University of Wollongong Research Online University of Wollongong esis Collection University of Wollongong esis Collections 1990 Punching shear strength of reinforced concrete flat plates with spandrel beams Masood Falamaki University of Wollongong Research Online is the open access institutional repository for the University of Wollongong. For further information contact Manager Repository Services: [email protected]. Recommended Citation Falamaki, Masood, Punching shear strength of reinforced concrete flat plates with spandrel beams, Doctor of Philosophy thesis, Department of Civil and MIning Engineering, University of Wollongong, 1990. hp://ro.uow.edu.au/theses/1260
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University of WollongongResearch Online
University of Wollongong Thesis Collection University of Wollongong Thesis Collections
1990
Punching shear strength of reinforced concrete flatplates with spandrel beamsMasood FalamakiUniversity of Wollongong
Research Online is the open access institutional repository for theUniversity of Wollongong. For further information contact ManagerRepository Services: [email protected].
Recommended CitationFalamaki, Masood, Punching shear strength of reinforced concrete flat plates with spandrel beams, Doctor of Philosophy thesis,Department of Civil and MIning Engineering, University of Wollongong, 1990. http://ro.uow.edu.au/theses/1260
overall depth of the flexural members is a minimum and columns can often be
buried in the wall. This form of structure is popular in most countries.
In the design of reinforced concrete flat plate structures, the regions around
the columns always pose a critical design problem. Experimental data on the
performance of slab-column connections at the edges and comers are very limited,
especially for slabs with spandrel beams. Fig. l(l)t shows a typical reinforced
concrete flat plate structure with spandrels beam. It may be seen that at the edges,
the slab load is transferred to the exterior columns through the spandrels.Thus they
are subjected to large torsional moments in addition to bending moments and shear.
The strengths of the spandrel beams have a significant effect on the punching shear
strengths and mechanisms of failure of the slab-column connections at the edges
and corners of building floors. However the strength behaviour of these
connections is not well understood and it calls for further research.
1.1 The Problem
Determination of the punching shear strength, Vu, of the slab-column-
spandrel connections of flat plates, at the edge- and comer-positions, has received
considerable attention by the engineering profession in recent years. A reliable
t Figures are given at the end of each chapter followed by tables.
3
method for the prediction of Vu, requires a general analytical method for a slab-
column-spandrel connection that can predict both the punching shear strength of the
connection and the mechanisms by which the load is carried. This problem may
also be expressed in terms of the following questions:
How would the size and location of the slab reinforcement affect the distribution of
moments and shears at the edge and comer-column positions?
How would the strength of the spandrel beams affect the magnitude of Vu?
Other relevant questions that might arise in the process of solving the above
problem may be listed as follows.
(i) What are the effects of the spandrel strength on the mechanisms of failure?
(ii) What is the most suitable critical perimeter?
(iii) What are the governing equilibrium equations?
(iv) How to quantify the restraining effects of the slab on the elongation and
rotation of the spandrel beams?
(v) How do torsion, bending and shear interact in the spandrel beam in the vicinity
of the connection?
Information regarding the behaviour of the slab-column-spandrel
connections near failure is reviewed and some of the assumptions of the existing
analytical methods are assessed in terms of how well they conform to the observed
behaviour. This is described in the next section.
4
1.2 Existing Analytical Methods and Experimental Data
Extensive reviews of the existing knowledge have been given previously
by the A C I - A S C E committee 426 (1974) in a state-of-the-art report, by Hawkins
(1974), and by Regan (1981). These literature reviews indicate that there have been
three different approaches to the problem. That is the existing analytical methods
for the prediction of the punching shear strength, Vu, may be classified as follows :
(i) methods based on a linear distribution of shear stress on some critical
perimeters, which do not consider the effects of reinforcement and its applicability
in the post-cracking stage.
(ii) methods based on elastic plate theory. This classification includes the finite
element analysis which may account for cracking and plastic behaviour. However
these methods do not account for any distribution of the stress caused by the
cracking of concrete and yielding of the steel bars. Finally,
(iii) methods based on beam analogies, which describe a slab-column connection as
the junction of orthogonal beam elements contained within the slab. Each beam is
assumed to be able to develop its ultimate bending, torsion and shear, making due
allowance for interaction effects, at the critical sections near the column faces. The
strength of the connections is calculated by summing the contributions of the
strengths of the beams.
From 1981 onward, and especially in the last three years, the following
contributions have been made by other researchers on the prediction of the
punching shear strength for slabs without spandrel beams.
5
Regan (1981) developed an equation for the calculation of Vu. Regan's
shear perimeter for rectangular columns was a "rounded rectangle" located 1.25d
out from the column. Jiang et al. (1986) developed a theoretical solution for the
punching shear strength of concrete slabs. In this approach the problem is treated as
a three-dimensional axisymmetrical one,and the material assumed to be rigid-
plastic. Chen (1986) developed a procedure for the prediction of the punching
shear strength of flat plates without shear reinforcement while transferring shearing
force only. Solanki and Sabinis (1987) presented a simple design approach for the
calculation of V u for the curved/shell concrete structures. Rankin and Long (1987)
developed a method for the estimation of V u from rational concepts of the various
modes of failure. This method is an extension of the method proposed by Long
(1975) for the prediction of Vu. Bazant and Cao (1987) were primarily concerned
with size effects, but they did propose a formula for the prediction of Vu. Gilbert
and Glass (1987) proposed a method for predicting Vu, which is based on the shear
criterion of failure. This method was then extended to cover the use of shear head
reinforcement by redefining the critical-area term. Alexander and Simmonds (1987)
in their paper proposed that punching shear failure could be represented by a truss
analogy and that failure is due to the concrete cover failing to contain the out-of-
plane components of force between the reinforcement and the concrete compression
stmts. Gonzalez et al. (1988) based on a nonlinear finite element analysis,
developed an analytical method for the prediction of V u , in which failure is
governed by the tensile strength of the concrete. Moehle et al. (1988) proposed an
expression for shear strength in the absence of significant moment transfer, as well
as three alternative procedures for the computation of the strength under combined
shear and moment transfer.
All of the above prediction procedures are for the case of slab-column
connections of flat plates without spandrel beams. Thus none of these works has
6
any direct relation to the present study which concentrates on flat plates with
spandrel beams.
A review of the existing publications also indicates that experimental data
on the performance of the slab-spandrel-column connections of flat plate slabs are
very limited. Hatcher, Sozen and Siess (1961) tested a multi-panel flat plate
containing spandrel beams. However, the punching shear failure occurred at a
column away from the corners and edges. Rangan and Hall (1983) tested a series
of four half-scale models with spandrel beams. In their models "3" and " 4 "
punching shear occurred at an edge column. N o corner column failure data were
available from their work. Rangan (1987) published a method for the prediction of
Vu. This method also allows for the prediction of the punching shear strength of
the slab-column connections with spandrel beams.
1.3 Codes of Practice
The design provisions incorporated in the various building codes are a
direct result of the empirical procedures derived from experimental studies.
However in the U.K., U.S.A. and Australia the development of the design
recommendations have followed different routes. The British code (BS8110-1985)
is based primarily on the work of Regan (1974), the American code (ACI318-83),
on the work of M o e (1961), and the new Australian Standard (AS3600-1988), on
the work of Rangan (1987). Note that the recommendations of the European code
(CEB-FIB-1978) and that of the Canadian code (CSA A23.3-M84) are in general
similar to those proposed by ACI318-83.
Among the abovementioned codes only AS3600-1988 provides a
prediction procedure for the punching shear strength, Vu, for slab-column-spandrel
connections at the edge- and comer-column positions. However an early
7
examination of the code procedure (Falamaki and Loo, 1988) indicated that these
proposed formulas overestimate the punching shear strength values, especially at
the comer positions.
In a separate report (Falamaki and Loo, 1990) the inadequacy of the code
formulas has been attributed to the use of : (i) incomplete set of equilibrium
equations, (ii) inadequate interpretation of the restraining effects of the slab on the
strength of the spandrel beams, and (iii) inadequate assumptions for the
distribution of shear force along some critical perimeters. These Australian
Standard formulas also do not consider the effects of the size and location of the
slab reinforcements on the magnitude of Vu. Further, the effects of bending
moment are not included in the assumed interaction equation for the spandrel beam.
1.4 Size of the Model Structures
To investigate the punching shear strength of the slab-spandrel-column
connections at the edge- and comer-positions theoretically or experimentally it is not
practical to deal with the whole building. Thus a localized portion in the vicinity of
the connections is considered. Of course the localized model should be adopted in
such a way so as to ensure that the distribution of the total unbalanced moment and
the shear force transferred from the slab to the column is the same as in the whole
building. O n the other hand, in an experimental study, adoption of a larger region
of the structure may require a smaller model and size effects may then be a problem,
which is one of the salient aspects of fracture mechanics.
According to fracture mechanics (Bazant and Cao, 1987) size effects
decreases as the structure size increases. Therefore by the adoption of large scale
model structures, the problem of size effects can be eliminated. It is important to
note that the strength of the beam and slab elements at the various sides of the
8
connection is affected by the deformational restraints provided by the surrounding
slabs of the building. Thus the model structure should be large enough to cover
the full length of these elements.
Regarding the above discussion, a sound analytical model not only should
be based on physical behaviour and test data of large scale test models with proper
boundary conditions but also account for the variation in each of the following
parameters:
(1) the overall geometry of the connection,
(2) the concrete strength,
(3) the size and location of flexural reinforcement of the slab,
(4) the slab restraint on the spandrel, and
(5) the enhanced strength of the slab-column connections due to membrane effects.
1.5 Objectives
The existing analytical methods for the prediction of Vu have been
summarized in Sections 1.2 and 1.3. For the case of slab-column-spandrel
connections of flat plates at the edge- and corner-positions there is still no reliable
procedure for the prediction of Vu. Thus the main objective of the present study is
to develop an analytical method for the prediction of V u for these types of
connections. Needless to say, the development of a sound analytical method for the
prediction of V u requires the test results from large-scale models with proper
boundary conditions. Experimental work of this nature is a highly expensive and
labour intensive task.
9
The objectives of the experimental phase of the present investigation are to
observe the behaviour of flat plate slabs with spandrel beams of different depths
and steel ratios, and to obtain essential data to use for the establishment of the
prediction procedure for Vu. A total of five cast-in-situ half-scale flat plate models
representing two adjacent panels at the corner of a real structure have to be tested up
to failure, under a uniformly distributed vertical load. Also to accelerate the
construction, in the design of flat plate models, instead of concrete columns,
prefabricated steel sections (with equivalent stiffnesses) may be used. The
instrumentation and test procedure ought to be designed in such a way as to provide
the required data for the analytical phase of the investigation.
The analytical studies which led to the development of the prediction
procedure for V u are mainly based on the behaviour and the experimental results
obtained from the present five half-scale models plus those tested by Rangan and
Hall (1983).
