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Civil Engineering Journal
(E-ISSN: 2476-3055; ISSN: 2676-6957)
Vol. 7, No. 11, November, 2021
1918
Limitations on ACI Code Minimum Thickness Requirements
for Flat Slab
Bilal Ismaeel Abd Al-Zahra 1, Maitham Alwash 1, Ameer Baiee 1, Ali A. Shubbar 2, 3
*
1 Lecturer in Civil Engineering, University of Babylon, Hilla, Babil, Iraq.
2 Research Assistant in Civil Engineering, Liverpool John Moores University, Liverpool, United Kingdom.
3 Researcher in Department of Building and Construction Technical Engineering, College of Technical Engineering, The Islamic University,
54001, Najaf, Iraq.
Received 03 August 2021; Revised 25 September 2021; Accepted 06 October 2021; Published 01 November 2021
Abstract
Reinforced concrete two-way flat slabs are considered one of the most used systems in the construction of commercial
buildings due to the ease of construction and suitability for electrical and mechanical paths. Long-term deflection is an
essential parameter in controlling the behavior of this slab system, especially with long spans. Therefore, this study is
devoted to investigating the validation of the ACI 318-19 Code long-term deflection limitations of a wide range of span
lengths of two-way flat slabs with and without drop panels. The first part of the study includes nonlinear finite element
analysis of 63 flat slabs without drops and 63 flat slabs with drops using the SAFE commercial software. The
investigated parameters consist of the span length (4, 5, 6, 7, 8, 9, and 10m), compressive strength of concrete (21, 35,
and 49 MPa), the magnitude of live load (1.5, 3, and 4.5 kN/m2), and the drop thickness (0.25tslab, 0.5tslab, and 0.75tslab).
In addition, the maximum crack width at the top and bottom are determined and compared with the limitations of the
ACI 224R-08. The second part of this research proposes modifications to the minimum slab thickness that satisfy the
permissible deflection. It was found, for flat slabs without drops, the increase in concrete compressive strength from
21MPa to 49MPa decreases the average long-term deflection by (56, 53, 50, 44, 39, 33 and 31%) for spans (4, 5, 6, 7, 8,
9, and 10 m) respectively. In flat slab with drop panel, it was found that varying drop panel thickness t2 from 0.25 to
0.75 decreases the average long-term deflection by (45, 41, 39, 35, 31, 28 and 25%) for span lengths (4, 5, 6, 7, 8, 9
and 10 m) respectively. Limitations of the minimum thickness of flat slab were proposed to vary from Ln/30 to Ln/19.9
for a flat slab without a drop panel and from Ln/33 to Ln/21.2 for a flat slab with drop panel. These limitations
demonstrated high consistency with the results of Scanlon and Lee's unified equation for determining the minimum
thickness of slab with and without drop panels.
Keywords: Long-term Deflection; Allowable Deflection; Flat Slab; Drop Panel Thickness; Concrete Compressive Strength; Crack
width; Span Length.
1. Introduction
A flat plate slab (or known also as a flat slab without a drop panel) is a two-way reinforced concrete slab that
transfers loads directly to the supporting columns without the aid of beams or drop panels or capitals. In case of the
presence of column capitals, drop panels, or both the slab is called a flat slab. The flat plate, that is common in
*
Corresponding author: a.a.shubbar@ljmu.ac.uk
http://dx.doi.org/10.28991/cej-2021-03091769
© 2021 by the authors. Licensee C.E.J, Tehran, Iran. This article is an open access article distributed under the terms
and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1919
residential building, has several advantages such as cost savings due to low story height and simple/quick construction
and formwork, and flat ceiling that has high fire resistance (few sharps corners for concrete spalling) and less
obstruction to light diffusion. The flat slab is satisfactory for long spans and heavy loads, in particular, the flat slab is
economical for parking, warehouses, and industrial buildings [1-3].
The deflection is a crucial issue in the design of flat slabs with or without drop panels. Most Standards like ACI
318-19 [4], CSA A23.3-04 [5], AS 3600 [6] and Euro code 2 [7] propose two alternative ways for the control of
deflection. The first approach is to calculate the deflection and to compare the calculated deflection with the allowable
limits. The second approach controls indirectly the deflection by limiting minimum slab thickness or maximum
span/depth ratio.
The flexural stiffness EI (E: the concrete modulus of elasticity and I: the moment of inertia) of a flexural member
is an essential variable in the calculation of deflection. For the reinforced concrete members, the amount of section
cracking affects significantly the moment of inertia and consequently, this effect must be considered in the analysis of
deflection [8]. Generally, there are two different methods for considering the cracking effect: the effective moment of
inertia method [9] and the mean curvature method [10]. Furthermore, creep and shrinkage have important effects on
the long-term deflection, and therefore literature provides several ways for considering this effect, the most famous
one is the ACI 318 method. The analysis for deflection can be done by using a range of refined methods [11], like a
non-linear analysis or finite element analysis [12]. Recent work has used the Artificial Neural Network approach [13]
for the prediction of deflection. However, the approaches for calculating the deflection in flat slabs are complicated
and involving several approximations due to complex behavior at the service load stage (cracking, time-dependent
effect, tension stiffening). Therefore, the direct calculation of deflection for the typical situations is impractical and
engineers prefer to control the deflection using the minimum slab thickness or maximum span/depth ratio approach.
The minimum slab thickness or maximum span/depth ratio approach is the focus of many researches for decades.
