Two-Way Flexural Behavior of Donut-Type Voided Slabs
International Journal of Concrete Structures and Materials
Two-Way Flexural Behavior of Donut-Type Voided Slabs
Voided slab systems were developed using segmented void formers such as spherical or oval plastic balls for two-way slab applications. This type of slab is expected to behave like a general two-way reinforced concrete slab because it has segmented voids, rather than the continuous voids of hollow core slabs. However, the structural behaviors of two-way voided slabs with segmented voids have not been clearly verified. Therefore, this paper analyzes the possibility of applying a donut-type two-way voided slab, which was investigated with a 12-point two-way bending test focused on global behaviors, including its load bearing capacity, flexural stiffness, ductility, deflection, and load distribution. In addition, the design method of a donut-type two-way voided slab was reviewed through the yield line method. The test results showed that one donut-type two-way voided slab acted like a conventional two-way reinforced concrete slab with the load distributed evenly between the different directions; however, another donut-type two-way voided slab with different characteristics showed uneven load distribution with different crack patterns. In addition, the yield line method could predict the load bearing capacities of the donut-type voided slabs with approximately 95% accuracy.
voided slab; two-way flexural behavior; experimental investigation and load distribution
A voided slab is a reinforced concrete slab in which voids
reduce the slab’s weight. In this slab, lightweight void
formers are placed between the top and the bottom
reinforcements before concrete casting to replace concrete in the
middle of the slab. In the early twentieth century, voided slab
systems were developed using segmented void formers such
as spherical or oval plastic balls for two-way slab
applications. The segmented void formers are expected to eliminate
the slab’s directivity and reduce its weight while maintaining
its flexural capacity. One of these slabs can reduce the slab
weight by as much as 35% compared to a solid slab with the
same flexural capacity
. For the advantages of
voided slabs using segmented void formers, the concept and
practice of voided slabs have been used, and various types of
voided slabs have been developed currently.
Many tests have been conducted to evaluate the flexural
capacities of voided slabs. In previous studies
Technology 2008; Kim et al. 2009; Chung et al. 2010, 2014;
, voided slabs showed similar strength and
slightly lower stiffness compared to that of a solid slab with
equal depth in analyses of one-way bending. Corey (2013)
reported that the flexural strength of a voided slab with
spherical voids is the same as that of a solid slab with equal
depth if the compression block used to apply the bending
force to the slab sections does not enter the void zone; and
the flexural stiffness of this voided slab is approximately
80–90% of that of the solid slab due to the cross-sectional
loss caused by voids.
These results were also demonstrated in analyses of
twoway flexural capacities of voided slabs. As reported by
Ibrahim et al. (2013)
, a voided slab with spherical voids
behaved like a conventional two-way solid slab. The voided
slab carried 89–100% of the ultimate load of a solid slab
with equal depth, and showed slightly less stiffness than the
conducted finite element
analysis of a voided slab with spherical voids, and reported
that the in-plane bending stiffness of the voided slab
decreased by 20% compared to that of a solid slab with equal
depth. However, these researches focused on only the
twoway flexural strength and stiffness of the voided slabs,
without considering the load distribution through the
different load-carrying directions.
The general assumption of a two-way slab design that the
load will be distributed load-carrying directions has limited
the acceptance of such two-way voided slabs since the
twoway load distribution of the voided slabs has not been
verified yet. Therefore, this study investigates the directional
load distribution in a two-way donut-type voided slab. In
addition, this study introduces the effects of donut-type
voids and the fixing device, which holds void formers in
place, on the voided slab’s structural behavior under
2. Experimental Program
2.1 Configuration of Two-Way Donut-Type
Voided Slab Specimens
The objective of the flexural test was to evaluate the
possibility of applying the donut-type voided slab as a
twoway slab by comparing the resulting the load distribution,
crack patterns, load-bearing capacity, flexural stiffness, and
deflection with those of a solid slab with same tensile
reinforcement ratio and dimensions.
To establish voids in the voided slab specimens, a
donuttype void former was used, as shown in Fig. 1. The
donuttype void former was a hexahedron with rounded edges and
a hole penetrating the center. The void height and width
were 140 and 270 mm, respectively. The hole diameter was
50 mm, and the distance between the voids was set to
30 mm in both the longitudinal and transverse directions.