To establish the prediction procedure for Vu, the tasks for the analytical studies are:
(i) determination of the total unbalanced moment and total shear force distribution
along some critical perimeters at the edge- and comer-positions, and
(ii) determination of the strength of the spandrel beam and slab elements joined to
the different faces of the edge- and comer-columns, with the aid of semi-empirical
formulas.
Note in (ii) that for the determination of the forces and moments in the
spandrel beam an interaction equation is to be developed for the combined effects of
torsion, shear and bending. Also for the determination of the strength of the slab
the effects of the size and location of the slab reinforcement, clear span of the slab
10
(in a direction perpendicular to the slab edge) and the in-plane forces in the slab are
to be considered. It is worth mentioning that for development of the interaction
equation for the spandrel beams, the restraining effects of the slab on the rotation
and elongation of the spandrels are to be studied first.
Thus the prediction equations for Vu that may be obtained from the above
study will then cover the cases of the slab-column-spandrel connection under axial
force and biaxial bending moments, at the edge- and comer-positions. This study
also investigates the effects of column width on the magnitude of V u at the comer
positions.
It is important to note that the abovementioned study is mainly for the case
of the slab-column connections with spandrel beams. However to obtain a better
picture for the effects of the spandrels on the behaviour of the slab, the last of the
five half-scale models is designed as a slab with torsion strips (but without closed
ties) at its edges.
1.6 Outline of Thesis
One of the requirements of the analytical study for the development of the
prediction formulas for V u is the determination of the shear force distribution along
some critical perimeter within the slab. This is presented in Chapter 2. The second
requirement is the determination of the torsion, shear, and moment interaction for
the spandrel beams which is discussed in Chapter 3. The last requirement is the
development of a computational procedure for the calculation of the slab bending
moments from the flexural reinforcement strain data. This is expanded in Chapter
4. In Chapter 5 the experimental programme is described in detail. Behaviour of
the test models and modes of failure are also presented herein.
11
A n outline of the research scheme for the prediction of the punching shear
strength, Vu, is presented in Chapter 6. The contributions of the discussions in the
other chapters in relation to the development of the proposed prediction procedure
for V u are also discussed in this chapter.
In Chapter 7, some semi-empirical formulas are developed. These
formulas may be used for the determination of the distribution of moment and shear
along the critical perimeter. The bases of the formulas are the analytical and
experimental studies carried out in Chapters 2,4 and 5.
The results of the experimental study presented in Chapter 5, have also
been used for the calibration of the interaction equation developed herein for the
spandrel beams. This is discussed in Chapter 8.
The formulas of the proposed prediction procedure for Vu are presented in
Chapter 9. The prediction method recommended in the AS3600-1988 is also
included in this chapter, where in the light of the experimental results reported
herein a comparative study is carried out. Finally, conclusions and
recommendations for further study are given in Chapter 10.
It should be noted that for each of the chapters, the figures are given at the
end of the text followed by the tables (if they exist).
12
t
Walls
Jf
• • •
•
•
• D D •
a) Plan view
Spandrel Beams
Yr I T Yr it Jil Jyt Jjui
t
ii
b) Section 1-1
Fig. 1(1) Typical flat plate structure with spandrel beams
CHAPTER 2
TRANSFER OF FORCES IN SLAB-COLUMN CONNECTIONS
OF FLAT PLATES
CHAPTER 2
TRANSFER OF FORCES IN SLAB-COLUMN
CONNECTIONS OF FLAT PLATES
2.1 General Remarks
The question of the transfer of shear force .and bending moments between
a slab and the column of a flat plate and their distribution along some critical
perimeter has always been a design problem, especially at the edge- and comer-
locations. In order to quantify the distribution of these forces and moments the
behaviour of the slab at the exterior panels should be investigated first. This
chapter expands the fundamentals of the slab-column-spandrel behaviour and
derives the useful equilibrium equations.
In Section 2.2 the effects of the size of the spandrel beams as well as the
loading pattern on the deflected shape of the exterior panels of the flat plates are
discussed. According to this discussion, at the ultimate state and under certain
specified conditions the deflected shape of the exterior panels of the flat plates may
be assumed similar to that of the one-way slabs. The effects of the spandrel's
strength on the distribution of forces in the vicinity of the slab-column connections
are investigated on the basis of this assumption.
The strength of the spandrel beams also affects the failure mode of the
slab-column connections.This is described in Section 2.3. Based on the expected
failure mechanism for the slab-column connections with shallow spandrels the
equilibrium equations of both the edge - and corner-connections are derived in
Section 2.4.
15
The above hypothesis then leads to the development of a new technique for
the determination of the actual distribution of the total shear force between various
faces of the edge- and comer-columns. This is detailed in Section 2.5. It should be
noted that this new technique is verified in Section 7.4.
2.2 Spandrel Beams and Slab Behaviour
2.2.1 One-way slab action
The behaviour of the present Models Wl to W5 which represent the two
adjacent panels at the comer of a typical flat plate floor is reported in Section 5.7.
The deflected shape of the slabs as well as the slab crack patterns all indicated a
one-way slab action at the ultimate state. Fig. 2.2(1) shows the deformed shape of
a typical slab of the present model structures after failure. Note that all the model
structures (Wl to W 5 ) failed under a uniformly distributed load. They all had
spandrel beams at the free edge*, except Model W 5 . This model also exhibited one
way slab behaviour.
Further, Simmonds (1970) tested a one-third scale model of a flat plate
structure. It consisted of square panels and rectangular columns with cross sections
elongated in one direction. H e found that the model behaviour changed from
essentially two-way to one-way slab action. Fig. 2.2(2) shows the top crack pattern
of this slab.
Furthermore, Hatcher et al. (1961) studied a quarter-scale reinforced
concrete flat plate model. The structure consisted of nine square panels with
spandrel beams at the discontinuous edges. Tests up to failure were also conducted
* Edges not stiffened by walls or other bracings (see Fig. 1.1)
16
by Rangan and Hall (1983) on half-scale models simulating the edge panels of flat
plate floors with spandrel beams. The bottom crack patterns of all the above flat
plate models indicate a one-way slab action in the exterior panels. A typical bottom
crack pattern is depicted in Fig. 2.2(3).
The above observations indicate that the comer and edge panels of the flat
plate slabs with rectangular panels would have a one-way slab action at the ultimate
state. This is true provided that the rotational stiffness of the slab-column
connections in one direction is higher than that in the other direction. This
condition may be attained by :
(i) using rectangular columns with cross sections elongated in one direction ( See
Fig 2.2(2)),
(ii) provisions of spandrel beams at the free edges, or
(iii) loading the slab on alternate panels to provide maximum unbalanced moments
at the slab-column connections ( See Fig. 2.2(4)).
Note that in (iii) at the ultimate state, the higher rotational stiffness of the
uncracked (adjacent) slab would help to create one-way action in the failed slab (see
the behaviour of Model W 5 in Section 5.7).
In summary, at the ultimate state, under certain specified conditions the
exterior panels of flat plate floors would have a one-way slab action. This type of
behaviour can be used as a basis to investigate the effects of the strength of the
spandrel beams on the distribution of forces in the vicinity of slab column
connections.
17
2.2.2 Distribution of forces along the spandrels
Fig. 2.2(5)a shows a one-way slab, treated as a series of narrow
individual slab strips spanning in a direction perpendicular to the spandrel, in which
the slab resistance against twisting is ignored. The slab is under a uniformly
distributed vertical loading and is assumed to be cast monolithically with the
supporting columns. It is further assumed that the column bases are fixed and the
vertical deflections of the spandrel beam are small .and may to be neglected.
Theoretically, at the ultimate state, the magnitudes of the bending moment
and shear force of each slab strip (at a section located at the face of the supporting
spandrel) is proportional to the magnitudes of the strains in the top steel bars of the
slab (in the corresponding section) in a direction parallel to the slab strips. The
magnitudes of the steel strains are in turn proportional to the angle of twist of the
spandrel with respect to the exterior columns supporting the spandrel.
In slabs with deep spandrel beams and very rigid columns, both the
spandrel and the columns provide near full bending restraint for the connecting
slab. Therefore the angle of twist of the spandrel in relation to the columns reduces
to zero and the slab will deform in the same manner all along the spandrel. In this
case provided the slab reinforcements are designed for a practical ultimate load, the
reinforcement strains at the face of the spandrel would all attain their maximum
values (or yield strains). Consequently a uniform distribution of bending moment
along the spandrel is expected. This is illustrated in Fig. 2.2(5)b. It may be seen
that the variation of the slope of th? bending moment diagram (i.e. the shear force),
and that of the torsional moment (as a result of the above bending moment and the
shear force) are both straight lines.
18
W h e n the spandrel is shallowt , the bending restraint provided by the
spandrel for the connected slab is less than that provided by the columns (which is
assumed to be rigid). Therefore the bending of the slab tends to rotate the spandrel
beam with respect to the columns. Fig. 2.2(5)c shows the effect of the spandrel
twist on the distribution of the slab reinforcement strains along the spandrel. In
other words (depending on the strength of the spandrel) the full bending restraint
provided at the column face reduces as w e get closer to the panel centerline.
Therefore a non-uniform variation of the bending moment and the shear force
(similar to that of the steel strains (See Fig. 2.2(5)c) would be expected. In this case
the variation of torsional moment will not be a straight line, but increase sharply
near the columns.
For slabs with no spandrel or (torsional strip), a variation similar to that of
the slabs with shallow spandrels is expected, but with a higher concentration of the
moment and shear in the vicinity of the columns (see Fig. 2.2(5)c).
In summary the angle of twist of the spandrel and its adjacent edge- and
comer-column depends on the strength of the spandrel beams. This observation is
used to investigate the possible mechanisms of failure (see Section 2.3).
2.3 Spandrel Beam and Modes of Failure
As discussed in Section 2.2, the effects of the strength of spandrel beams
on the failure mechanisms of the slab-column connections may be expressed in
terms of the angle of twist of the spandrel and its adjacent edge- and comer-
columns. For slabs with deep spandrels the angle of twist tends to be zero and
t The differences between the shallow and the deep spandrel beams are discussed in Section 8.5
19
consequently, at the ultimate state a negative yield line would occur along the face
of the spandrel and the slab-spandrel connection fails in negative bending.
For shallow spandrels, again as discussed in Section 2.2, due to the full
bending restraint provided by the (rigid) column a yield line would first develop at
the ultimate state across the front face of the edge - and comer-columns. Further
increases in loads increase the angle of twist of the spandrel in relation to its
adjacent columns. This continues until the spandrel-column connection fails. In
this process, because of the concentration of torsion and shear at the side face(s) of
the column, failure occurs by the formation of inclined spiralling cracks in the
spandrel. Similar failure mechanisms prevail in the case of connections without
spandrel or torsion strip.
2.4 Equilibrium Equations
2.4.1 Definitions
Fig. 2.4(1) shows the freebody diagrams of typical slab-column
connection of flat plates with spandrel beams. The following features should be
noted.