Several studies [14-18] have proposed different expressions for the maximum allowable span/depth ratio for slabs
(including flat plate and flat slabs) considering the effects of different factors such as sustained load, aspect ratio,
reinforcement ratio, support condition, concrete modulus of elasticity, target maximum permissible incremental
deflection and long-term deflection effects.
Vollum and Hossain [19] have studied the span/depth rules given in Euro code 2 and they have found that the
deflections calculated in flat slabs dimensioned with span/depth rules of Euro code 2 can be excessive in external and
corner panels since the rules fail to allow for the effect of cracking during construction. Lee and Scanlon [20] have
compared the minimum slab (one-way and two-way) provisions of various Standards (ACI 318-08, Euro code 2, BS
8110-1:1997, and AS 3600-2001 and the unified equation proposed by Scanlon and Lee [15]) by performing a
parametric study to evaluate the effects of several relevant design parameters. The results show that ACI 318
conditions need a revision to cover the range of the affected design parameters. Furthermore, applicability limitations
require to be added to ACI provisions, especially for flat slab provisions which seem to be sufficient for the limit of
L/240 (for typical loading and spans) but insufficient in many cases for the limit of L/480. Bertero [21] has
investigated the effectiveness of ACI 318 provisions for minimum thickness of two-way slabs for controlling the
deflection to be within the allowable limits. This study evaluates (from a statistical viewpoint) the calculated
deflections for two-way slabs having minimum thickness specified according to the ACI 318-14 requirements and as a
result, it provides recommendations for upcoming ACI code revision. Hasan and Taha [22] have investigated the
effects of several parameters (aspect ratio, live load, concrete strength) on the long-term deflection of flat plate slabs
without edge beams (corner panels). They have highlighted the effect of not account for the aspect ratio in five
Standards (ACI 318-14, CSA A23.3-14, AS 3600, BS8110, Euro code 2) provisions for the minimum slab thickness.
Moreover, the applicability of the ACI 318-14 requirements for the thickness of flat plat slab without edge beam
appeared to be sufficient to satisfy the permissible deflection limits L/360 and L/240 for typical spans and concrete
strength while they were insufficient in many cases for the limit of L/480. Sanabra-Loewe et al. [23] have assessed the
ACI 318 code and Eurocode 2 methods for the minimum slenderness ratio of R.C. slabs. The evaluated factors were:
load, span, and permissible deflection. The results highlight the shortcoming of the Eurocode 2 and ACI 318 code
provisions. Al-Nu'man & Abdullah [24] have developed a simulation model that considering the materials and loads
uncertainties and along with the sensitivity analysis of results. The results indicate that the ACI 318-14 minimum
thickness requirements are adequate for 4m and 6m span or less for flat plate and flat slab respectively. Depending on
the characteristic strength of concrete, the redistribution factor, and the total steel ratio, Santos and Henriques [25]
have proposed new span/depth limits satisfying both deflection and ductility requirements. However, these limits are
restricted to the cases of beams and one-way slabs.
From the above review of literature, it is clear that there is a common consensus that the minimum thickness
provisions required by ACI 318 code for flat slabs cannot ensure the deflection to comply with the maximum
permissible limits for all flat slabs. Therefore, the objective of the present paper is to study the domain of applicability
of ACI minimum thickness provisions for flat slab for controlling the long-term deflection and to provide the
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1920
community of engineers the limitations for these provisions. The present paper addresses this issue by selecting the
slab thickness according to the ACI 318-19 provisions, then, calculating the deflections using the Nonlinear Finite
element Analysis for 126 case studies of flat slabs (with and without drop panels) for a range of span lengths and
practical selected values of several influencing parameters (live loads, materials strengths, and drop panel thickness)
and comparing the computed deflections with the ACI 318-19 permissible values (L/240, L/480).
2. Nonlinear Finite Element Analysis
The methodology of the present study is devoted to calculate the long-term deflection of flat slabs with thicknesses
that determined according to ACI 318-19 Code minimum thickness requirements and to compare the calculated
deflections with ACI 318-19 Code permissible limits. To achieve this goal, a nonlinear Finite Element Analysis was
performed to investigate the long-term deflection in flat slabs. The SAFE software was considered here for this
purpose. The long-term deflection was calculated according to the procedure illustrated in [26]. This procedure
includes the calculation of deflection for three cases:
Case 1: the immediate deflection due to short-term loads: DL + SDL + LL,
Case 2: the immediate deflection due to sustained loads: DL + SDL + ΨL LL,
Case 3: the long-term deflection due to sustained loads: DL + SDL + ΨL LL.
Where DL, SDL and LL represent the slab self-weight, superimposed dead load and live load applied on the slab
respectively. ΨL is the percentage of live load considered to be sustained.
Using SAFE software analysis options, the nonlinear (cracked) analysis was performed for cases 1 and 2, instead,
for case 3 the nonlinear (long-term cracked i.e. with creep and shrinkage effects) analysis was carried out.
The value of long-term deflection was determined as a linear combination of case 3 + case 1 - case 2, where the
difference between case 1 and case 2 represents the incremental deflection (without creep and shrinkage) due to non-
sustained loading on a cracked structure.
Two layouts of the flat slabs were considered for analysis in the present study. both cases consist of three equal
spans in each direction without edge beams, however, the first one is without drop panels (i.e. flat plate), see Figure 1,
and the second layout with drop panels as shown in Figure 2. The drop panel dimensions were selected to comply with
ACI 318-19 requirements for the drop panel as detailed in Figure 3.