To hold the donut-type void formers in place keeping them
in the center of the slab’s depth, two types of fixing methods
were used: the spacer and the merged type, as shown in
Fig. 2. The spacer-type consisted of void formers with
protrusions, which acted as the spacers between the top and
bottom rebars without requiring additional steel cages to
hold the void formers. The merged-type held void formers
using a steel cage, which was fabricated by welding the top
and bottom rebars with D6 diagonal rebars.
The spacer-type fixing device does not affect the
reinforcement ratio in either the longitudinal or transverse
directions. In contrast, as shown in Fig. 3c, the legs of the
merged-type fixing device increase the tensile reinforcement
ratio in the B–B0 section, creating a relatively strong
direction and a relatively weak direction in the voided slab
specimen. Previous studies (Sagaseta et al. 2011;
et al. 2012
) have reported that different tensile reinforcement
ratios in the longitudinal and transverse directions influence
the flexural behavior of a slab under two-way bending.
These reports motivated this investigation of the effect of the
fixing methods on the structural behavior of the donut-type
Three test slabs were designed to investigate these effects:
a conventional solid reinforced concrete slab (solid), a
donut-type voided slabs with the spacer-type fixing method
(TF–D–S–P.P), and a donut-type voided slabs with the
merged-type fixing method (TF–D–M–P.P). The specimens
were designed as square slabs with symmetric rebar
arrangements. The widths and lengths of the slab specimens
were 3300 mm, and their thicknesses were 250 mm. Twenty
D10 and D13 rebar were symmetrically arranged in both the
X- and Y-directions as the top and bottom rebars,
respectively. In general, voided slabs are vulnerable to shear
strength deterioration; hence, the slab specimens were
designed to have low tensile reinforcement ratios (q) of
0.353% to induce flexural failure prior to shear failure. The
merged-type fixing device was placed in the X-direction of
the specimen. Detailed specifications of specimens are
presented in Table 1 and Fig. 3.
2.2 Specimen Materials
Concrete used in all slab specimens came from one batch.
The design strength of the concrete was 24 MPa, and the
mixing ratio is summarized in Table 2. Five concrete
cylindrical specimens were made with dimensions of
100 mm (diameter) 9 200 mm (height), and then cured
under the same conditions as that of the slab specimens. The
concrete strength test conducted immediately before the
structural test showed an average strength of 24.2 MPa,
which was essentially equivalent to the design strength of
For the rebar, D10 and D13 rebar with yield strength
grades of 400 MPa were used as the top and bottom rebars,
and D6 rebar with a yield strength grade of 440 MPa was
used to fabricate the merged-type fixing devices. Tensile
tests were conducted on the rebars, and the results are
summarized in Table 3.
2.3 Loading and Measurement Set-Up
(Sagaseta et al. 2011; Matesˇan et al.
2012; Fall et al. 2014)
that conducted two-way bending tests
used a pointed load at the center of the slab. However, with a
pointed load, an unexpected punching shear failure could
occur because of the concentration of shear stress around the
loading point. Therefore, the application of multiple loading
conditions was considered to provide better results,
particularly in the case of voided slabs: voided slabs are
vulnerable to the shear strength deterioration because they use of
less concrete in the slab web and therefore less concrete is
available to resist shear. For this reason, Ibrahim et al. (2013)
tested a voided slab with spherical voids under a five-point
loading system using five hydraulic jacks to avoid
unexpected punching shear failure.
In this study, a 12-point loading system was used to
apply the two-way bending test, as shown in Fig. 4, in
order to avoid unexpected punching shear failure. The
12-point loading system consisted of one square and four
triangular steel plates, twelve loading plates with
dimensions of 200 mm 9 200 mm, and sixteen steel ball
bearings. To rotate freely with the slab deformation, the
triangular steel plates were supported at only three points,
each of them was connected with a ball bearing. The load
generated by the actuator was first transferred directly to
the center of the square steel plate, then transferred equally
to the geometric centers of the four triangular steel plates,
and then each of those loads was finally transferred equally
to the three loading plates associated with each of the
The slab was supported at all edges with line-type reaction
hinges with 1800 mm length to minimize experimental error
from support conditions, such as the generation of fixed end
moment, stress concentration, etc. The reaction hinges were
located 225 mm apart from each end with a clear span of
2850 mm. Each reaction hinge was set up on a rubber sheet
with a thickness of 10 mm above two steel frames that were
arranged with 400 mm distances, and a load-cell was located
between the two steel frames, as shown in Figs. 5 and 6.