(i) The critical perimeter for the direct transfer of the slab bending moment and
shear force to the column is also shown in Fig. 2.4(1). The front segment of the
critical perimeter is located at a distance 0.5d from the front face of the column,
where d is the effective depth of the slab. The side segments of the critical perimeter
are located at the column side face(s). Note that the present definition of the critical
perimeter, instead of the critical shear perimeter prescribed by AS3600-1988, leads
to better predicted results for Ml and Vx. This is discussed in Appendix IV.
20
(ii) The point of contraflexure at the edge- and comer-columns is assumed at a
distance L 2 from the center of the slab (see Fig. 2.4(l)c). In this figure Fh and V u
are respectively the horizontal and vertical column reactions at the contraflexure
point.
(iii) The forces and moments of the spandrel beam at the left and right sides of the
slab-column connection are respectively shown as V 2 L T 2 > L and M 2 > L , and V 2 R,
T 2 R and M 2 R where V denotes shear, T denotes torsion, and Mdenotes moment.
It is important to note that in this study V2L and V2R are assumed to be
equal to V2, and T 2 L and T 2 R equal to T2. When both the slab panels adjacent to
the edge connection are similar, M 2 L is equal to M 2 R and consequently the total
unbalanced moment in the transverse direction, M C 2 , is zero. When one panel is
slightly stiffer, the bending moments M 2 L and M 2 R would no longer be equal. In
this case the unbalanced bending moment with respect to the centroid of the
spandrel is designated as M 2 and the corresponding total unbalanced bending
moment as M C 2 . Note that M C 2 is obtained by taking moments with respect to point
O t (See Fig. 2.4(l)c).
The horizontal column reaction, Fh, that is to be resisted by the slab's
inplane forces is shown in Fig. 2.4(1). By considering the equilibrium of forces at
the slab-column connections in the horizontal direction, it is obvious that, part of Fh
is to be resisted within the width C 2 at the front face of the column and the
remainder within the width bounded by the panel center line(s) adjacent to the
column. In the derivation of the equilibrium equations, the portion of Fh resisted
within the width C 2 is ignored. This is because at the ultimate state, the
development of a negative yield line over the front width of the column (see section
2.3) would cause the formation of a wide crack across this width.
21
Based on the above definitions and discussions the derivation of the
equilibrium equations for the slab-column connections of flat plates with spandrel
beams is carried out in the next section.
2.4.2 Formulas
The definitions given in Section 2.4.1 can now be used for the derivation
of the equilibrium equations. The freebody diagram of the slab-column connections
at the edge- and comer-locations are shown in Fig. 2.4(1). In a comer connection,
the equilibrium of forces in the vertical direction (at the ultimate state) may be
expressed as
VU = V2 + V! 2.4(1)
Enforcing the equilibrium of forces with respect to the center of the
spandrel (point O ) , in the main and transverse moment directions, while
incorporating the above definitions gives:
M C 1 = T 2 + M 2 + yx(°l2 d ) + M T 2-4(2)
"<*'' D M ? - D ^ " ^
2Li
Similarly for the edge-connections we have
VU = 2V2 + V! 2.4(4)
MC1 = 2T2 + Mx + v/bl2+ dl + MT 2.4(5)
22
M2
**=' D l - D r 2A^
' 2Lj
In the above equations, M C 1 and M C 2 are respectively the total moments in
the main and transverse moment directions (see Fig. 2.4(1)) with respect to point
Oj; V u is the total shear at the column centerline; T2, V 2 and M 2 are respectively the
torsion, shear and bending moment at the side face of the critical section; M1 and Vj
are respectively the bending moment and shear force at the front face of the critical
section. And finally,
MT'Tff^'l+Vj-V^l 2.4(7)
Note that in Eq. 2.4(6) M 2 is the unbalanced moment with respect to the
centroid of the spandrel. Obviously, M C 2 = 0 if the two panels adjacent to the edge
connection are identical.
For the general case in which the width of the spandrel is the same as that
of the column, Eq. 2.4(7) reduces to :
MT ^(^^JMci 2-4(8)
Obviously MT = 0 if the depth of the spandrel and the slab are similar.
Also for the particular case in which the spandrel beam is projecting upward, M T
becomes
M T ..^Bl^syvf-l^X) 2.4(9)
23
2.5 Distribution of Shear Force
2.5.1 Assumptions
Flat plate is an indeterminate system. Therefore the measurement of the
forces in the vicinity of the slab-column connections requires a sophisticated
analytical process and proper instrumentation. Magnitude of the total shear force at
the column centerline, V u, which is the ultimate shear strength may be measured
directly by means of vertical load cells in the support system. A fraction of V u is
resisted along the front segment of the critical perimeter by V j (see Fig. 2.4(1)),
and the remainder by V 2 at the side face(s). Therefore by the development of an
experimental method for the measurement of Vl 5 the shear force V 2 may be readily
calculated.
The proposed procedure for the measurement of V1 is mainly based on the
assumption that at the ultimate state, and under certain specified conditions (see
Section 2.2.1) the edge- and corner panels of flat plate slabs have a one-way slab
action. The other assumptions used may be expressed as follows. (Note that to
clarify the understanding of the procedure some of the assumptions of Section
2.2.1 are repeated here).
(i) Similar to one-way slabs, the flat plate is treated as a series of narrow individual
slab strips, spanning in a direction perpendicular to the spandrel (see Fig. 2.5(l)a),
in which the slab restraint against twist is ignored.
(ii) The slab is under a uniformly distributed vertical loading and is assumed to be
cast monolithically with the supporting columns.
24
(iii) The column bases are fixed and the vertical deflections of the spandrels are
small and therefore negligible.
(iv) Variation of the bending moment along each slab strip is parabolic.
2.5.2 The procedure
Details of the proposed procedure for the measurement of the shear force
Vj are described below.
(i) According to assumption (iv) of Section 2.5.1, for the slab strip i, the bending
moment per strip width is equal to
Mi = AfX2 + BiX + Q 2.5(1)
in which the magnitude of the bending moment at each section of the strip is
proportional to the slab reinforcement strains in the corresponding section.
(ii) For slab strip i, the magnitude of the shear force per strip width (i.e. the slope
of bending moment diagram) is also a function of the slab's steel strain. Therefore
the first derivative of Eq. 2.5(1) gives the shear force
^ = 2AiX + Bi 2.5(2)
Eq. 2.5(2) indicates that the magnitude of the shear force per strip width at X = 0 is
equal to B{ (see Fig. 2.5(l)b).
(iii) To determine the three parameters Ai5 B^nd Q of Eq. 2.5(1) for slab strip i,
three measured slab strip moments are to be substituted into that equation.
25
Therefore the three measuring stations 1, m, and 3 respectively at distances X = 0,
X = Xjn and X = X 3 were considered for the determination of moments with the aid
of experimental strain data. Note that X = 0 corresponds to a distance equal to d/2
from the face of the spandrel beam (see Fig. 2.5(l)a), where d is the effective depth
of the top steel bars of the slab.
(iv) According to step (iii), to measure the slab strip moments at the measuring
stations 1, m , and 3, strain gauges have to be attached to selected slab
reinforcement at the corresponding distances X = 0, X = X m and X = X3. The
slab bending moments may then be obtained, using the measured strains of the slab
steel bars, with the aid of an established moment-strain relationship.
(v) Substituting the coordinates of the three measured moments of the measuring
stations 1, m and 3 of each slab strip i, namely (0, M H ) , ( X m , M 3 i) and ( X3,M3i)
into Eq. 2.5(1) and solving for Bj gives
Bi = xm(x^3- xm)
[Mmi + Mli] + x3(x^m xm)
[M3i"Mli] 2'5(3)
where according to (ii) above, Bj is the shear force per unit width of strip i at X = 0
(i.e. at the measuring station 1); and M H , Mmi and M 3 i are the absolute values of
moments per strip width. Note that the subscript i stands for the strip numbers and
subscripts l,m and 3 respectively refer to the measuring stations l,m and 3.
(vi) Magnitude of the total shear force at the center of column support, Vu, may be
measured directly by means of the vertical load cells. Subtracting from the Vu, the
self weight of the column and the portion of the slab (including the spandrel) that is
represented by the shaded area in Fig. 2.5(l)a, gives the magnitude of the total
shear force along the critical section 1. This force may be designated as Vul. Note
26
that the shaded area, as shown in Fig. 2.5(l)a is bounded by the panel centerlines
from two sides, and the measuring station 1 (i.e. lines pq) from the third side.
(vii) Vul may also be determined with the aid of the data obtained from the strain
gauges attached to the slab reinforcement. Thus dividing the same region of the slab
defined in step (vi) into n slab strips perpendicular to the spandrel (see Fig.
2.5(1 )a), w e have
Vi = ZBi 2.5(4) i = l
where Bj is the measured shear force (using Eq. 2.5(3)) at X = 0.
(viii) The portion of the total shear force Vu which is resisted along the front
segment of the critical perimeter may now be calculated as
Vi = -^l-Vui 2.5(5)
IBi i= l
where Bj is the measured shear force (using Eq. 2.5(3)) of the slab strip located in
the front of the column, with a width C2.
Eq. 2.5(5) indicates that the proposed procedure for the measurement of
V j is based on the data obtained from the strain gauges attached to selected slab
reinforcing bars, and the vertical load cells at the column supports. The reliability of
this measuring system is discussed in Chapter 7.
27
Or
J .
s .3 a 03
"3 -a o
s "a es
"is
a
Cui
• PN
28
Positive yield line
Columns
Loading points
Fig. 2.2(2) The top surface crack pattern of a one-third scale flat plate model tested by Symmonds (1970)
29
(NUMBERS ON THE CRACKS ARE LOAD IN kN/m2)
Fig. 2.2(3) Typical soffit crack pattern of the exterior panels of the flat plate models tested by Rangan and Hall (1983)
30
a) Plan View
b) Elevation
Fig. 2.2(4) Typical flat plate loaded on alternate spans
Slab strips
Spandrel beams
a) Typical one way slab
Column face
Variation of:
Beam
^nferMne Column face
Moment
Column face
Variation of
Shear
Torsion
Moment
Shear
Torsion
Beam center,,ne
Column face
b) Deep spandrel beams c) Shallow spandrel beams
+ i.e. strain in top steel bars of the slab in the main moment direction
Fig. 2.2(5) Theoretical variation of moment, shear and torsion along the spandrel beams
32
c 'uS cu cu
^cu
"S >
c o
c
cu
c C o cu
•a W
c "u*uJ
cu Or
e e o cu
r-
C r»
o eg
cu cu
c C o u->
I r-
Or
c J-
o cu •a
c es •
cu WO T3 CU r-
E ea r-
Wj
es r>.
o cu cu u [a.
IN WD • —
33
Interior Face of the Spandrel Beam
Panel Centerline Measuring Station 1
Strip 1
Spandrel Beam ^^«
Slab Edge;—&°
Shaded Area
&•;•*?».