The ACI code provisions for the minimum thickness of flat slab take into account only two effects: span length and
yield strength of steel . However, this paper considers the effects of several factors on the long-term deflection and
as a result on the minimum thickness requirements, these are: span length L, concrete compressive strength , service
live load, and drop panel thickness t2. The range of values for each one of these factors was selected to be consistent
with that used in the real practice and with available ACI 318-19 provisions. The selected values were: span length L
(4, 5, 6, 7, 8, 9, 10) m, concrete compressive strength
՛ (21, 35, 49) MPa, service live load (1.5, 3, 4.5) kN/m2, and
drop panel thickness t2 (0.25, 0.5, 0.75). On the contrary, the other parameters were considered fixed
through the analysis and their specified values were:
Steel reinforcement properties: yield strength = 420 MPa (Grade 60), Modulus of elasticity Es = 200 GPa,
Modulus of elasticity of concrete Ec = .
Superimposed dead load = 2 kN/m2,
Dimension of squared columns supporting flat slabs with span length 4, 5, 6, 7, 8, 9 and 10 m are 300, 300, 350,
400, 450, 500 and 550 mm respectively,
The percentage of live load that considered to be sustained ΨL = 25%.
The time-dependent factor or creep coefficient = 2, i.e. for sustained load duration five years or more as
specified in ACI 318-19.
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1921
Figure 1. Two-way flat slab without drop panel
Figure 2. Two-way flat slab with drop panel
Figure 3. Drop panel detail
Consequently, in total 126 case studies of flat slabs were analyzed to study the effects of factors considered in this
paper. These case studies were divided equally into two main groups. The first one includes 63 case studies of the flat
slab without drop panels and the second one comprises 63 case studies of flat slabs with drop panels. The two groups
were similar in the range of values for span length and live load (values stated above), however, the concrete
compressive strength was varied in the first one and had a fixed value =21 MPa in second group. Furthermore, the
range of values for drop panel thickness (given above) was considered in the second group only.
3. Results and Discussion
Using the nonlinear Finite Element Analysis, the long-term deflection was investigated at different points of 126
case studies of flat slabs. Figures 4 and 5 show the resulting long-term deflection for two extreme case studies of the
flat slab without drop panel having the same concrete strength (=21MPa) and live load (LL=4.5 kN/m2) but with
different values for span length (L=4m for Figure 4 and L=10m for Figure 5). Figures 6 and 7 illustrate the long-term
deflection for another two case studies similar to that shown in Figures 4 and 5 respectively but for the flat slab with a
drop (=0.25). From these four figures, it is clear that the maximum long-term deflection occurs at corner panels
and nearly at the midpoint of the diagonal line between the corner and interior columns. The same finding was drawn
from all other cases and therefore the long-term deflection given in the next sections will be at the midpoint of the
diagonal line between the corner and interior columns for the corner panels.
Due to the large campaign of case studies considered in the present paper, it is convenient to discuss the results
into two subsections, firstly for the cases of flat slabs without drop panel and secondly for the cases of flat slabs with
drop panel.
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1922
Figure 4. Long-term deflection for flat slab without drop (L=4m,
LL=4.5 kN/m2, t=315 mm and )
Figure 5. Long-term deflection for flat slab without drop
(L=10m, LL=4.5 kN/m2, t=315 mm and )
Figure 6. Long-term deflection for flat slab with drop (L=4m,
LL=4.5 kN/m2, , t=290 mm and =0.25)
Figure 7. Long-term deflection for flat slab with drop (L=10m,
LL=4.5 kN/m2, , t=290 mm and =0.25)
3.1. Two-way Flat Slab without Drop Panels
Analysis results of maximum long-term deflection for the 63 case studies of the flat slab without drop panel are
given in Table 1 and shown graphically in Figures 8, 9 and 10. As shown, the results were obtained from analyzing
flat slabs having span lengths varied from 4 to 10 m, and for three values of concrete compressive strengths (21, 35,
49) MPa and three values of live loads (1.5, 3, 4.5) kN/m2. The resulting maximum long-term deflections were
compared with the ACI 318-19 allowable deflection limits: L/480 (roof or floor construction supporting or attached to
non-structural elements likely to be damaged by large deflections) and L/240 (roof or floor construction supporting or
attached to non-structural elements not likely to be damaged by large deflections). Although the slab thickness was
dimensioned according to ACI 318-19 minimum thickness requirements (Ln/30) for all cases, the calculated
maximum long-term deflection exceeds one or both allowable limits in many cases. As an example, for the cases with
LL=1.5 kN/m2, the calculated deflections exceed the limit of L/240 when the span length is larger than 4, 6, 8 m for
՛
values of 21, 35, 49 MPa respectively. Furthermore, Figures 8, 9 and 10 show a nearly linear increase in maximum
long-term deflection as the span length changes from 4 to 10 m, but with a slope that becomes steeper for weak
concrete strength. In other words, improving the concrete compressive strength from 21 to 49 MPa reduces the
maximum long-term deflection by an average of (56, 53, 50, 44, 39, 33 and 31%) for spans (4, 5, 6, 7, 8, 9 and 10 m)
respectively. These percentages indicate that the efficiency of using stronger concrete (=49 MPa) is the highest
when the slab span length is 4 m. Regarding the effect of live loads, as expected, changing the live load from 1.5 to 4.5
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1923
kN/m2 leads to more deflection, however, this effect is more pronounced for a small span length of 4 m and is
diminished gradually for a larger span length. This behavior can be explained by referring to any short-term deflection
elastic equation (for example wL4/384EI) where the span length L has power 4 while the loads w has power 1 and
consequently the effect of the increase in span length is dominated.
