def1~ction curves. For
Slab 1, the theory over-predicted the 10ad by 6% and for Slab 2 under
predicted the value by 8%. Thus it wou1d seem that whi1st the 10ad -
def1ection characteristics predicted by the theory were gross1y in disagreement
with those measured, the cracking 10ad 'can be predicted fàir1y accurate1y.
4. Yie1d 1ine theory predic~ed u1timate loads of approximate1y
50% of the actua1 fai1ure 16ads of a11.s1abs .. Thus it can be conc1uded
that yie1d 1ine theory is of 1itt1e he1p in predicting the actua1 fai1ure
10ads of two way prestressed concrete slabs on simple supports. However,
the yie1d 1ine theory 10ad i8 conservative for these boundary conditions
and hence cou1d be of some value in the absence of other analyses.
The yië1d 1ine theory 10ads were in error main1y because the
theory does not a110w for the fo110wing factors:
a) 'Tensi1e membrane action, which at high deformations reduces
the compressive stresses in the central region. Additiona1 10ad must be
app1ied to bring the strains, further out from the centre, to their u1timate
capacity. SO whereas yie1d 1ine theory assumes the u1timate strains will
be first reached at the centre the influence of membrane action resu1ts
in maximum strains at or near the corners. A1so the uniform prestressing
resists the membrane action more than a reinforced concrete section.
Thus a 1arger 10ad is required to produce a rotation large enough to com
p1ete1y crush the concrete.
- 82 -
b) Yield line theory assumes a rigid-plastic stress-strain
curve for the reinforcement. ~restressing steel does not have this pro
perty. As the load is applied the steel strains and the force in the wire
increases thus increasing the moment at that section. Thus prestressing
steel i~ibits the section rotating more than a similar sized mild steel
reinforced section.
c) Other factors include the fact that the yield criteria is
conservative for reinforcement not perpendicular to a yield line and does
not take into accoti~t the increase in concrete strength under biaxial com
pression., Yield line theory also assumes bonded reinforcement and thus
the yield moment of an unbonded section should be used in calculations.
This latter factor has not be thoroughly investigated but it is generally
believed to decrease the moment capacity (5).
5. The cracking pattern, for loads below 75% of the ultimate
load, was the familiar diagonal pattern for square simply supported slabs.
At incipent failure a square crack pattern occurred as a result of tensile
membrane action. The failure lines occurred from the corner, along the
diagonal until the intersection with the central square pattern whence the
failure occurred on one or more of the sides of this square crack pattern
until a mechanism formed.
6. The corners of aIl slabs deflected. For loads in the elastic
region these deflections were very small (ie. less than 0.01 inch). For
loads past the cracking load the corner deflections increased substantially
and were 0.2 inches for Slabs 1 and 2 and 0.15 inches for Slab 3 at failure.
The rate of deflection decreased with the onset of membrane action as did
the deflection at every other point in the slab.
7. Central deflections were large at failure (three times the
effective depth of the slabs). Failure occurrèd gradually if defiried'i,as
large deflections and much cracking. However, the final failure occurred
abruptly and with litt le warning, as a result of complete crushing of the
concrete.
8. The finite element method developed at McGill,which assumed
that the slab behaved incfementally elastic gave results in very close agree
ment with the experimental results for aIl ranges of load. The method
- 83 -
predicted fai1ure within 8%, 0% and 5% for Slabs 1, 2 and 3 respective1y
and was a1ways 10wer than or equa1 to the actua1 resu1ts. The finite e1e
ment resu1ts were 10wer because they did not direct1y a110w ~or membrane
action. The steel stresses used in the finite e1ement program were obtained
from the experimenta1 vâ::lues. The 10ad def1ection curves were very close
except at incipient fai1ure where the method predicted def1ections 6%, 15%,
and 23% greater than those measured for Slabs 1, 2 and 3 respective1y.
These discrepancies were again due to the fact that; the me"thod did not
direct1y account for membrane action.
9. The 10ad-def1ection curves, u1timate 10ad, cracking 10ad,
cracking pattern and u1timate def1ections of Slabs 1 and 2 were a1most
identica1. Thus the resu1ts of Slab 2 can be said to verify those of Slab 1.