Column
a) Plan view
d/2
Critical Slab Strip
Panel Centerline strip n
Measuring tation m
•Measurin Station 3
Column
Typical Slab Strip i
Bj
b) Elevation
,
t VmHUHIfft
M li
M li
X=0
c) Moment diagram
Fig. 2.5(1) Variation of moment along the slab strips
CHAPTER 3
INTERACTION OF TORSION, SHEAR AND BENDING
IN SPANDREL BEAMS
35
CHAPTER 3
INTERACTION OF TORSION, SHEAR AND
BENDING IN SPANDREL BEAMS
3.1 General Remarks
In slab-column connections of flat plates with spandrel beams, the
spandrels are under the combined effect of torsion, shear and bending. To quantify
these forces an interaction equation needs to be developed. However, because of
the slab restraining effects and consequently increase in the strength of the
spandrels, the calibration of any semi-empirical interaction equation requires a
substantial amount of test data.
A theoretical investigation of the restraining effects of the slab on the
elongation and rotation of the spandrel beams is described in Section 3.2.
According to this investigation, the slab restraining effects may be expressed in
terms of an increase in the longitudinal and transverse steel bars of the spandrel.
By the determination of the restraining effects of the slab on the strength of the
spandrels (compared to the isolated beams) the following procedure m a y be used
for the development of the interaction equation for the spandrel beams.
It is believed that the most complete interaction surface for isolated beams
under the combined effects of torsion, shear and bending was developed by Elfgren
et al. (1974). In Section 3.3 the applicability of this interaction surface for the
spandrel beams is investigated. To do so, the deformational restraint provided by
the slab m a y be ignored. Also on the basis of the physical. observations it is assumed
that the skew failure surface occurs on the sides and top of the beam while the
compression zone is located at the bottom. The analysis leads to the determination
36
of an interaction surface for a beam with the same loading condition as the
spandrels.
In Section 3.4 the slab restraining effects as discussed in Section 3.2 are
incorporated into the interaction equation proposed in Section 3.3. This led to an
interaction equation for spandrel beams, Eq. 3.4(15). The reliability of this
equation is discussed in Chapter 8.
3.2 Slab Restraint and the Enhanced Strength of Spandrels
Theoretically when compared to isolated reinforced concrete beams the
restraining effects of the slab provide a higher strength for the spandrels. This
increase in strength is a result of the slab restraint on i) the elongation and ii) the
rotation of the spandrel beams. Also it is expected that the spandrel beams joining a
slab-column connection with a wider column, yield a higher punching shear
strength for the connection. These are elaborated in the following sections.
3.2.1 Slab restraint on the elongation of spandrels
In 1972, Onsongo and Collins reported on the results of the tests on a
series of longitudinally restrained reinforced concrete beam elements subjected to
torsion. According to their results, any longitudinal restraint on the beam
elongation, increases its torsional capacity. This increase in strength may be
computed by expressing the restraint in terms of an equivalent area of additional
longitudinal steel.
The enhanced strength of the spandrel due to the restraining effects of the
adjoining slab was first reported by Rangan and Hall (1983). In their report based
on the work of Onsongo and Collins (1972) Rangan and Hall analysed the spandrel
37
beams of some half-scale flat plate models and found that the the torsional strength
of the spandrel increases by a factor of 4 to 5 when compared to isolated beams.
Rangan (1987) later revised the above conclusion and suggested that the restraining
effects of the slab increase both the shear and torsional strengths of the spandrel by
a factor of 4.
Rangan's proposal which has been incorporated in the new Australian
Standard for Concrete Structures (AS3600-1988), is not supported by any test data
on slab-column connections of flat plates with realistic spandrel beams. This fact
helps to explain the shortcomings of the Australian Standard approach (see Section
9.4). It also calls for the development of a more general procedure for the
determination of the slab restraining factor, \j/.
To investigate qualitatively the restraining effects of the slab on the
elongation of the spandrels, let us examine the behaviour of the flat plates with
spandrel beams at the ultimate conditions. A n isolated beam increases in length
when subjected to torsion. A spandrel beam in a building floor will also tend to
increase in length under load. This tendency causes a tensile force, P, to develop in
the adjoining slab, at the face of the spandrel. The reaction, therefore, is a
compressive force (equal to P) in the spandrel itself (see Fig. 3.2(1)). This
compressive force reduces the expected magnitude of the tensile forces in the
longitudinal bars of the spandrel. Thus it is similar to increasing the strength of
these bars from A, f, to \\fA, f, . O n the other hand, the magnitude of the induced
compressive force, P, may be expressed as a function of the angle of twist of the
spandrel beam in relation to its adjacent column. Increase in the strength of the
spandrel provides more bending restraint to the rotation caused by the adjoining
slab and consequently reduces the spandrel rotation with respect to its adjacent
columns. This decrease in rotation reduces elongation of the spandrel and
consequently the induced compressive force, P. Therefore, with the assumption
38
that deep spandrel beams provide near full bending restraint for their adjoining
slab, the induced compressive force, P, would then reduce to zero. Consequently
the slab restraining factor \|/, tends to unity.
In summary, the slab restraint on the elongation of the spandrel beam
enhances its load carrying capacity. This enhanced strength decreases as the
strength of the spandrel beam increases. The enhanced strength m a y be expressed
in terms of the strength of the longitudinal steel bars of the spandrel beam, i.e.
Aj fj increases to \|/A|f, .
3.2.2 Slab restraint on the rotation of the spandrels
In normal design practice, the center of twist of the spandrel is below the
horizontal centroidal axis of the slab. W h e n the spandrel beam is twisted, the
horizontal displacement of the top portion of the spandrel will be restrained by the
large horizontal stiffness of the slab. The spandrel rotation will also produce a
vertical displacement at the spandrel-slab interface. This vertical displacement will
be restrained by the vertical stiffness of the slab.
In other words, the restraining effects of the slab on the rotation of the
spandrel increases its strength in the transverse direction. This enhanced strength
m a y be considered similar to the provision of more transverse reinforcement in the
spandrel beam. Therefore it m a y be assumed that the transverse strength of the
spandrel beam, co''" is increased to co + coo, where co0 is defined as the additional
transverse strength of the spandrel beam.
f co = Awsfwy/s, where fwy and A w s are respectively the yield strength and the cross-sectional area of the bars from which lies are made. The spacing of the closed ties is s.
39
It is expected that an increase in the area of the spandrel-slab interface
(which is proportional to the overall depth of the slab) increases the magnitude of
C0o. However for the present study the slab depth is constant. Note that the
investigation of the effects of abovementioned interface area is beyond the scope of
this thesis.
3.2.3 Effects of column width
Fig. 3.2(2) shows schematically the effects of column width on the slab
restraining factor, \i/. Fig. 3.2(2)a illustrates the plan view and the angles of twist
of the two corner columns A and C of dimensions 2a x b, and 2c x b
respectively (where c > a). Note that in this figure 9 A and 0c are respectively the
angles of twist of the spandrel beams (at critical sections located near the face of the
columns) in relation to the centerline of columns A and C.
According to Fig. 3.2(2)a, as the distance of the above mentioned critical
sections from the column centerline increases, the angles of twist of the spandrel
cross-section in relation to the column also increase. In Section 3.2(1) it has been
concluded that increase in the angle of twist of the spandrel in relation to its adjacent
columns increases the induced compressive forces in the spandrels and
consequently, the slab restraining factor y.
Fig. 3.3(2)a shows that the angle of twist of column C is larger than that
of column A. Therefore, the slab restraining factor for column C (i.e. \|/c) is greater
than that of column A (i.e. \J/A).
Based on the observation made in Section 3.2.1 the slab restraining factor,
y, decreases as the spandrel strength increases. This is schematically presented in
Fig. 3.2(2)b for the case of column positions C and A. It may be seen that the
40
relationship between \|/A and \|/c can be expressed as \j/c = X + \J/A , where X is
defined as the column width factor. Calibration of X is described in Section 8.3.2.
3.3 Interaction Surface for Isolated Reinforced Concrete Beams
(Truss Analogy)
3.3.1 Historical review
A study of the interaction of torsion with bending and shear may be based
on truss analogy. The pioneering work on reinforced concrete members subjected
to torsion was carried out by Rausch(1929). H e assumed that a concrete member,
reinforced with longitudinal and transverse reinforcement, acts like a tube, so that
the applied torsional moments is resisted by the circulatory shear flow in the walls
of the tube. Furthermore, the tube is assumed to act like a space truss in resisting
this circulatory shear flow.
The space truss analogy has been generalized by Lampert and Thurlimann
(1969) for members subjected to torsion or to combined torsion and bending.
Since in their analytical model the angle of the concrete struts was not restricted to
45°, they called their theory the variable-angle truss model. This trass model was
further applied by Elfgren (1972) to members subjected to torsion, bending and
shear.
A review of the existing literature by Hsu (1984) indicates that the most
general and complete interaction surface for the isolated reinforced concrete beams
under the combined effects of torsion, bending and shear is the one developed by
Elfgren et al. (1974). They observed that for rectangular beams with closed ties:
41
(i) The ultimate strength in combined torsion, bending and shear, after some
simplifying assumptions, can be evaluated from a study of the equilibrium of
external and internal forces on the inclined failure surfaces.
(ii) The concrete compression zone can form in the top, in the bottom, or in one of
the vertical sides of the beam. This leads to three different modes of failure (i.e.
modes t,b and s). Then corresponding to each mode of failure they developed an
interaction surface.
(iii) The interaction surfaces for the three modes together form an interaction surface
which governs the load-carrying capacity of a beam.
The interaction surface established by Elfgren et al. (1974) is used herein
as a basis for the derivation of the interaction equation for the spandrel beams. This
is discussed in the ensuing sections.
3.3.2 The interaction equation
Fig. 3.3(1) shows a typical slab-column connection at the comer of a flat
plate with spandrel beams, under a uniformly distributed vertical loading. It may be
seen that the spandrel is under the combined effects of torsion, bending and shear.
The resulting skew failure surface for the spandrel beam, under the above
loading condition, at the ultimate state is shown in Fig. 3.3(2)a. The corresponding
internal forces at the spandrel support, i.e. the torque T2, the bending moment M 2 ,
and shear force, V 2 are depicted in Fig. 3.3(2)b. D u e to the different diagonal
tensile stresses in the different faces of the beam, the inclination of the cracks and
that of the concrete compression struts will vary from face to face. According to
Fig. 3.3(2) the failure surface on three sides is defined by an inclined spiralling
42
crack and on the fourth side, the bottom of the beam, the ends of the cracks are
joined by a compression zone. The above failure surface is defined on the basis of
the observed behaviour of the slab-column connections of the present half-scale
flat-plate models, at the edge- and corner-positions, at the ultimate state. A typical
punching shear failure of the present model structures at the corner position, W4-C,
is shown in Fig. 3.3(3).
To investigate the interaction surface due to the internal forces and
moments T2, V2, and M2, we may first ignore the effects of the adjoining slab on
the strength of the spandrel. This allows us to compare the spandrel beam with an
isolated reinforced concrete beam for which the failure surface on the top and side
faces is defined by an inclined spiralling crack, while a compression zone occurs on
its bottom.