Table 1 also compares the maximum cracks width at the top and bottom faces of the slab with the ACI 224R-08
[27] allowable limit of 0.3 mm that corresponds to the exposure condition: humidity, moist air and soil. From these
analysis results, there is a clear trend of increasing the crack width with the increase in span length and as a result
exceeding the allowable limits 0.3 mm for span length more than 7 m.
Table 1. Analysis results for flat slab without drop panel with different values of spans length, concrete compressive
strength and live loads
Two-way flat slab without drop panels,
LL=1.5 kN/m2
Span
(L), m
mm
allowable
deflections
(mm)
Allowable
crack
width mm
long term def
mm
maximum crack width
mm
long term
def
mm
maximum crack width
mm
long term
def
mm
maximum crack
width mm
top face
bott face
top face
bott face
top face
bott face
4
125
12.6
0.12
0.15
8.1
0.17
0.12
6.1
0.16
0.12
8.3
16.6
0.30
5
160
22.9
0.18
0.17
12.7
0.19
0.17
10.5
0.20
0.17
10.4
20.8
0.30
6
190
34.8
0.22
0.21
19.5
0.23
0.21
16.3
0.24
0.22
12.5
25.0
0.30
7
220
47.2
0.25
0.25
30.7
0.26
0.26
22.2
0.26
0.26
14.5
29.1
0.30
8
255
55.7
0.29
0.32
44.8
0.30
0.32
30.1
0.31
0.33
16.6
33.3
0.30
9
285
66.6
0.32
0.38
53.2
0.32
0.38
41.2
0.31
0.37
18.7
37.5
0.30
10
315
79.1
0.31
0.38
67.0
0.32
0.38
53.0
0.31
0.37
20.8
41.6
0.30
LL= 3 kN/m2
Span
(L), m
mm
allowable
deflections
(mm)
Allowable
crack
width mm
long term def
mm
maximum crack width
mm
long term
def
mm
maximum crack width
mm
long term
def
mm
maximum crack
width mm
top face
bott face
top face
bott face
top
face
bott face
4
125
18.8
0.15
0.14
10.8
0.15
0.13
7.7
0.15
0.13
8.3
16.6
0.30
5
160
31.6
0.18
0.19
19.7
0.18
0.18
12.4
0.19
0.18
10.4
20.8
0.30
6
190
40.4
0.21
0.23
28.3
0.22
0.22
17.7
0.22
0.22
12.5
25.0
0.30
7
220
52.1
0.24
0.27
40.0
0.23
0.26
29.0
0.24
0.26
14.5
29.1
0.30
8
255
60.0
0.30
0.33
45.9
0.29
0.32
37.4
0.29
0.33
16.6
33.3
0.30
9
285
70.2
0.30
0.35
57.8
0.30
0.35
48.3
0.30
0.34
18.7
37.5
0.30
10
315
82.2
0.32
0.39
70.7
0.31
0.39
55.4
0.31
0.39
20.8
41.6
0.30
LL=4.5 kN/m2
Span
(L), m
mm
allowable
deflections
(mm)
Allowable
crack
width mm
long term def
mm
maximum crack width
mm
long term
def
mm
maximum crack width
mm
long term
def
mm
maximum crack
width mm
top face
bott face
top face
bott face
top
face
bott face
4
125
22.5
0.14
0.14
15.0
0.16
0.14
9.5
0.15
0.14
8.3
16.6
0.30
5
160
33.9
0.17
0.18
23.8
0.18
0.18
18.1
0.18
0.18
10.4
20.8
0.30
6
190
44.9
0.21
0.22
33.1
0.21
0.22
26.4
0.21
0.22
12.5
25.0
0.30
7
220
54.3
0.24
0.26
43.1
0.24
0.26
35.5
0.24
0.26
14.5
29.1
0.30
8
255
62.1
0.29
0.31
51.6
0.26
0.32
41.2
0.29
0.32
16.6
33.3
0.30
9
285
73.3
0.31
0.34
63.7
0.30
0.34
50.3
0.30
0.35
18.7
37.5
0.30
10
315
85.2
0.35
0.35
74.3
0.35
0.35
60.7
0.35
0.36
20.8
41.6
0.30
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1924
Figure 8. Long-term deflection versus span length at different
concrete compressive strength for flat slab without drop (LL=1.5
kN/m2)
Figure 9. Long-term deflection versus span length at different
concrete compressive strength for flat slab without drop (LL=3
kN/m2)
Figure 10. Long-term deflection versus span length at different concrete compressive strength for flat slab without drop
(LL=4.5 kN/m2)
3.2. Two-way Flat Slab with Drop Panels
From the analysis of 63 case studies of the flat slab with drop panel, Table 2 provides the resulting maximum long-
term deflections, also, Figures 11, 12, and 13 show these results graphically. The variables in this analysis were the
span lengths (varied from 4 to 10 m), live loads (1.5, 3, 4.5) kN/m2 and drop panel thickness t2 (0.25, 0.5,
0.75). Since the effect of concrete compressive strength became clear from the above analysis of the flat slabs
without drop panel, a fixed value of
՛=21MPa was considered here for the analysis of flat slabs with drop panel. The
resulting maximum long-term deflections were compared with the ACI 318-19 allowable deflection limits L/480 and
L/240. In spite of the slab thickness was selected to comply with ACI 318-19 minimum thickness requirements