Loading was implemented with a 2000 kN static-dynamic
hydro-actuator with a loading speed of 1 mm/min. The load
distributed toward each support was measured by a 1000 kN
load cell installed beneath the center of the line-type reaction
hinge. The deflection was measured by nine linear variable
differential transformers (LVDTs) placed under the
intersection points of the slab’s quartering lines in both directions, as
shown in Fig. 6. The strain gauges were placed at the bottom
rebars. For such simply-supported square slabs, the largest
moments are generated along the diagonal axis, similar to the
yield line. Therefore, the strain gauges on the bottom rebars
were placed at the center and four corners of each slab
specimen following the assumed yield line, as shown in Fig. 6.
2.4 Estimation of Load Bearing Capacity of Two-Way Slab Specimens
The ultimate load bearing capacities of slab specimens
under two-way bending can be estimated based on the yield
line theory. The yield line method uses rigid plastic theory to
compute the failure loads corresponding to given plastic
moment resistances in various parts of the slab
1972; Hillerborg 1996; Wight and Macgregor 2012)
yield line method is a powerful method for predicting the
failure load of reinforced concrete slabs
Famiyesin et al. 2001; Foster et al. 2004)
. Therefore, in this
study, the ultimate load bearing capacities of specimens were
estimated by the yield line method based on rigid plasticity
theory as follows.
Failure mechanisms such as crack patterns and failure
modes must be assumed to calculate the ultimate
loadbearing capacities of specimens. In this study, considering
the square shape of the slabs, the yield lines were assumed to
form X-shapes along the diagonals between the unsupported
corners, and the square slab was eventually divided into
four-triangular parts, as shown in Fig. 7.
The external work (WE) generated by the 12-point load is
formulated by multiplying the external loads and
displacements, as Eq. (1).
0:8 þ 8
Here, Pu is the ultimate load; du is the deflection at the center
of slab under the ultimate load.
The internal work (WI) generated by the in-plane moment
along the yield line is formulated by multiplying the in-plane
moment, the yield line length, and the rotation angle
between two triangular plates along the yield line. The
inplane moment (mb) per unit length along the yield line can
be calculated by considering the moment equilibrium over
per unit element of the slab, as shown in Eq. (2).
mb ¼ mx sin2ðaÞ þ my cos2ðaÞ
Here, a is the angle of the yield line; mx and my are the
inplane moment per unit length resisted in the X- and
Both mx and my can be calculated based on their respective
section properties per unit length, following Eq. (3)
mx ¼ qxdxfyðdx
ax=2Þ and my ¼ qydyfyðdy
The value of a can be assumed to be 45 because the specimens
have square shapes and symmetric rebar arrangements; the
assumed to be the same in both X- and Y-directions because
their difference in these directions are small. Based on the full
slab length (Lf), the total yield line length can be defined as
2pffi2ffi Lf. The rotation angle (hu) between two triangular plates
along the yield line can be calculated using Eq. (4), assuming
the angle is small (see Fig. 7).
Here, Ln is the net length between simple supports, as shown
in Fig. 7.
Through Eqs. (2)–(4), the internal work (WI) can be
calculated by Eq. (5).
WI ¼ 2pffi2ffi Lf
hu ¼ 8mb
tensile reinforcement ratio (q), effective depth (d), and the
depth of the equivalent rectangular stress block (a) can be
According to the energy conservation principle, the
external work and internal work should be equal; therefore,
the specimen’s ultimate load bearing capacity (Pu) is
calculated to be 1119 kN using Eq. (6).
Pu ¼ 15mb Lf
3. Test Results and Discussion
3.1 Global Failure Behavior
All of the slabs showed typical flexural behavior under
twoway bending: maintaining an elastic state until cracking,
inelastic behavior after cracking, and failure with concrete
crushing at the top of the slab surface after the bottom rebars
As shown in Fig. 8, the solid slab showed ductile flexural
behavior with yielding of bottom the rebars, and the
donuttype voided slabs also showed ductile flexural behavior
regardless of the fixing method. The displacement-ductility
ratio of the donut-type voided slabs (l = 3.4 and 4.1) were
also comparable to that of the solid slab (l = 4.4).