A comparison between the spandrel's mode of failure (as discussed above)
and the modes of failure suggested by Elfgren et al. (1974) (see Section 3.3.1),
indicates that the spandrel failure mode is similar to the proposed failure mode t.
The corresponding interaction formulas for this mode of failure may be derived
from the following equilibrium equation:
2M2 +rT2V s ut +fV2Y s 2di _1 3 3(1) Aisfiydi lv2AtJ Awsfwy Akfiy |^2diJ Awsfwy Aisfiy
where At and ut are respectively the area and the perimeter of the rectangle defined
by the longitudinal bars in the comers of the closed ties; fjy and Au are respectively
the yield strength and total area of the longitudinal steel bars, fwy and Aws are
respectively the yield strength and cross-sectional area of the ties; and s is the
spacing of the ties. Note that in this equation the vertical distance between the
43
longitudinal steel bars is assumed to be equal to d b where di is the effective depth
of the spandrel beam.
From Eq. 3.3(1) the load carrying capacity of the isolated reinforced
concrete beams can be evaluated for pure bending, Mu s, for pure torsion, Tus, and
for pure shear Vus. They are
Mu-.-jAj-fjyd! 3 3 ( 2 )
Tus = 2A lo> v/^fix utco •V 3.3(3)
v-=^»V?F" ^ where
w ~ s 3.3(5)
Substituting M u s , Tus, and V u s respectively from Eq. 3.3(2) to 3.3(4) into Eq.
3.3(1) gives
us J M u s TusJ "IVusJ +W:=1 3.3(6)
Eq. 3.3(6) corresponds to mode t of the interaction surface proposed by Elfgren et
al. (1974). This is shown in Fig. 3.3(4) schematically .
Fig. 3.3(4) also shows the straight-line shear-torsion, shear-bending, and
torsion-bending interaction. While simple to use, this straight-line variation appears
44
slightly conservative for isolated reinforced concrete beams. The corresponding
interaction equation may be expressed as
?^ + ^=l 3.3(7) xus vus iVius
It is important to note that in this analysis, the first order interaction
equation, Eq. 3.3(7), is used as a basis for the derivation of the interaction equation
for the spandrel beams and consequently for the prediction of the punching shear
strength, Vu. This is done simply because this equation, as compared to the second
order interaction equation, (Eq. 3.3(6)), leads to more accurate and consistent
values for Vu. This is discussed in Appendix H
3.4 Proposed Interaction Equation for Spandrels
The interaction equation for the spandrels may now be obtained by the
incorporation of the restraining effects of the slab in the first order interaction
equation for isolated beams, Eq. 3.3(7). That is the strength terms YAlsfiy and
co + co0 are respectively substituted for the corresponding terms Alsfiy and co into
Eq. 3.3(3) and 3.3(4) (see Sections 3.2.1 and 3.2.2). Expressing the load carrying
capacities of the spandrel beams for pure bending, pure torsion and pure shear by
Msp, Tsp, and Vsp respectively, we have
M s P = ifAlsflydl(^) 3'4(1)
lsp — 2 A ,(CO + C O 0 ) A / - ^ ^ - 3.4(2)
45
_ n Uu (co+ co0) A / —
\ ut(
Vsp = J2cLu7 (co + co0) \l ¥ lsly 3.4(3)
' "t(CO + co0)
Substituting M s p, Tsp, and V s p respectively for Mu s, Tus, and V u s in Eq. 3.3(7)
gives
T2 V2 M2
Eq. 3.3(4) may now be considered as the interaction equation for the
spandrel beams. This equation may also be expressed in terms of the spandrel
parameter, co0, and the slab restraining factor, \\f. To do so the following
definitions are proposed.
(i) Isolated reinforced concrete beam with minimum reinforcement - This is a
reinforced concrete beam with minimum practical transverse and longitudinal
reinforcement, i.e. one Y12 longitudinal bar at each comer, and 4 mm hard-drawn
wire stirrups at a maximum spacing as specified in Clause 8.3.8(b) of AS3600-
1988. Note that the Y12 designation is for a 12 mm diameter deformed bar. The
tensile test results on the hard-drawn wires and deformed bars vary between 400 to
550 MPa. In Australia the average strengths of the 4 mm and 12 mm bars are
assumed to be equal to 480 and 450 MPa respectively.
(ii) Longitudinal steel ratio, a - This is the ratio of the spandrel longitudinal
strength, A, f, , to that of an isolated reinforced concrete beam with minimum
longitudinal strength. That is
A f ass7KT^ 3-4(5)
(, ls ly^min
46
Substituting for (Aj £") from definition (i) into Eq. 3.4(5), gives ^ •''min
mm
Alsfl a "200000 3-4<6)
(in 1 Transverse steel ratio, p - This is similar to the definitions for a which may be
expressed as
ut(co + COQ) P =-? x 3.4(7)
(utco) . v /min
or _ut(co + co.) 3^4(8)
utf Aws
wy s I min
where ut is the perimeter of the rectangle defined by the longitudinal bars in the
comers of the closed ties; fwy and Aws are respectively the yield strength and the
cross-sectional area of the closed ties, and s is the spacing of the ties. Substituting
for | ut fwv ~~~ 1 . from definition (i) into Eq. 3.4(8) gives ^ ' s ^min
ut(co + COQ) 13 _ 50000 K }
(iv) Spandrel strength parameter (o) - This is the product of the longitudinal steel
ratio, a, the transverse steel ratio, (3, and the ratio of di/d, where di and d are
respectively the effective depths of the spandrel and the slab. Thus
8 = ccP^ 3.4(11)
where a and P respectively are defined in Eqs. 3.4(6) and 3.4(9).
47
The interaction equation for spandrel beams, Eq. 3.4(4), may now be
expressed in terms of co0 and y with the aid of the above definitions. Assuming
0.5 Akfiydi as the yield moment of the top steel bars of the spandrel at the face
of the column support, M y , and substituting Alsfly and ut(co + co0) respectively
from Eqs. 3.4(6) and 3.4(9) into Eqs. 3.4(2) and 3.4(3) give
Msp = ¥My 34(12)
Tsp = 200 000^(apx|/)1/2
3>4(13)
Vsp = 200 00o|^apV)1/2
3.4(14)
Substituting Eqs. 3.4(12) to 3.4(14) into Eq. 3.4(4), we have
^ + =200000 f Y'V,,, " . =200 000 P^ At\ fch I \j/ U J V2ut
\y-%) •
Note that Eq. 3.4(15) contains the undetermined co0 and co which are to be
calibrated experimentally (see Chapter 8).
48
es u. •4-1
cu u -O
CU
cu
a CZJ
c
a ur*)
CU
-c
CU
cu u
*(^ CU # >
Cfl CU
a
o cu cu cu 3 T3 S
do
49
Spandrel beam
Spandrel beam
Column centerline
Critical section near the side face of column A
Critical section near the side face of column C
Distance
a) Angles of twist of the spandrel in relation to the corner columns A and C
¥ C = ^ + ¥ A
¥
b) Variation of V and the corresponding spandrel strength
Fig. 3.2(2) Column width and the slab restraining factor, ¥
and reactions, and making certain measurements of the deformed structure.
This section is concerned with a brief description of how the load was ap
plied, how the load and reactions were measured, what measurements of the
structure were made, and what procedure was followed in carrying out a test.
82
5.6.1 Loading system
The flat plate models were cast-in-situ and were supported within a re
action frame. The reaction frame is shown diagrammatically in Fig. 5.6(1); as
depicted in Fig. 5.6(2) the reaction frame consists of six vertical steel sections
(pedestals), each supporting one of the six columns of the flat plate model. The
steel pedestals were tied to each other (both on the top and bottom) to resist the
lateral forces that would be induced during the test.
One 20-tonne double acting hydraulic ram was located at the centre of each
panel of the two-panel test structure. The load from each ram was distributed
through a whiffle-tree loading system to sixteen 100 mm square pads on the top of
each panel to simulate a uniform load. The downward (tension) load applied by the
hydraulic rams to the slab was resisted by the main reaction frame (or test rig). In
this system the rams' hinge-supports were bolted to the reaction frame; their other
ends were connected to the whiffle-tree loading system via a specially designed
electrical tension load cells. These load cells provided the means of measuring the
load density in each panel.
Due to the membrane effects in flat plate models (see Appendix I) a failure
load much higher than the design ultimate load was expected. Therefore the beams
and bolts of the whiffle-tree loading system as well as the main reaction frame were
designed for a load of approximately 3 times the design ultimate load. The plan
view and elevation of the loading system as well as some connection details are
shown in Fig. 5.6(3).
Synchronization of the rams was achieved by pumping through one single
hydraulic control system. Details of the hydraulic system are given in Fig. 5.6(4).
83
5.6.2 Testing procedure
To get the initial data (just before the application of the dead load by
stripping the slab) all the load cells and strain gauges were connected to the data
acquisition control system. Then the initial data were recorded onto a floppy disk
via a Hewlett Packard microcomputer (see Section 5.6.3). After removing the
formwork and hanging the whiffle-tree loading system the second set of load data (
due to dead load + weight of the loading system, approximately equal to 3.05 kN /
m^) were recorded. Also at this stage the zero deflection and rotation readings were
recorded.
For the third set of data onward the test loads were applied in increments
of approximately 5 percent of the expected failure load. To apply a load increment,
the hydraulic pressure in the loading rams was raised to a desired level. The load
was then held constant for a few minutes. During this time data for deflection ,
angles of twist of the spandrel, column reaction, and load density were recorded.
The new cracks were also identified.
The loading was continued up to failure. In all models immediately after
failure, there was a significant drop in load. At this stage all the recorded data were
transferred to floppy disks and the hydraulic system were disconnected. About
three hours were required to conduct each test.
5.6.3 Reaction measurements
As shown in Fig. 5.4(3) the steel columns simulating the in-situ concrete
columns were supported by pedestals via electrical compression load cells. With the
aid of these specially designed load cells, the three reaction components at the pin
84
supports (contraflexure points) at the bases of the six steel columns were measured.
Construction details for a typical load cell is illustrated in Fig. 5.6(5).
A Hewlett Packard 3054A data acquisition control system with a capacity
of 50 channels and a Hewlett Packard 9826 computer were used to record the
strains and other electrical signals. The first 20 channels were used for logging the
load cell readings; the remaining 30 channels were connected to strain gauges
attached to selected reinforcing bars of the slab.
To obtain the abovementioned electrical signals from the strain gauges and
load cells (via the data acquisition control system) and convert them to steel strains
and column reactions a computer program has been developed. The functions of
this program includes:
(i) printing the zero strain data;
(ii) instantaneous scanning of all the 50 channels, and printing the strain data at each
stage of loading;
(iii) reporting the faulty strain gauges by printing "wires disconnected" in front of
the gauge number;
(iv) printing magnitudes of shear force, bending moment, and transverse moment at
each column support for all stages of loading;
(v) printing the magnitudes of load in each ram as well as the load density for all
stages of loading;
85
(vi) drawing and printing the variations of shear force, bending moment, and
transverse moment versus load density at each column support (from the beginning
of the test) for each stage of loading (if required);
(vii) recording all the data on to a floppy disk operated by the computer.