(Ln/33) for all cases, the computed maximum long-term deflection exceeds one or both allowable limits in many
cases. As an example, for the cases with LL=1.5 kN/m2, the calculated deflections exceed the limit of L/240 when the
span length is larger than 4, 5, 7 m for drop panel thickness t2 of 0.25, 0.5, 0.75 respectively. Moreover,
Figures 11, 12 and 13 show a nearly linear relation between the resulting maximum long-term deflection and the span
length. However, the slopes of these relations reduce as the drop panel becomes thicker. In other words, varying drop
panel thickness t2 from 0.25 to 0.75 decreases the average long-term deflection by (45, 41, 39, 35, 31, 28 and
0
10
20
30
40
50
60
70
80
90
4 5 6 7 8 9 10
Long - term deflection (mm)
Span (m)
fc=21MPa fc=35MPa
fc=49MPa L/240
L/480
0
10
20
30
40
50
60
70
80
90
4 5 6 7 8 9 10
Long- term deflection (mm)
Span (m)
fc=21MPa fc=35MPa
fc=49MPa L/240
L/480
0
10
20
30
40
50
60
70
80
90
4 5 6 7 8 9 10
Long- term deflection (mm)
Span (m)
fc=21MPa fc=35MPa
fc=49MPa L/240
L/480
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1925
25%) for span lengths (4, 5, 6, 7, 8, 9 and 10 m) respectively. These percentages show that the positive effect of drop
panel thickness is important for small spans and it becomes less significant for larger spans. Concerning the live load
effect, a similar finding to that drawn above for flat slab without drop was found here i.e. increasing the live load leads
to larger long-term deflection but this effect becomes less important with the increase in span lengths.
In addition to the maximum long-term deflection, Table 2 shows the resulting maximum cracks width at the top
and bottom faces of slab. These results exhibit a logical increase in the width of the cracks as the span length varies
from 4 to 10 m. The comparison of the resulting maximum cracks width with the ACI 224R-08 [27] allowable limit of
0.3 mm (that corresponds to the exposure condition: humidity, moist air and soil) indicates that the crack width fails to
comply with the allowable limit (0.3 mm) when the span length is more than 7 m.
Table 2. Analysis results for flat slab with drop panel with different values of spans length, drop panel thickness and live loads
Two-way flat slab with drop panels, ,
LL=1.5 kN/m2
Span
(L)
m
mm
=0.25
=0.5
=0.75
Allowable
deflections
(mm)
Allowable
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum crack
width mm
top
face
bott
face
top
face
bott
face
top
face
bott
face
4
115
29
12.8
0.18
0.11
58
9.1
0.21
0.10
87
7.2
0.24
0.10
8.3
16.6
0.30
5
145
37
24.9
0.21
0.16
73
18.8
0.23
0.15
109
13.1
0.28
0.15
10.4
20.8
0.30
6
175
44
34.8
0.24
0.20
88
25.1
0.28
0.19
132
20.5
0.31
0.19
12.5
25.0
0.30
7
200
50
46.1
0.26
0.23
100
37.1
0.29
0.22
150
26.7
0.31
0.22
14.5
29.1
0.30
8
230
58
55.4
0.29
0.27
115
43
.9
0.31
0.27
173
35.3
0.32
0.26
16.6
33.3
0.30
9
260
65
65.7
0.31
0.33
130
53
.7
0.31
0.33
195
45.7
0.32
0.29
18.7
37.5
0.30
10
290
73
78.2
0.32
0.36
145
63.8
0.34
0.38
218
55.9
0.34
0.37
20.8
41.6
0.30
LL=3 kN/m2
Span
(L)
m
mm
=0.25
=0.5
=0.75
Allowable
deflections
(mm)
Allowable
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum crack
width mm
top
face
bott
face
top
face
bott
face
top
face
bott
face
4
115
29
18.5
0.15
0.12
58
13.5
0.18
0.12
87
9.8
0.21
0.11
8.3
16.6
0.30
5
145
37
29.7
0.19
0.16
73
22.2
0.23
0.16
109
18.3
0.24
0.16
10.4
20.8
0.30
6
175
44
40.1
0.23
0.20
88
29.1
0.27
0.20
132
23.5
0.28
0.20
12.5
25.0
0.30
7
200
50
50.7
0.26
0.23
100
42.3
0.27
0.23
150
32.6
0.30
0.23
14.5
29.1
0.30
8
230
58
60.1
0.26
0.27
115
50.4
0.28
0.27
173
42.1
0.31
0.27
16.6
33.3
0.30
9
260
65
70.2
0.32
0.30
130
58.5
0.30
0.32
195
50.6
0.31
0.32
18.7
37.5
0.30
10
290
73
80.7
0.32
0.36
145
69.1
0.33
0.35
218
60.9
0.34
0.34
20.8
41.6
0.30
LL=4.5 kN/m2
Span
(L)
m
mm
=0.25
=0.5
=0.75
Allowable
deflections
(mm)
Allowable
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum
crack width
mm
mm
long
term
def
mm
Maximum crack
width mm
top
face
Bott.