Slab 3, which had 33.3% 1ess steel than slabs 1 and 2, fai1ed at an u1timate
10ad of 13% 1ess than the average 10ads of Slabs l::~nd 2. Thus a considerab1y
10wer steel percentage did not resu1t in a corresponding decrease in the
u1timate 10ad. This is attributed to the fact that the steel does not
yie1d at a constant value. So whi1st the steel percentage was 1ess, each
of the wires in Slab 3 contributed a 1arger force than those in Slabs 1
and 2. Another important reason for this phenomena is the fact that the
slab was subjected to large tensi1e membrane stresses. So long as the wires
did not fai1 before the concrete, the capacities as membranes wou1d be of
simi1ar magnitude. This phenomena was noticed by Taylor et al. (13) for
reinforced concrete slabs. Slab 3 def1ected more than the other slabs under
the same 10ad since the steel had to strain more to mobi1ize the same tensi1e
forces. The u1timate central def1ection of Slab 3 was on1y 12% greater than
the average u1timate def1ection of Slabs 1 and 2. Strains can be considered
a derivative of def1ections and thus it wou1d seem that fai1ure occurred
once the u1timate compressive strain of concrete had been reached. That is,
Slab 3 had to def1ect more than Slabs 1 and 2 to produce the same concrete
strains. This fact is accentuated further as a resu1t of Slab 3 having
10wer initial strains due to a 10wer 1eve1 of prestress in the concrete.
Slab 3 had the same initial stiffness as the other slabs as wou1d be ex
pected for the identica1 sections when uncracked. Cracking in Slab 3 occur
red at a lower load 1eve1 as a resu1t of the lower prestressing force.
- 84 -
10. The experimenta1 setup worked extreme1y we11. Loading the
slabs uniform1y using air pressure was a very quick, c1ean and simple pro
cedure.
- 85 -
REFERENCES
1. Scorde1is, A.C., Pister, K.S., and Lin, T.Y. "Strength of a Concrete Slab Prestressed in Two Directions". A.C.I. Journal V.28 No.3, Sept. 1956 pp. 241-256.
2. Wood, R.H, "Plastic and E1astic Design of Slabs and Plates" 'rhames and Hudson, London 1961.
3. Johansen, K.W. "BrudHnieteorier" Copenhagen 1943. Eng1ish edition "Yie1d Line Theory" London Cement and Concrete Association.
4. prager and Hodge "Theory of Perfect1y Plastic Solids". Wi1ey.
5. Lin, T.Y. "Design of Prestressed Concrete Structures". Wiley 1963.
6. Robinson, G.S. "Behaviour of Concrete in Biaxia1 Compression" Journal of the Structural Division A.S.C.E. Vo1.93 Pt.1 Feb. 1967
7. Kemp, K. "Yie1d of a Square' Reinforced Concrete Slab on Simple Supports A110wing for Membrane Forces" Structural Engineer, Vo1.45 No.7 Ju1y 1967.
8. Jacobson, A. ''Membrane F1exura1 Fai1ure Modes of Restrained Slabs" Journal of the Structural Division A.S.C.E. Vol. 93 Pt.2 Oct. 1967.
9. Christiansen, K.P. "The Effect of Membrane" Stresses on the U1timate Strength of the Interior panel of a Reinforced Concrete Slab". Structural Engineer Vo1.41 No.8 Aug. 1963.
10. Bondy, K.B "Rea1iâtic Requirements for Unbonded Pretensioning Tendons" Journal of the Prestressed Concrete Institute Vo1.15 No.1 Feb. 1970.
11. Burns, N.H. '~oment Curvature Re1ationships for Partia11y Prestressed Concrete Beams" Journal of the Prestressed Concrete Institute. Vol. 9 No.1 Feb. 1964.
12. Park, R. "Tensi1e Membrane Behaviour of Uniform1y Loaded Rectangu1ar Reinforced Concrete Slabs with Fully Restrained Edges". Magazine of Concrete Research Vo1.16 No.46 March 1964 Eng1and, pp. 39-44.
13. Taylor, R., Mâher,5"B~~, and Hayes, B. "Effect of the Arrangement of Reinforcement on the Behaviour of Reinforced Concrete Slabs". Magazine of Concrete Research Vo1.18 No.55 June 1966.
14. Magura, D.D., Sozen, M.A., and Siess, C.P. "A Study of Stress Relaxation in "Prestressing Reinforcement" Journal P.C. I. April 1964 Vol. 9 No.2.
15. Neville, A..M. "l.'roperties of Concrete" Chapter 6. Sir Isaac Pitman and Sons, London.
16. Jaeger, L.G. "E1ementary Theory of E1astic Plates" Pergamon Press Great Britain.
17. Youssef, A. McGi11 Thesis 1971, Dept. of Civil Engineering.