It is worth mentioning that the time spent for the development of the
program was about three months of continuous work.
Fig. 5.6(6) shows the direction of the main slab reinforcement and the
bending moment for the flat plates. Variations of the column reactions of Models
Wl to W5 at the edge- and corner-positions are shown in Figs. 5.6(7) to 5.6(19).
The column reactions of the column positions Wl-C and W4-B are excluded simply
because they did not fail.
5.6.4 Strain measurement
Steel strains were measured at numerous locations in the test structures.
The purpose of measuring steel strains was two-fold. To determine the moments in
the slab and to determine the distribution of shear force along the critical perimeters
of the columns (see Section 2.5).
The strains were measured with electrical resistance strain gages (TML-
PLS-10-11) with a gauge factor between 2.06 to 2.11. A total of 30 gauges were
used on Model Wl, the number was increased to 40 for the remaining models.
Note that the extra 10 strain gauges provided in Models W2 to W5 were monitored
manually.
Fig. 5.6(20)a shows the locations of the strain gauges along the spandrel
beam AB. They were attached to the top steel bars of the slab and located along
86
measuring station No. 1 at a distance d/2 from the face of the spandrel, where d is
the effective depth of the slab. For each of the flat plate model structures (Wl to
W5), the readings of all the strain gauges were recorded up to failure. The strain
data corresponding to the last loading stage or the ultimate state is represented in
Fig. 5.6(20)b"l" . Similar experimental plots are also provided for the slab spans
BC, FG, and GH (see Appendix III). These strain profiles were then used as a
basis for the calculation of the slab bending moments.
The slab strain data may also be used as a basis for the definition of the
column critical perimeter. This is discussed in Appendix IV.
5.6.5 Measurement of deflections
In Model Wl the vertical deflections of the slab were measured at the
centre of each panel and the column line "BG". For the remaining models, to
provide a sound picture of the deflected shape of the test models, vertical
deflections of the slab were measured at 11 locations(see Fig. 5.6(21)a.
As shown in Fig. 5.6(2l)b, steel rulers were hung at those specified
locations with the aid of small hooks which in turn were glued to the slab. A level
was used to book the rulers, where the hanging rulers were functioning as the
leveling staffs with an accuracy of 0.5 mm. The level was located at a distance to
book all the rulers at each loading stage.
The deflection data for the centers of the north and south panels as well as
the column line BG (see Fig. 5.6(21)a) of Models Wl to W5 respectively are
t Note that Model W2 with a deep spandrel beam at its edges failed by the formation of a negative yield line at the face of the spandrel. At the time of failure the strain gauges, located along the measuring station 1 (see Fig. 5.6(2Q)a), were either disconnected or were indicating reinforcement strains more than the yield strain (i.e about 0.0044). Therefore in Fig. 5.6(20)b a constant strain of 0.005 was considered for this model.
87
presented in Figs. 5.6(22) to 5.6(26). It may be seen that, in all the models the
deflections at the center of the north panel, at the ultimate state, are higher than the
remaining measurement points. This may be attributed to the lower stiffnesses of
the slab-column connections of the north panel, and consequently the lower in-
plane forces (due to membrane effects).
A comparison of the slopes of the load-deflected curves of Models Wl to
W 5 indicates that (up to the ultimate state) the load gradients for the slabs with
spandrel beams are steep. Whereas for the slab with torsion strips, this slope
became flat near the ultimate state (see Fig. 5.6(26)). Thus in the absence of the
spandrel beams, the flat plate structures reached its maximum loading capacity first.
Further deformation then led to the failure of the system.
5.6.6 Measurement of the angle of twist of the spandrels
In this section the instrumentations used for the measurement of the angles
of twist of a typical spandrel beam in relation to its supporting column, Column
" B " of Fig. 5.2(2), are illustrated. The behaviour of the spandrels up to the
ultimate load are also discussed. The angles of twist of the spandrel beams of
Models W l to W 5 were measured with dial gauges of 0.001 in. graduation (see
Fig.5.6(27)). Fig 5.6(28) shows the locations of the dial gauges, installed at
Sections 1, 2, 3, and 4 along the spandrels. These gauges were located on the side
of the beam, two on each section, at a vertical spacing of 340 m m . The distance
between the sections is 450 m m , i.e. each set is located at one-sixth point. The
angle of twist of Sections 2, 3 and 4 of the spandrel in relation to Section 1 (located
at the center of the adjacent column) may then be calculated by means of the test
data obtained from the dial gauges.
88
The present models consisted of two comer columns "A" and " C " and one
edge column "B". Fig. 5.6(29) shows the variation of the angles of twist of the
spandrel beam of Models W l , W 2 , W 3 and W 5 in relation to the edge column "B".
It may be seen that by increasing the load density, the angle of twist also increases.
The maximum twist occurs between Sections 1 and 2, and as it gets closer to the
panel centreline (at a distance of 1350 m m ) the rate of twist decreases. A similar
variation has also been observed for comer location "A".
A comparison of the angles of twist of these models at the ultimate state
(see Fig. 5.6(29)) indicates that as the depth or the strength of the spandrels
decrease, the corresponding angle of twist increases.
The maximum angle of twist of Model W2 (with deep spandrels) is about
0.004 radian, whereas that of Model W 5 (without spandrel beam) is 0.049 radians,
which is almost 12 times that of Model W 2 .
The above discussion leads to the conclusion that the angle of twist of the
deep spandrel beams in relation to their adjacent columns are negligible. However a
larger angle of twist occurs in the absence of the spandrel.
5.7 Cracking and the Ultimate Load
The behaviour of all the models was the same. At the ultimate state they
all sustained loads far in excess of the design ultimate load.
In flat plate models with spandrel beams, at the ultimate state, the first
positive yield line occurred at midspan parallel to the spandrel. Further increase in
load led to the development of negative yield lines along the front faces of the edge-
and comer-columns.
89
Model W l reached a maximum load of 30.63t kPa. At this load both the
corner column "A" and the edge-column "B" failed suddenly and violently (see
Figs. 5.7(1) and 5.7(2)). In both column positions "A" and "B", spiralling skew
cracks occurred at the faces of the spandrel adjacent to the column. The cracks
were clearly the result of torsion in the spandrel.
Model W2, with the largest spandrel beam in the test series, carried a
maximum load of 28.91 kPa. Again, at the ultimate state individual negative yield
lines developed at the front faces of the edge- and corner-columns. Further
increases in load led to the joining up of these yield lines, at which instant a sudden
and violent failure occurred (see Fig. 5.7(3)). A postmortem examination of the
test model showed that the negative flexural reinforcement of the slab across the
face of the spandrel beams were fractured after necking of the bars. The same was
true for the bars across the positive yield line.
Model W3 had the smallest practicable spandrel beams in this test series.
The maximum load carried by this model was 24.69 kPa. This model had a similar
behaviour to that of Model Wl. In this model punching shear failure occurred at all
the column positions ("A", "B" and "C") simultaneously (see Figs. 5.7(4) to
5.7(6)).
The overall depth adopted for the spandrel beam of Model W4 was 50 mm
thinner than that of Model W2, and the load at which failure occurred was 28.95
kPa. Model W4 also had a behaviour similar to that of Models Wl and W3. In this
model punching shear failure occurred at the corner column positions "A" and "C"
simultaneously (see Figs. 5.7(7) and 5.7(8)). The behaviour of this model led to
t The slab self-weight is also included,
90
the conclusion that Model W 2 had the weakest deep spandrel beam for which the
mode of failure was flexural.
Model W5 had no closed ties in its torsion strips along the edges. The
failure was initiated by the formation of negative yield line segments along the front
face of the edge- and corner-columns. The maximum load carried by the slab was
19.01 kPa. In this model, the first punching shear failure occurred at column "B";
and further application of load led to the punching shear failure at column positions
"A"and"C".
In all the column positions, "A", "B" and "C", of Model W5, 45° cracks
occurred in the upper face of the spandrels adjacent to the columns. The cracks
were clearly the result of torsion in the torsion strip (see Figs. 5.7(9) to 5.7(11)).
Apart from Model W5, the edge column positions R-3A and R-4A of the flat plate
models tested by Rangan and Hall (1983), also had torsion strips. In all these
models the ultimate steel strains of the positive slab reinforcement in the main
moment direction (at midspan) were about 0.0022, whereas the corresponding yield
strains are on average 0.0044. Thus no positive yield line at mid-span preceded the
punching shear failure.
Typical crack pattern of the present model structures are depicted in Figs.
5.7(12) and 5.7(13). Further details may be obtained from a parallel study by Latip
(1988).
5.8 Concluding Remarks
The experimental work carried out herein has produced data on punching
shear failure at 13 edge and corner column positions namely, W2-A to W5-A,
Wl-B to W3-B, W5-B and W2-C to W5-C. The data thus obtained together with
91
the observed behaviour of the test models up to failure provide useful information
for the development of the proposed prediction procedure for the punching shear
strength, Vu. This procedure is discussed in the next chapter.
DISTRIBUTION OF MOMENT AND SHEAR ALONG THE CRITICAL PERIMETER
164
CHAPTER 7
DISTRIBUTION OF MOMENT AND SHEAR
ALONG THE CRITICAL PERIMETER
7.1 General Remarks
Chapter 2 covers the theoretical background for the distribution of the total
moment, Mci, and the total shear, V u along some critical perimeters, at the edge-
and corner-column positions. The distribution is ready determinable provided
magnitudes of M x and Vi can be accurately predicted.
To develop semi-empirical formulas for the prediction of Mi and Vi, these
forces are to be measured first. The measurement procedure used has been
summarized in the block diagram of Fig. 6.3(2). According to this procedure the
slab m a y be divided into a series of parallel slab strips perpendicular to the
spandrel. The bending moment diagram along each slab strip may then be
determined with the aid of the slab reinforcement strain data. By establishing the
moment diagrams, its slope (i.e. the shear force) can also be calculated. These are
further discussed in Section 7.2.
Theoretically the measured values of Mi should be proportional to the
corresponding slab yield moments. Therefore by the calibration of the measured
M i values, some semi-empirical formulas can be developed. This is detailed in
Section 7.3.
The semi-empirical formulas developed for the prediction of Vi, are
presented in Section 7.4. In these equations Vi is expressed as a function of (i) the
locations and the strengths of the slab reinforcement, (ii) the clear span of the slab
165
in a direction perpendicular to the slab edge, and (iii) the stiffness of the spandrels
(at the side face(s) of the columns) and the slab.
7.2 Measurement of the Internal Forces at Slab-Column Connections
To measure the internal forces at the slab-column connections of the flat
plates, magnitudes of M i and Vi are to be measured first. As discussed in Section
2.5, the proposed measurement procedure is mainly based on the measured slab
strip moments.