face
top
face
bott
face
top
face
bott
face
4
115
29
23.1
0.15
0.13
58
17.2
0.18
0.12
87
12.8
0.21
0.12
8.3
16.6
0.30
5
145
37
35.7
0.19
0.17
73
26.2
0.22
0.16
109
22.4
0.23
0.16
10.4
20.8
0.30
6
175
44
43.2
0.23
0.20
88
35.1
0.26
0.20
132
27.6
0.25
0.20
12.5
25.0
0.30
7
200
50
54.3
0.23
0.23
100
46.2
0.25
0.23
150
39.3
0.27
0.23
14.5
29.1
0.30
8
230
58
64.1
0.25
0.27
115
54.5
0.26
0.27
173
47.6
0.35
0.27
16.6
33.3
0.30
9
260
65
73.5
0.27
0.31
130
62.8
0.29
0.31
195
55.2
0.36
0.31
18.7
37.5
0.30
10
290
73
84.5
0.30
0.35
145
75.1
0.32
0.33
218
65.3
0.36
0.31
20.8
41.6
0.30
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1926
Figure 11. Long-term deflection versus span length for different
thickness of drop panels (LL=1.5 kN/m2)
Figure 12. Long-term deflection versus span length for different
thickness of drop panels (LL=3 kN/m2)
Figure 13. Long-term deflection versus span length for different thickness of drop panels (LL=3 kN/m2)
4. Proposed Minimum Thickness of Flat Slab
4.1. Modifications of the ACI-318 Code Limitations
Based on the above discussion of the results obtained in the present study, it is clear that for the control of
deflection the use of a single formula for the minimum thickness for all flat slabs without a drop (Ln/30) or with drop
(Ln/33) as specified by ACI code (for = 420 MPa, exterior panel without edge beam) is a serious issue. The main
shortcoming of the ACI formulas is its restriction to a single variable (span length) and the ignoring of other
influencing factors like concrete compressive strength, applied live loads, and drop panel thickness.
Consequently, the 126 case studies considered here were re-analyzed using the nonlinear finite element analysis in
order to specify, for each case, the appropriate minimum thickness that can ensure the complying of long-term
deflection with the allowable limit of L/240. For this purpose, the re-analysis was performed with a gradual increase in
the slab thickness (increments of 5 mm) for each case and then the maximum long-term deflection was investigated
and compared the limit L/240. According to ACI 318-19 code, in any case, the flat slab thickness should be at least
125 mm for slab without a drop and 100 mm for slab with a drop, therefore these values were considered as the
starting values for the slab thickness in the analysis.
Table 3 gives the analysis results for the 63 cases of the flat slab without a drop. It shows, for each case study, the
resulting appropriate minimum slab thickness and the corresponding maximum calculated long-term deflection. Based
0
10
20
30
40
50
60
70
80
90
4 5 6 7 8 9 10
Long- term deflection (mm)
Span (m)
t2=t/4 t2=t/2
t2=3t/4 L/240
L/480
0
10
20
30
40
50
60
70
80
90
4 5 6 7 8 9 10
Long-term deflection (mm)
Span (m)
t2=t/4 t2=t/2
t2=3t/4 L/240
L/480
0
10
20
30
40
50
60
70
80
90
45678910
Long- term deflection (mm)
Span (m)
t2=t/4 t2=t/2
t2=3t/4 L/240
L/480
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1927
on these results, a new proposed formula for minimum slab thickness that corresponds to each case study was
proposed and provided in Table 3. As shown, these formulas vary from Ln/30 to Ln/19.9 which is a wide range as
compared with the single formula provided by ACI code (Ln/30).
Regarding the re-analysis of the 63 cases of the flat slab with a drop, the analysis results were given in Table 4.
These results include, for each case study, the investigated appropriate minimum slab thickness, the maximum
computed long-term deflection and as a result the proposed new formula for the minimum slab thickness. As shown,
the proposed formulas for the cases of slab with drop panel have a range from Ln/33 to Ln/21.2 which provides
evidence that the single ACI code formula (Ln/33) cannot be satisfactory for all cases.