3.2 Load Bearing Capacity
As expected, the load bearing capacities of donut-type
voided slabs were similar to that of the solid slab. As shown
in Table 4, the yield load and initial cracking load of TF–D–
S–P.P were similar to those of the solid slab. TF–D–M–P.P
was similar in the yield load but higher in the initial cracking
load compared to the solid slab. The yield load was defined
as the load at which one of the bottom rebars reached the
yield strain of 0.24%. The initial cracking load was defined
as the load at which the strain on one of the bottom rebars
increased suddenly in the elastic state.
The ultimate load-bearing capacities of TF–D–S–P.P and
TF–D–M–P.P were equivalent to 95 and 99% of that of the
solid slab, respectively. Table 4 shows that the ultimate load
bearing capacity of TF–D–M–P.P was approximately 5%
higher than that of TF–D–S–P.P. Although this is only a
slight difference, it is inferred to be enabled by the legs of
the merged-type fixing device resulted in an increased tensile
reinforcement ratio. The legs of the merged-type fixing
device increase the tensile reinforcement ratio by 0.40% in
the longitudinal direction of the merged-type fixing device
(X-direction of specimen).
If the effect of the merged-type fixing device on the
increased tensile reinforcement ratio is considered, the load
bearing capacity of TF–D–M–P.P calculated by the yield line
method is 1172 kN, an approximately 5% increase over
1119 kN, which is the load bearing capacity of TF–D–S–P.P
calculated by the same method. In other words, the
calculations show the same tendency as the experimental result.
Therefore, the increased tensile reinforcement ratio due to
the merged-type fixing device needs to be considered with
regard to the load bearing capacity of this type of voided
When the load bearing capacity of the donut-type voided
slab was evaluated by the yield line method, the load bearing
capacity could be predicted with approximately 95%
accuracy. Therefore, the yield line method can be applied to
calculate the load bearing capacity of the donut-type voided
slab under two-way bending, as in similar analyses of the
conventional solid slab.
3.3 Flexural Stiffness
As shown in Fig. 8, the flexural stiffness of the donut-type
voided slabs decreased compared to that of the solid slab,
although the flexural stiffness of the donut-type voided slab
was improved slightly with the merged-type fixing method.
For each specimen, the flexural stiffness and the effective
moment of inertia (Ie) were compared to evaluate the flexural
stiffness of the donut-type voided slab. The flexural stiffness
was compared through the secant stiffness (K) under a yield
load and was calculated by Eq. (7).
As shown in Table 4, the secant stiffness of the donut-type
voided slabs were lower than that of the solid slab, with
stiffness decreases of 27 and 23% for TF–D–S–P.P and TF–
D–M–P.P, respectively. The decreased secant stiffness of the
K ¼ dy
donut-type voided slabs is attributable to the cross-sectional
loss caused by voids inside the slab. However, the secant
stiffness of TF–D–M–P.P and TF–D–S–P.P also differed,
even though these slabs had the same cross section. The
secant stiffness of TF–D–M–P.P was 7% higher than that of
TF–D–S–P.P, which may have been caused by the increased
the tensile reinforcement ratio due to the merged-type fixing
To confirm this conjecture, the effective moment of inertia
of the specimens were compared under a yield load. The
effective moment of inertia was initially proposed by
, with the intention of reflecting the loss of concrete
cross section according to crack propagation. As shown in
Eq. (8), the effective moment of inertia is calculated using the
uncracked moment of inertia (Ig), the cracked moment of
inertia (Icr), and the flexural cracking moment (Mcr).
Ie ¼ Icr þ
As shown in Fig. 9, the effective moment of inertia of the
voided slab was calculated for the cross section passing
through the center of the donut-type voids. In the case of
TF–D–M–P.P, the increased tensile reinforcement ratio due
to the merged-type fixing device was considered to calculate
the effective moment of inertia.
The uncracked moment of inertia of the donut-type voided
slab (Ig,D) was calculated using Eqs. (9) and (10), in
consideration of the cross-sectional loss of concrete due to the
voids. The cracked moment of inertia of the donut-type
voided slab (Icr,D) was assumed to be 90% of the cracked
moment of inertia of the solid slab (Icr,S) based on the results
n K lSo
C Pu louS;
of previous researches
(BubbleDeck Technology 2008;
, as shown in Eq. (11).
Ig;D ¼ Ig;S
N ð IDÞ
In these equation, Ig,S is the uncracked moment of inertia
of the solid slab’s cross section considering rebar; ID is the
moment of inertia of a void; Icr,S is the cracked moment of
inertia of the solid slab’s cross section; N is the number of
voids in the calculated section; and the others values are
shown in Fig. 9.