7.2.1 Slab strip moments
Fig. 7.2(1) shows the size and location of the slab strips of Models Wl to
W 5 . It may be seen that the slab regions bounded by the panel center line(s)
adjacent to the edge- and corner-columns are divided into a series of parallel slab
strips.
Theoretically, for the determination of the moment diagram along the slab
strip i, at least three measured moments are required. Therefore along each slab
strip three measuring stations 1, m and 3 respectively at distances X = 0, X = X m
and X = X 3 were considered (see Fig. 7.2(1)). Note that X = 0 corresponds to a
distance equal to d/2 from the face of the spandrel beam, where d is the effective
depth of the top steel bars of the slab. The absolute values of the slab strip
moments of strip i at the measuring stations 1, m and 3 are respectively M n , M 3 i
and Mmi, where the subscript i stands for the measuring station (i.e. 1, m or 3).
To obtain a better picture of the variation of bending moments and shear
forces in the vicinity of columns, the width of the slab strips adjacent to the
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columns were reduced (see Fig. 7.2(1)). This is to capture the very steep strain
gradients in the vicinity of the columns (see Fig. 5.6(20)b).
As a result of a separate experimental and analytical investigation (see
Chapter 4), a relationship between the slab reinforcement strains and the bending
moments acting in a reinforced concrete slab up to the ultimate state has been
established. This relationship was used for the measurement of the slab strip
moments.
Based on the data obtained from the strain gauges at the measuring stations
1, m and 3, and with the aid of the above moment-strain relationship, magnitudes of
Mn, M3i and M^ for all the slab strips of the Models Wl to W5 are calculated. A
computer program has been developed for the calculation of the slab strip moments.
The calculation results are presented in Appendix VII. These data are used for the
determination of the shear force Vi in Section 7.2.3.
7.2.2 Slab moments Mx and Mm
Fig. 7.2(1) shows the plan view of a typical flat plate floor at a corner
location. In this figure the slab strips E and C respectively located opposite the
edge- and corner-column positions are defined as the critical slab strips; the widths
of these strips (as discussed in Appendix IV) are equal to the widths of the columns
(i.e. C2).
Mi and Mm are respectively defined as the slab moments at the measuring
stations 1 and m of the above critical slab strips. Note that the location of the
measuring station 1 overlaps with that of the front segment of the critical perimeter
(i.e. at a distance d/2 from the column face).
167
Obviously, the measurement procedure for the determination of the slab
strip moments Mi and Mm of the critical slab strips E and C would be similar to
that used for the other slab strip moments (see Section 7.2.1). Magnitudes of Mi
and Mm are then measured for all the following positions:
(i) the A series corner column, i.e. column positions Wl-A to W5-A,
(ii) the C series corner column, i.e. column positions W2-C to W5-C,
(iii) the B series edge column, i.e. column positions Wl-B to W5-B , and
(iv) the edge columns R-3A and R-4A tested by Rangan and Hall (1983).
The measured values of Mi and Mm can then be calibrated against their
respective slab yield moments. This leads to the establishment of the semi-
empirical formulas for Mi and Mm, which are discussed in Section 7.3.
7.2.3 Shear force Vi
For each flat plate model structure the measured slab strip moments as
defined in Section 7.2.1., and the data obtained from the vertical load cells at the
center of the column support systems are used for the calculation of Vi. That is, the
substitution of these data in the calculation procedure developed in Section 2.5 led
to the determination of Vi.
The measured values of Vi are then calibrated in Section 7.4. This leads
to the determination of the semi-empirical formulas for Vi. The accuracy of the
proposed measurement system is also discussed in section 7.4.
168
7.3 Distribution of the Total Bending M o m e n t
Distribution of the total bending moment between the front and side faces
of the columns m a y be determined by the prediction of M i (see Section 6.3). In
addition to M i , the predicted values of M m are also required in the prediction
procedure for Vi. Therefore in this section both for the prediction of M i and M m
semi-empirical formulas are developed.
The measured Mi and Mm and the calculated values of their corresponding
slab yield moments are calibrated separately for the same column positions reported
in Section 7.2.2.
Note that in the calculation of the yield moment of the slab strips, each
strip is considered as an isolated beam. Then the corresponding yield moments are
calculated. The calculation results are listed in Appendix VII.
A simple regression analysis is then performed to draw the "best fit"
straight line through the scatter graph of the above test data. The results are
discussed in the following subsections.
7.3.1 Edge column positions
Fig. 7.3(1) shows the variation of the measured Mi values versus their
corresponding slab yield moments, M i y , for the edge column series reported in
Section 7.2.2. The semi-empirical formula of the "best fit" straight line through the
scattered data may be expressed as
Mi,edge = 0.83 M1> y i e l d+ 0.18 7.3(1)
169
This expression may then be simplified as
Mi,edge = 0.83M1>yield 7.3(2)
where Mledge and Mlyield are respectively the estimated and yield moments of the
slab at the front segment of the critical perimeter of the edge columns.
According to Fig. 7.3(1) the coefficient of determination (R2) for the
fitted curve is 0.997, indicating a reliable fit and confirming the accuracy of the
semi-empirical formula (Eq. 7.3(2)).
As discussed in Section 5.7, for all the models with spandrel beams
(Models Wl to W4), failure started by the formation of a positive yield line at mid-
span (parallel to the spandrels). Further increase in load led to the punching shear
failure of the edge- and/or corner-column positions. This indicates that at the
ultimate state, the positive slab reinforcement corresponding to the measuring
station m of the critical slab strips C and E also yielded (see Fig. 7.2(1)). Therefore
Mm would be equal to the slab yield moment.
In Section 5.7 it is also shown that in the flat plate models which had no
realistic spandrel beams, the punching shear failure occurs prior to the formation of
a positive yield line at mid-span.
Note that the spandrel beams of these models are either without any closed
ties (Model W5), or with a di/d ratio less than 1 (Models R-3A and R-4At), where
di and d are respectively the effective depths of the spandrel and the slab.
t Rangan and Hall (1983)
170
The measured M m values for the above model structures are presented in
Table 7.3(1), and the corresponding slab yield moment are reported in Appendix
VII. A comparison of these data indicates that in the absence of the spandrels, the
measured Mm values are 0.70 times their corresponding slab yield moments.
This discussion thus shows that for slabs with spandrel beams Mm is equal
to the corresponding slab yield moment at mid-span, whereas for slabs without
spandrels this value reduces to 70% of the slab yield moment.
7.3.2 Corner column positions
Fig. 7.3(2) shows, the variation of the measured Mi values versus their
corresponding slab yield moments (Mly) for the A and C series comer columns,
i.e. column positions Wl-A to W5-A, and W2-C to W5-C.
The semi-empirical formula of the "best fit" straight line through the
scattered data may be expressed as
Ml,corner = Ml,yield 7-3(3)
where Mlcomerand Mlyield are respectively the estimated and the yield moments of
the slab at the front segment of the critical perimeter of the comer columns.
According to Fig. 7.3(2) the coefficient of determination (R2) for the fitted
curve is 1, again indicating a reliable fit and confirming the accuracy of the semi-
empirical formula, i.e. Eq. 7.3(3).
For the prediction of Mm, again a discussion similar to that of the edge-
columns may be followed (see Section 7.3.1). That is, for slabs with a spandrel
171
beams M m is equal to the corresponding slab yield moment at mid-span, whereas
for slabs without spandrels this value reduces to 7 0 % of the slab yield moment.
7.3.3 Accuracy of results
Both for the edge- and corner-column positions, the ratio of the measures
to predicted M i and M m values are presented in Table 7.3(1), respectively in the
columns titled Mutest/ Mi,predicted, and Mm,test/ Mm,predicted. The ideal ratio is
unity, where the predicted value is equal to the corresponding measured one. It
may be seen that the semi-empirical equations are accurate and consistent in their
prediction with a mean test/predicted ratio of 1.02 and a standard deviation of 0.06
for the Mi,test/ Mi,preciicted ratio. The corresponding values for the ratio of Mm,test
/ Mm,predicted are respectively 1.00, and 0.02.
In summary, according to the present procedure the estimated Mi value for
the corner column position is equal to the corresponding slab yield moment along
the front segment of the critical perimeter. This value reduces to 0.83 times the slab
yield moment at the edge locations. Also for slabs with realistic spandrel beams
(di/d > 1), M m is equal to the corresponding slab yield moment, whereas for the
slabs with torsion strips (di/d < 1), with or without closed ties, M m reduces to 7 0 %
of the corresponding slab yield moment. Note that the slab yield moment (in all
cases) is measured over a width C2 of a slab, where C2 is equal to the column
width.
172
7.4 Distribution of the Total Shear Force
7.4.1 Theoretical background
As discussed in Section 6.3, with the prediction of Vlt the distribution of
the total shear force can also be determined. Vi may be predicted by the
development of some semi-empirical formulas. Theoretically the measured values
of Vi should be proportional to the slope of the bending moment diagrams of the
critical slab strips. These slopes, as per Section 2.5, may be expressed in terms of
Mb Mm and Lc, where Mi and Mm are respectively the critical slab strip moments
at the measuring stations 1 and m, and Lc is the clear span (see Fig. 7.4(1)).
The effects of the stiffnesses of the slab boundaries on the ultimate loading
capacity of the slab is discussed in Appendix I. According to this discussion, in the
flat plates with spandrel beams, the restraint provided by the spandrels and columns
against the horizontal displacements of the slab, causes a portion of the load to be
carried by an arch or dome action, which is able to utilize the strength of the
materials much more efficiently than the normal slab actions. And as a result the
loading capacity of the slab as well as the ultimate bending and shear capacities of
the slab-column connections significantly increase.
Therefore if the di/d ratio represents the relative stiffness of the spandrel
beam in relation to the slab, due to the dome action, Vi should increase as the di/d
ratio increases.
In summary, to determine the distribution of the total shear force along the
critical perimeter, some semi-empirical formulas for Vi should be developed. In
these formulas Vi may be expressed as a function of Mi, Mm, Lc and di/d. Note
173
that Mi and M m reflect the contribution of the size and location of the slab flexural
reinforcements as well as the concrete strength.
7.4.2 The formulas
Fig. 7.4(1) shows the plan view of a typical flat plate. The shaded area
represents the critical slab strips C and E, in which the variation of the bending
moment due to vertical loading is expressed as
M = AX2 + BX + C 7.4(1)
Assuming Vi as the slope of the moment diagram at X = 0, we have
Vi = B 7.4(2)
Substituting for B from Eq. 7.4(2) into Eq. 2.5(3) gives
V 1 • x m ( x*3- X m ) <
M ™ + M - > + X 3 ( X ^ X.) - M'> 7 4 ( 3 )
where Mi, M m and M 3 respectively are the absolute values of the moments at X =0,
X = Xm and X = X3 (see Fig. 7.4(1)).