Table 3. Proposed minimum thickness of flat slab without drop panels that satisfies the ACI limit L/240
Two-way flat slab without drop panels,
L.L=1.5 kN/m2
Span
(L)
m
Allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term def
(mm)
4
125
12.6
125
8.1
125
6.1
16.6
5
170
16.7
160
12.7
160
10.5
20.8
6
215
23.8
190
19.5
190
16.3
25.0
7
265
29.1
225
26.1
220
22.2
29.1
8
325
33.2
275
29.7
260
30.1
33.3
9
375
37.5
325
35.4
35.1
37.5
10
445
41.4
385
39.0
350
37.1
41.6
L.L=3 kN/m2
Span
(L)
m
Allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term def
(mm)
4
130
15.1
125
10.8
125
7.7
16.6
5
180
20.2
160
19.7
160
12.4
20.8
6
220
25.0
200
24.2
190
17.7
25.0
7
280
27.5
240
29.1
220
29.0
29.1
8
340
32.0
285
33.3
265
33.3
33.3
9
390
37.5
340
36.2
310
36.0
37.5
10
460
41.5
400
40.0
355
41.6
41.6
L.L=4.5 kN/m2
Span
(L)
m
Allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term def
(mm)
4
140
15.0
125
15.0
125
9.5
16.6
5
190
20.4
170
18.7
160
18.1
20.8
6
235
25.0
210
24.6
195
24.6
25.0
7
290
28.8
255
29.1
240
28.0
29.1
8
350
33.3
305
32.1
275
33.3
33.3
9
405
36.7
355
36.7
325
37.1
37.5
10
475
41.6
410
40.2
370
40.7
41.6
290
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1928
Table 4. Proposed minimum thickness of flat slab with drop panels that satisfies the ACI limit L/240
Two-way flat slab with drop panels, ,
L.L=1.5 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term
def (mm)
4
115
29
12.8
115
58
9.1
115
87
7.2
16.6
5
150
38
21.7
145
73
18.8
145
109
13.1
20.8
6
200
50
25.0
185
93
24.6
175
132
20.5
25.0
7
235
59
29.0
220
110
29.1
200
150
26.7
29.1
8
290
73
33.3
260
135
31.5
240
180
32.0
33.3
9
340
85
37.5
305
153
37.5
280
210
36.5
37.5
10
405
102
41.1
370
185
41.0
340
255
40.0
41.6
L.L= 3 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term
def (mm)
4
120
30
16.4
115
58
13.5
115
87
9.8
16.6
5
165
42
20.8
145
73
22.2
145
109
18.3
20.8
6
205
52
24.1
185
93
24.5
175
132
23.5
25.0
7
245
62
29.1
225
113
28.7
210
158
28.7
29.1
8
305
77
31.0
270
135
33.2
250
188
32.3
33.3
9
360
90
37.2
325
163
35.5
295
222
36.3
37.5
10
425
103
41.6
380
190
41.6
350
263
41.4
41.6
L.L=4.5 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
allowable
deflections mm
long term def
long term def
long term def
mm
long term
def (mm)
mm
long term
def (mm)
mm
long term
def (mm)
4
130
35
15.4
120
60
15.4
115
87
12.8
16.6
5
175
44
20.8
165
83
19.8
145
109
20.7
20.8
6
215
54
24.7
200
100
24.0
180
135
25.0
25.0
7
260
65
29.1
240
120
28.4
220
165
29.1
29.1
8
315
79
32.2
285
143
33.2
265
199
33.1
33.3
9
375
94
36.5
340
170
36.0
305
229
37.5
37.5
10
445
112
41.3
410
205
41.0
370
278
41.4
41.6
4.2. Scanlon & Lee Unified Slab Thickness Equation
In 2006, Scanlon & Lee [15] presented a unified equation to estimate the minimum thickness for non-prestressed
one-way and two-way slabs and beams. The proposed equation takes into account many parameters relating to the
geometrical and material characteristics of the flat slab, such as the support conditions, the existence of drop panel,
aspect ratio for edge supported slab, and the modulus of elasticity of concrete. The general form of the proposed
equation is as follows:
(1)
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1929
where: : is the clear span in mm; : is the minimum thickness in mm; : edge support coefficient (for slab without
edge support equals to 1.0 and for edge supported equals to long span / short span); : is the targeted
incremental deflection which equals to 1/480 for flat slab; : drop panel coefficient which equals to 1.0 for slabs
without drop and 1.35 for slabs with drop panels; : is the modulus of elasticity of concrete; : is the strip width
which equals to 1000mm; : is the coefficient of end support condition which equals to 1.4 for both ends continuous,
2.0 for one end continuous and 5.0 for both ends continuous; : is the coefficient column supported condition of two
way slabs which equal to 1.35 for column supported and 1.0 for other cases; : is the edge supported condition
which equals to 0.2 + 0.4 for edge supported slabs and 1.0 for other cases; : is the time-dependent factor of
sustained loads according to ACI 318-14 Code. : is the sustained load in kN/m2 which equals to the self-weight plus
superimposed dead load plus 0.25 of the live load; and : is the additional live load in kN/m2 which equals to
0.75 of the live load.
For comparison reasons, Equation 1 is implemented on the investigated cases of slabs with and without drop
panels. The results were listed in Tables 5 and 6. In general, high consistence was found between the results of the
Scanlon and Lee equation and the proposed limitation especially for slabs without drop panels. Higher thickness was
recorded by using the equations of Scanlon and Lee than the proposed limitations and the ACI-318 Code limitations.
All the output of the equation and the proposed limitations were satisfied the required allowable deflection that
indicated by the ACI-318 Code. That demonstrated the efficiency of the proposed limitations by means of agree with
the results of the equation and at the same time satisfying the allowable deflection requirements. Moreover, the
proposed limitations considered effect of thickness of the drop panels which is neglected in the Scanlon and Lee
equation.