Table 4 presents the effective moment of inertia about
Y-axis of each slab under a yield load. Comparing the
specimens, the effective moments of inertia of TF–D–S–P.P
and TF–D–M–P.P were lower than that of the solid slab by
22 and 14%, respectively. In addition, the effective moment
of inertia of TF–D–M–P.P was approximately 10% higher
than that of TF–D–S–P.P. These results of the effective
moment of inertia show the similar tendency as the
experimental result of the secant stiffness. Therefore, the
crosssectional loss caused by voids and the increased tensile
reinforcement ratio due to the merged-type fixing device
need to be considered with regard to the flexural stiffness of
this type of voided slab.
3.4 Crack Pattern
Figure 10 shows the crack patterns on the bottom surfaces
of the slabs. In general, diagonal cracks formed in an
X-shape ranging from the center of the slab toward the
unsupported corners, following the assumed yield lines.
In the solid slab, several diagonal cracks with large widths
were observed between the unsupported corners and
multiple diagonal cracks with small widths were diffused beside
the large diagonal cracks. Very few orthogonal cracks with
large widths were observed; only some orthogonal cracks
with small widths were formed in the center of the slab and
near the supports. The number of cracks in the solid slab was
relatively higher than those in the donut-type voided slabs;
however, most of these cracks had small widths, as shown in
In TF–D–S–P.P, the crack pattern was similar to that of the
solid slab: several large-width diagonal cracks between the
unsupported corners. However, only a few small-width
diagonal cracks were observed in TF–D–S–P.P, and the
spacing of the large-width diagonal cracks was relatively
large in comparison to that of the solid slab, as shown in
In TF–D–M–P.P, the diagonal crack pattern was similar to
that of TF–D–S–P.P. TF–D–M–P.P also showed large-width
diagonal cracks and a few small-width diagonal cracks
between the unsupported corners, with relatively large crack
spacing. However, large-width orthogonal cracks were
observed parallel to the merged-type fixing device direction,
as shown in Fig. 10c. These cracks were inferred to be
enabled by the merged-type fixing device. In general, in a
square slab with a symmetrical rebar arrangement,
orthogonal cracks do not propagate. Only diagonal cracks
propagate along the yield line, as occurred in the solid and TF–D–
S–P.P slabs. In contrast, in a slab with an asymmetrical rebar
arrangement, the cracks propagate perpendicular to the weak
(Sagaseta et al. 2011; Matesˇan et al. 2012; Fall
et al. 2014)
. The legs of the merged-type fixing device
increase the tensile reinforcement ratio by 0.40%, creating a
strong direction and a weak direction, relative to one another.
As a result, the orthogonal cracks propagated parallel to the
direction of the merged-type fixing device in TF–D–M–P.P.
3.5 Strain of Bottom Reinforcement Bars
In reinforced concrete flexural members, the rebar
behavior can show the failure mechanism clearly. Therefore,
the strains of the bottom rebars in both the X- and
Y-directions at five points, located at the center and four corners
of each slab specimen (along with the assumed yield lines),
were examined. The load-rebar strain relationships of all
specimens are presented in Fig. 11.
In the solid slab, there was little difference in the rebar
behavior for rebars arranged in different directions; however,
the rebar behavior could be divided into two groups
according to location. Rebars located at the center of the slab
deform earlier with cracking and start to yield at a
significantly lower load, compared to rebars at the corners of the
slab. Therefore, the deformation of the slab can be deduced
to begin at the center, and then spread toward the corners,
and the solid slab behaved symmetrically in both the X- and
Y-directions under two-way bending.
In TF–D–S–P.P, the rebar behavior was very similar to that
of the solid slab: the behavior of rebars could also be divided
into two groups by location, and the rebar behaviors in both
X- and Y-directions were almost the same. There was just
one noticeable difference between the solid slab and TF–D–
S–P.P: in TF–D–S–P.P, the rebars located at the corners
began to yield at a slightly lower load than in the solid slab.
However, this difference was minor in comparison with the
ultimate load. Therefore, the donut-type voided slab with the
spacer-type fixing method can be considered to behave
symmetrically in both the X- and Y-directions, like a
conventional solid slab under two-way bending.