In general M3=Mi, taking X3 and Xm as fractions of the clear span, Lc, measured
from face to face of supports. Eq. 7.4(3) may be rewritten as
Vl = y(Ml
T+ Mm) 7.4(4)
174
where y is a constant with respect to M i and M m-
According to Eq. 7.4(4), V1 is proportional to Mi, M m and Lc, where M i
and M m are in turn proportional to the strength and the location of the flexural bars
of the slab. Therefore Vi is a function of the size and location of the slab
reinforcement, the clear span Lc, and the compressive strength of the concrete.
In the following subsections, the influence of the effective depth ratio
(di/d) on the magnitude of the parameter y will be examined for the edge- and
comer-columns.
7.4.3 Edge column positions
Eq. 7.4(4) is a theoretical formula for the prediction of Vi in terms of Mi,
M m , L c and the parameter y. To determine this parameter Eq. 7.4(4) may be
rewritten as
Y =(LcVl,edge)
Tedge Mi + M m '" W
Theoretically yedge should be proportional to the effective depth ratio di/d
(see Section 7.4.1). Thus, to develop a semi-empirical formula for the estimation
of this parameter, the measured values of M i , M m , Vj>edge (i.e. Vi for edge
columns) and L c against the di/d ratio are plotted for the B series edge columns.
These include column positions W l - B to W 5 - B and those tested by Rangan and
Hall (1983), i.e. column positions R-3A and R-4A.
175
A simple regression analysis is then performed to draw the "best fit"
straight line through the scatter graph of the above test data (see Fig. 7.4(2)).
This gives:
Yedge = Lr(3.19+1.56-kj ? 4(6)
where
L c \0.85 L r = ( 2 3 7 7-4(7)
By substituting L,. from Eq. 7.4(7) into Eq. 7.4(6), and Yedge from Eq.
7.4(6) into Eq. 7.4(5), and solving for V^ge, we have:
V.-4,.- 0.75(2.04 + ^-±#-- 7.4(8)
According to Fig. 7.4(2) the coefficient of determination (R2) for the fitted
curve is 0.988, indicating a reliable fit and confirming the accuracy of Eq. 7.4(8).
In this equation Mi and Mm are in kN-m, Lc in m, and Vi>edge^ in kN.
7.4.4 Corner column positions
A procedure similar to that in Section 7.4.3 may be used for the
determination of the semi-empirical formula for Vi,corner (i-e- Vi f°r corner
columns). Therefore the magnitude of Vi,corner again may be expressed as a
* Note that V l e d g e refers to the total shear force acting along the front segment of the critical perimeter over the slab width C2 (see Fig. 7.4(1))
176
function of the measured slab moments Mi, M m , Lc and the effective depth ratio
(di/d). This leads to
Vi„ = 0.24(6.90 + ^^-^ 7.4(9)
In Eq. 7.4(9) again Mi and Mm are in kN-m, Lc in m, and Vi,corner in kN.
Note that the data used for the derivation of Eq. 7.4(9) are those from the results of
the model structures Wl-A to W5-A, and W2-C to W5-C.
7.4.5 Comparison of results
Both for the edge- and comer-column positions, the ratiosof the measured
to predicted values of Vi are presented in Table 7.4(1), in the column titled
Vi,test/Vi,Predicted- Again the ideal ratio is unity, for which the predicted value is
equal to the corresponding measured one. It may be seen that the semi-empirical
formulas are accurate and consistent in their prediction with a mean test/predicted
ratio of 0.99 and a standard deviation of 0.06 .
It is important to note that the developed semi-empirical formulas for the
determination of Vi (Eqs. 7.4(8) and 7.4(9)) lead to the determination of the actual
distribution of V u (i.e. is the total shear force at the column center) between various
sides of the critical perimeter.
In summary the proposed prediction formulas for the actual distribution of
V u along the critical perimeter incorporates the effects of the size and location of the
flexural reinforcement of the slab, the clear span, Lc, the compressive strength of
the concrete, and the restraint provided by the spandrel against the horizontal
displacement of the slab.
177
7.4.6 Discussion
Rangan and Hall (1983) presented some empirical formulas for the
estimation of Vi. Based on the concept of beam analogy, they divided the slab into
a series of parallel slab strips. Unit slab shears were then obtained through
calculations of the slopes of the unit moment curves which were derived from the
measured slab reinforcement strain data.
Later Rangan (1987) found that their proposed empirical formulas
underestimate the shear force Vi. Then he referred to the absence of reliable
information, and on that basis assumed that the distribution of the average shear
stress along the critical shear perimeter is uniform. This proposal, although without
a reliable basis, was adopted by the new Australian Standard AS3600-1988.
The presented semi-empirical formulas (see Sections 7.4.3 and 7.4.4)
cover directly the effects of the following significant variables which influence the
distribution of shear force over the critical perimeter:
(i) the size and location of the slab reinforcement,
(ii) compressive strength of the concrete,
(iii) clear span (Lc) in the main moment direction, and
(iv) the effective depth ratio (di/d), which is proportional to the induced
compressive membrane action in the slab.
178
According to the above discussion Rangan and Hall (1983) did not
consider the membrane effects in their analysis. This might be one of the reasons
for their underestimated Vi values.
It is worth mentioning that for the case of the slabs without spandrels,
Regan(1981)as well as Alexander and Simmonds(1987) both emphasized the
effects of the size and location of the slab steel bars on the distribution of the total
shear force Vu along the critical perimeter.
In summary, the semi-empirical equations developed herein for the
prediction of Vi leads to the determination of the shear force variation around the
critical perimeter. To be able to determine the distribution of the total bending
moment and shear force, as discussed in this chapter, means that part of the
prerequisites for developing the proposed prediction procedure for Vu are
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CHAPTER 10
CONCLUSIONS
227
CHAPTER 10
CONCLUSIONS
As a part of a long-term study on the strength behaviour of reinforced concrete
flat plates with spandrel beams a series of five half-scale models has been tested.
With the aid of the experimental results, a prediction procedure is developed for the
punching shear strength at the corner- and edge-column positions. Based on the
present study, conclusions can be drawn in the following five areas:
(i) failure mechanisms,
(ii) moment and shear transfer between the slab and the columns,
(iii) derivation of an interaction equation for spandrel beams,
(iv) development of a prediction procedure for the determination of the punching
shear strength, Vu, and
(v) the accuracy of the proposed prediction procedure.
They are given in Sections 10.1, 10.2, 10.3, 10.4 and 10.5 respectively
with recommendations for further study enumerated in Section 10.6.
10.1 Failure Mechanisms
At the exterior slab-column connections of flat plates with spandrel beams
or torsion strips, depending on the strength of the spandrel or torsion strip at the
side faces of the column, failure could occur in one of two modes:
228
Mode 1- Development of a negative yield line across the front faces of the edge and
corner columns followed by the formation of inclined spiralling cracks in the
spandrel beam.
Mode 2- Flexural failure due to the formation of a negative yield line along the
spandrel-slab connection.
Note that in spandrel-slab connections, when the spandrel strength
parameter (8) is equal to 23 or more, failure would follow Mode 2. The spandrels
of these connections are called deep spandrel beams. For slabs with shallow
spandrel beams or torsion strips failure would be similar to Mode 1.
10.2 Moment and Shear Transfer Between Slab and Columns
To develop a prediction procedure for the punching shear strength, Vu, it
was necessary to determine the distribution of the total bending moment and the
total shear force transferred to the column center. Based on pilot studies carried
out by the author a procedure was formulated for the measurement of the individual
forces acting on a slab-column connections. The experimental data are then used
for setting up semi-empirical formulas for the prediction of M i and Vi. After
determining these strength parameters, all the forces and moments along the critical
perimeter of the column may be calculated by means of statics (see Section 6.3).
10.3 Proposed Interaction Equation for Spandrel Beams
To complete the prediction procedure for the punching shear strength, Vu,
an interaction equation for spandrel beams has been developed. In this equation the
restraining effects of the slab on the rotation and elongation of the spandrels
229
respectively are incorporated by means of co0 and y. The parameter co0 and the
factor \|/ have been calibrated experimentally.
It is worth mentioning that in the proposed procedure for the case of comer
columns, the effects of the column width on the \|/ values are expressed in terms of
the column width factor, X.
10.4 Prediction of Vu
A simple procedure is presented for the prediction of the punching shear
strength of slab-column connections of flat plates at the edge- and corner-locations.
The procedure involves
(i) determination of Mi and Vi with the aid of the proposed semi-empirical
formulas( Eqs. 9.2(8) and 9.2(9) for the comer column positions, and Eqs. 9.2(17)
and 9.2(18) for the edge column positions); and
(ii) determination of the slab restraining factor, V, as a function of the spandrel
strength parameter, 8, using the proposed semi-empirical formulas(Eq. 9.2(11) for
the corner column positions, and Eq. 9.2(19) for the edge column positions).
Then using (i) and (ii) together with the derived equilibrium equations, the value for
V u can readily be computed.
A short computer program has been developed for the calculation of Vu.
This program contains simple and explicit equations which are suitable for adoption
in design codes.
230
10.5 Versatility and Accuracy of the Proposed Procedure
The proposed procedure covers the effects of the following significant
variables which influence the punching shear strength of slab-column connections
of flat plates:
(i) the overall geometry of the connection,
(ii) the concrete strength,
(iii) the strength and location of the flexural reinforcement of the slab,
(iv) the restraining effects of slab on the rotation and elongation of spandrel beams;
(v) the enhanced strength of the slab-column connections due to membrane effects.
In the light of the experimental results reported herein a comparative study
is carried out. This indicates that the present prediction procedure is more accurate
than that recommended in AS3600-1988. The later procedure also suffers, at
times, the serious drawback of considerably overestimating the punching shear
strength.
10.6 Recommendations for Further Study
Further studies of the behaviour of slab-column connections of flat plates
should aim at determining the effects of the following parameters on the punching
shear strength:
(i) the width and the steel ratio of the spandrel beam;
231
(ii) the width of the edge columns;
(iii) the depth of the slab;
(iv) the loading patterns (including concentrated load, line load and uniformly
distributed load).
Part of the above recommendations have already been examined with the
aid of the test results of models M2, M3 and M4, constructed during 1989 with
grants provided by the Australian Research Council to Professor Y. C. Loo. It is
expected that another 3 reinforced concrete flat plate models will be constructed
during 1990. Also as a result of the progress made to date, a third year of research
is planned for 1991 to concentrate work on prestressed post-tensioned flat plates.
REFERENCES
232
REFERENCES
ACI-ASCE Committee 426, (1974), The Shear Strength of Reinforced
Concrete Member-Slabs. Proc, ASCE, Vol. 100, ST8, pp. 1543-
1591.
ACI Committee 318, (1983), Building Code Requirements for Reinforced
Concrete. ACI 318-83, American Concrete Institute, Detroit, Mich,
111 pp.
ACI-ASCE Committee 352, (1988), Recommendations for Design of
Slab-Column Connections in Monolithic Reinforced Concrete