Table 5. Minimum thickness of flat slab without drop panels based on the proposed limitations, ACI-318 Code limitations
and Scanlon and Lee equation
Two-way flat slab without drop panels,
L.L=1.5 kN/m2
Span
(L)
m
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
125
125
130.0
125
125
118
125
125
110
5
160
160
174.0
160
160
157
160
160
147
6
220
190
222.0
190
190
200
190
190
187
7
260
220
271.0
225
220
245
220
220
228
8
310
255
325.0
275
255
292
260
255
273
9
360
285
381.0
330
285
342
300
285
319
10
425
315
440.0
385
315
394
355
315
368
L.L=3 kN/m2
Span
(L)
m
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
125
125
138.0
125
125
125
125
125
117
5
175
160
183.0
160
160
165
160
160
155
6
220
190
231.0
200
190
209
190
190
195
7
280
220
282.0
245
220
255
230
220
238
8
330
255
336.0
290
255
303
270
255
283
9
380
285
393.0
340
285
354
320
285
331
10
445
315
452.0
400
315
407
370
315
380
L.L=4.5 kN/m2
Span
(L)
m
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
135
125
144.0
125
125
131
125
125
123
5
185
160
191.0
170
160
173
160
160
162
6
235
190
240.0
210
190
217
200
190
204
7
290
220
292.0
260
220
264
245
220
247
8
340
255
347.0
305
255
314
290
255
293
9
400
285
405.0
355
285
365
335
285
341
10
460
315
465.0
410
315
419
390
315
391
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1930
Table 6. Minimum thickness of flat slab without drop panels based on the proposed limitations, ACI-318 Code limitations
and Scanlon and Lee equation
Two-way flat slab with drop panels, ,
L.L=1.5 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
115
115
116.0
115
115
116.0
115
115
116.0
5
150
145
154.0
145
145
154.0
145
145
154.0
6
200
175
196.0
185
175
196.0
175
175
196.0
7
235
200
240.0
220
200
240.0
200
200
240.0
8
290
230
287.0
260
230
287.0
240
230
287.0
9
340
260
366.0
305
260
366.0
280
260
366.0
10
405
290
387.0
370
290
387.0
340
290
387.0
L.L= 3 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
120
115
123.0
115
115
123.0
115
115
123.0
5
165
145
163.0
145
145
163.0
145
145
163.0
6
205
175
205.0
185
175
205.0
175
175
205.0
7
245
200
250.0
225
200
250.0
210
200
250.0
8
305
230
298.0
270
230
298.0
250
230
298.0
9
360
260
348.0
325
260
348.0
295
260
348.0
10
425
290
400.0
380
290
400.0
350
290
400.0
L.L=4.5 kN/m2
Span
(L)
m
=0.25
=0.5
=0.75
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and
Lee eq. t
mm
Proposed t
mm
ACI-318 t
mm
Scanlon and Lee
eq. t
mm
4
130
115
129.0
120
115
129.0
115
115
129.0
5
175
145
170.0
165
145
170.0
145
145
170.0
6
215
175
214.0
200
175
214.0
180
175
214.0
7
260
200
260.0
240
200
260.0
220
200
260.0
8
315
230
308.0
285
230
308.0
265
230
308.0
9
375
260
359.0
340
260
359.0
305
260
359.0
10
445
290
411.0
410
290
411.0
370
290
411.0
5. Conclusions
The nonlinear Finite Element Analysis was used in order to study the effectiveness of using ACI minimum
thickness provisions for flat slab for the controlling of long-term deflection. The analysis involved 126 case studies
that considered the effects of several influencing parameters: slab span length, concrete compressive strength, the
applied live load, and the thickness of the drop panel. From the analysis results, the main findings can be summarized
as follow:
ACI 318-19 minimum thickness provisions required for the control of deflection in flat slab (with or without
drop) cannot be satisfactory (i.e. to comply with the ACI allowable limits L/480 and L240) for all cases because
they consider the effects of only the span length and yield strength of steel and ignoring the effects of the other
influencing factors like the concrete compressive strength, live load, and the drop panel thickness. Therefore,
these ACI code provisions have a serious problem and need a real revision;
The effect of using high concrete compressive strength on reducing the long-term deflection was found to be
significant especially for small spans. It was observed that the increase in concrete compressive strength from
21MPa to 49MPa decreases the average long-term deflection by (56%, 53%, 50%, 44%, 39%, 33% and 31%) for
spans (4, 5, 6, 7, 8, 9 and 10 m) respectively;
Civil Engineering Journal Vol. 7, No. 11, November, 2021
1931
In flat slab with drop panel, the use of thicker drop panel has an important positive effect on the reduction of
long-term deflection especially for small spans. It was found that varying drop panel thickness t2 from 0.25 to
0.75 decreases the average long-term deflection by (45, 41, 39, 35, 31, 28 and 25%) for span lengths (4, 5, 6,
7, 8, 9 and 10 m) respectively;
Concerning the live load effect, it was observed that increasing the live load leads to larger long-term deflection
but this effect becomes less important with the increase in span lengths;
Formulas for calculating the minimum thickness of flat slab were proposed to vary from Ln/30 to Ln/19.9 for flat
slab without drop panel and from Ln/33 to Ln/21.2 for flat slab with drop panel;
High constancy was observed between the results of Scanlon and Lee equation and the proposed limitations of the
minimum thickness of slabs with and without drop panels.
6. Declarations
6.1. Author Contributions
Conceptualization, B.I.A., M.A., A.B. and A.A.S.; methodology, B.I.A., M.A., A.B. and A.A.S.; software, B.I.A.;
validation, M.A., A.B. and A.A.S.; formal analysis, B.I.A.; investigation, B.I.A., M.A., A.B. and A.A.S.; data
curation, A.B. and A.A.S; writing—original draft preparation, B.I.A. and M.A; writing—review and editing, A.B. and
A.A.S.; visualization, B.I.A., M.A., A.B. and A.A.S.; supervision, B.I.A., M.A., A.B. and A.A.S.; project
administration, B.I.A., M.A., A.B. and A.A.S. All authors have read and agreed to the published version of the
manuscript.
6.2. Data Availability Statement
The data presented in this study are available on request from the corresponding author.
6.3. Funding
The authors received no financial support for the research, authorship, and/or publication of this article.
6.4. Conflicts of Interest
The authors declare no conflict of interest.
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