In contrast, in TF–D–M–P.P, the rebar behavior was quite
different from that in the solid slab and TF–D–S–P.P. In TF–
D–M–P.P, the rebar behavior cannot be easily divided into
two-groups by location, even though the rebars at the center
of the slab yielded first. However, the rebar behavior in both
the X- and Y-directions were clearly different. The rebars in
the Y-direction had a tendency to yield at a relatively lower
load compared to that of the rebars in the X-direction. This
result is expected to be caused by the legs of the
mergedtype fixing device, which create relatively weak and strong
006141 SG-1SLG-1T SG-CSLG-CT 14S0G-2S1LG6-02T SSSSSGGGGG-----CC112LLTLT
SG-3SLG-3T SG-4SLG-4T SSGG--23LT
Yield strain of rebar : 0.24 % Bot:oSmteerleiSntfroaricneGmaeungte YAisesldumLeinde SSSGGG---443LTT 0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
directions, as discussed above. Hence, the deformation was
concentrated along the weak direction of the Y-axis in this
Based on these results, the failure behavior of TF–D–M–
P.P can be verified. At the initial loading stage before the
rebar yielded, the slab behaved symmetrically with diagonal
cracks like the solid slab, and then the rebars in the
Y-direction began to yield earlier than that in X-direction. Thus,
the load was redistributed to the unyielding rebars in
X-direction, and as a result, the load increased even more after
the rebars yielded in Y-direction.
3.6 Deflection Distributions in Both X- and Y
In order to investigate the slab deflections along both the
X- and Y-directions, measurements from the lines formed by
the LVDTs 4, 5, 6 and LVDTs 2, 5, 8 were compared at four
loading stages: 0.25Pu, 0.50Pu, 0.75Pu, and 1.0Pu. The
deflection distributions in both the X- and Y-directions as the
load increased are presented in Fig. 12.
The solid slab and TF–D–S–P.P showed almost equivalent
deflection distributions along both the X- and Y-directions
until the ultimate load. In contrast, TF–D–M–P.P showed
different deflection distributions in both the X- and
Y-directions. The deflection distributions of these two directions
were almost the same until the load reached 0.5Pu. However,
the deflection in the Y-direction was larger than that in the
X-direction after the yielding of the rebars arranged in the
Y-direction, and then at 1.0Pu, the deflection in the
Y-direction became significantly larger than that in the
X-direction, as shown in Fig. 12c. The difference in the deflection
distributions between the two directions was caused by the
early yielding of the Y-direction rebar. After the yielding of
the Y-direction rebar, the deflection would be increased
significantly along in the Y-direction with even a small load
3.7 Load Distributions in Both X- and Y
A two-way slab is established under the assumption that
the load will be distributed through both the X- and
Y-directions, and the load distribution ratio between the different
load-carrying directions is used in current design methods.
The load distribution ratio in two directions is simply
calculated under a few basic assumptions:
(1) Each load, which is distributed between different
loadcarrying directions, works only in one direction, although
the slab behavior under two-way bending results from
complex interactions between the flexural behaviors in each
direction; and (2) the deflection at the center of the slab
should always be identical regardless of the directions.
The deflection of a two-way slab is related to the amount
of load, the material’s elastic modulus (E), and geometrical
properties such as the length (L) and moment of inertia of a
section in the load-carrying direction (I). In general, the
deflection at the center of a slab (dc) along each direction
under a distributed load (x) is calculated with Eq. (12).
Assuming that the material properties are identical in both
directions, the load distribution ratio between the different
load-carrying directions can be obtained by Eq. (13) because
the deflection at the center of a slab in both directions should
dc ¼ 384EI
xx Ix Ly
xy ¼ Iy Lx
X-axis Deflection 2.8
measure the total distributed load at each support because the
hinges supported not only load-cells, but also steel beams to
secure stability. Therefore, the ratio of each load-cell’s
measured value to the sum of the measured values for all
load-cells were compared to evaluate the load distribution of
To summarize, the proportion of the distributed load in
each direction is directly proportional to the ratio of their
moment of inertia and inversely proportional to the ratio of
their length. The case of a square slab is more simplified
because the length ratio is unity. Therefore, the load is
theoretically distributed equally in both directions in the case of
a square slab with a symmetrical rebar arrangement, like the
solid slab and TF–D–S–P.P.
To evaluate the load distribution in the donut-type voided
slab, measurements from four load-cells placed under the
centers of the support hinges were compared at four loading
stages: 0.25Pu, 0.50Pu, 0.75Pu, and 1.0Pu. It is impossible to
(c) TF-D-M- P.P
Fig. 13 Load sharing ratios of two-way slab specimens.
the slab under two-way bending. The results are summarized
in Fig. 13.
The load-sharing ratios in both the X- and Y-directions
were almost equivalent to each other until the ultimate load
in the specimens with the symmetric rebar arrangement (the
solid slab and TF–D–S–P.P). In contrast, TF–D–M–P.P
showed different load-sharing ratios in the two directions
with increasing load. The load-sharing ratios in these two
directions were almost the same until the load reached
0.25Pu before cracking. However, the load-sharing ratio in
the X-direction was larger than that in the Y-direction after
the cracking, and then the load-sharing ratio in the
X-direction gradually increased until the load reached ultimate
load, as shown in Fig. 13c. The different load-sharing ratios
in the two directions are expected to be caused by the
different moment of inertia in the two directions. The
donuttype voided slab with the merged-type fixing device has a
larger effective moment of inertia in the X-direction
compared to that in the Y-direction. Therefore, the different
effective moment of inertia in both two directions is
expected to have caused the different load-sharing ratios in
To evaluate the possibility of applying the donut-type
voided slab as a two-way slab, the structural behavior of the
donut type voided slab, including its load bearing capacity,
flexural stiffness, ductility, deflection, and load distribution,
were investigated. The following conclusions can be drawn
from the results of the experimental tests of the donut-type
voided slab under two-way bending.
(1) The donut-type voided slab showed typical two-way
flexural behavior, with diagonal cracks formed in an
X-shape ranging from the center of the slab toward the
unsupported corners. Compared to a conventional solid
slab, the spacing of the diagonal cracks was relatively
large, and only a few small-width diagonal cracks were
observed in donut-type voided slab. In addition, the
donut-type voided slab with the merged-type fixing
device showed large-width orthogonal cracks, parallel
to the merged-type fixing device direction.
(2) In terms of load bearing capacity, the donut-type voided
slabs demonstrated comparable load bearing capacities
to that of the solid slab with the same reinforcement ratio.
The application of the merged-type fixing device
increased the load bearing capacity of the donut-type
voided slab by approximately 5%, which was attributed
to the improved tensile reinforcement ratio introduced by
the legs of the merged-type fixing device.
(3) In terms of flexural stiffness, the donut-type voided
slabs demonstrated lower flexural stiffness than that of
the solid slab with the same reinforcement ratio. The
decrease in the flexural stiffness of the donut-type
voided slab was caused by the reduced effective
moment of inertia introduced by the voids. In addition,
the merged-type fixing device affected the effective
moment of inertia of the donut-type voided slab by
changing the reinforcement ratio. Therefore, the
geometrical properties of the voids and the fixing device
should be considered when analyzing the flexural
stiffness of the donut-type voided slabs under two-way
(4) In terms of ductility, the donut-type voided slabs
showed displacement ductility ratios of 3.4–4.1. These
values were comparable to that of the solid slab.
Therefore, the two-way donut-type voided slabs were
considered to have sufficient ductility to perform as
general flexural members.
(5) In terms of the deflection distribution, the donut-type
voided slabs demonstrated equivalent deflection
distribution in both the X- and Y-directions. However, the
donut-type voided slab with the merged-type fixing
device demonstrated an asymmetric deflection
distribution, which could be regarded as a typical behavior
for two-way slab with asymmetric rebar arrangements.
(6) In terms of load distribution, the donut-type voided
slabs demonstrated equivalent load distributions in
both the X- and Y-direction. The donut-type voided
slab with the merged-type fixing device demonstrated
different load distributions in the X- and Y-directions,
which as with the deflection result, could be regarded
as a typical behavior for two-way slabs with
asymmetric rebar arrangements.
(7) In this study, the yield line method was reviewed for
designing a donut-type two-way voided slab. As a
result, the yield line method demonstrated a prediction
accuracy of as high as 95% in the ultimate load bearing
capacity. Therefore, the yield line method can be
applied to design of the donut-type voided slab under
two-way bending, as in similar analyses of the
conventional solid slab.
(8) Based on the test results and evaluations, the
donuttype voided slab can be applied as a two-way slab in
place of the conventional heavy solid slab.
This work was supported by the National Research
Foundation of Korea (NRF) Grant funded by the Korea
Government (MSIP) (No. NRF-2016R1C1B1012618).
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