Heavy duty long span floors
Long spanned floors with high live loads
Industrial and commercial buildings
Rui Pedro Frazão Barreiros
Abstract: The main purpose of this dissertation is to establish a comparative analysis among several
flat slabs, taking into account parameters such as the live load, the span, the material (normal or
lightweight concrete) and whether or not post-tensioning is adopted.
In the analyzed model, each slab is composed of 9 panels, supported by columns. The study is
organized into 4 groups. In group A, slabs with unbalanced spans are studied, which means that all 9
panels have the same dimensions in both directions. While the span in one direction is constant and
equal to 8m, the span in the other direction varies every 2 meters from 10m to 20m. In the next
group slabs show balanced spans, i.e. the dimension of the edge spans in each direction is 75% of the
interior span, the latter being either 16m or 20m. Slabs in group C only have unbalanced, 16m long
spans and are made of lightweight concrete, including classes 1.6, 1.8 and 2.0. All the above
mentioned slabs are post-tensioned with a 1-way band beam system, unlike those in group D where
the slabs are not post-tensioned and instead have drop panels in each column. The geometry of
these slabs is equivalent to those in group C. For each group and each parameter presented, the live
load ranged from 3 to 15 kN/m2.
The main conclusions of this thesis are: slabs with unbalanced 16m long spans are about 11% more
expensive than the balanced spans solutions; the current costs of lightweight concrete apparently
prevents slabs composed of this material from being competitive against slabs with normal concrete;
reinforced concrete slabs are about 51% more expensive than post-tensioned ones.
Keywords: Flat Slab, Band Beam, Post-Tensioned Concrete, Lightweight Aggregate Concrete
1
1. Introduction
1.1. General
The new architectural requirements regarding aesthetics and functionality of industrial and
commercial buildings have been dictating an increase in the span. Because of this, flat slabs are
becoming ever more a popular solution when it comes to designing theses types of buildings. The
goal of this paper is to understand how various factors such as the live load, the span, the material or
even the presence of prestressing influence the design and finally the cost of the above mentioned
slabs.
The requirement of longer spans and higher live loads naturally leads to the adoption of prestressing
steel in order to keep the slabs thin and avoid implementing heavy drop panels. Consequently, that
type of solution will be dealt with in more detail in this paper. Nevertheless, an economic and
structural comparison between reinforced and prestressed concrete will be presented.
Another aspect studied in this paper is the adoption of lightweight aggregate concrete (LWAC) to
replace the normal concrete used in the slabs. With a reduced self-weight, the adoption of this type
of concrete could prove to be more economical than its counterpart, reducing the required
reinforcement and prestressing steel needed to satisfy both ultimate and serviceability limit states.
1.2. Organization
The analyses are carried out on slabs composed of 9 panels, supported by columns.
The goal is to study the following aspects:

To study the influence of span lengths and live load values on the materials quantities of
structural concrete in industrial and commercial floors.

To assess the difference between balanced spanned slabs over unbalanced ones (a balanced
spanned slab being one in which the exterior spans have 75% the length of the interior ones)

To evaluate the potential economical benefit of adopting LWAC, including classes 1.6, 1,8
and 2.0

To assess the difference between reinforced concrete slabs and prestressed ones for such
applications
The analyses are organized according to the following approach:
2
Groups
A
10x10x10
B
18x18x18
12x12x12
14x14x14
20x20x20
12x16x12
16x16x16
C
15x20x15
Group A – slabs with unbalanced spans;

Group B – slabs with balanced spans

Group C – slabs made up of LWAC

Group D – reinforced concrete slabs (with drop panels)
12x16x12
16x16x16
1.6

D
1.8
2.0
For each case presented in the above diagram, the live load ranged from 3 to 15 kN/m2.
1.3. Structural System
The structural system adopted for post-tensioned and reinforced concrete slabs is shown in figures 1
and 2. The panels dimensions forced a solution of one-way band prestressed beams, for the posttensioned solution. As for the reinforced concrete solution, rectangular drop panels under each
support were adopted.
Figure 1 - Structural system adopted for the prestressed solutions
3
Figure 2 - Structural system adopted for the reinforced concrete solutions
2. Materials
The materials adopted in the study are summarized in tables 1 and 2. Note that the costs of the
materials are based on values provided by local suppliers.
Concrete
Reinforcing Steel Prestressing Steel
C30/37
A500
A1670/1860
ϕ
2,5
-
-
ν
0,2
-
-
γ (kN/m3)
25
78,5
78,5
fctm (MPa)
2,9
-
-
fk (MPa)
30,0
500
1860
fd
20,0
435
1452
εud (‰)
3,5
10,0
E (GPa)
33,0
200,0
205,0
Cost
100,0
0,9
3,1
(€/Kg)
(€/Kg)
1
3
(€/m )
Table 1 - Main characteristics of the materials prescribed in this dissertation
1
The value of the flexural tension resistance of the concrete was calculated in each case according to
*(
+
)
article 3.1.8(1) of the EC2 – Part 1.1 by the formula:
4
LC30/33
1,6
1,8
2,0
η1
0,84
0,89
0,95
ηE
0,53
0,67
0,83
ϕ
1,32
1,67
2,07
ν
0,2
γ (kN/m3)
17,5
19,5
21,5
flctm (MPa)
2,4
2,6
2,7
fck (MPa)
30,0
fcd (MPa)
20,0
εud (‰)
2,9
3,1
3,3
E (GPa)
17,5
22,1
27,3
Cost (€/m2)
180,0
166,0
131,0
Table 2 - Classes of the tested lightweight concrete
3. Design
3.1. Post-tensioning
Due to the fact that slabs are elements with a very reduced thickness, a system of bonded flat ducts
was adopted in order to maximize the tendons eccentricities. The geometry of these ducts is
presented in table 3.
Steel
Duct
Unit
6-4
h
1,8
H
2,1
b
7,2
B
7,5
s
0,03
(dimensions in cm)
Table 3 - Geometry of the tendons
5
In order to take full advantage of the beneficial effects of post-tensioning (mainly maximum
eccentricity, where the bending moments are higher), the following tendon layout was adopted in
each case:
Figure 3 - Tendon layout adopted
The immediate and time dependent losses are summarized in the following table:
Immediate Losses
15 %
Time dependent Losses
15 %
Total Losses
≈28 %
Table 4 - Prestressing losses
3.2. Serviceability Limit States
3.2.1. Deflections
The criterion adopted in this thesis regarding maximum deflections in slabs in serviceability
conditions is to limit them to span/500 for the quasi-permanent loads. The deflections were
calculated in the structural analysis software SAP2000™.
3.2.2. Cracking
Since cracking in slabs leads to a substantial increase in their deformability (caused by the reduction
of their rigidity), as well as a decrease in functionality and appearance, the criterion adopted was
that no slab would be cracked due to quasi-permanent loads (except very locally in the region of the
supports). This was accomplished by either increasing the prestress in tendons or the thickness of the
slab. Additionally, in order to guarantee cracking control where tension is to be expected, minimum
reinforcing steel was calculated in each slab according to the formula:
(1)
6
3.2.3. Vibrations
Due to the high levels of slenderness of the slabs in analysis, most of them with long spans and
reduced thicknesses, it is important to make reference to the problem of vertical vibrations which
may impair the slabs functionality. However, the task of accurately calculating the maximum vertical
acceleration that a slab is expected to be subject to, is still a complex one, mainly because there is
not yet a consensual definition of the appropriate load case, with which to assess those accelerations
for buildings in general. That being said, the vulnerability of a slab regarding vertical accelerations
can be roughly evaluated through the frequency associated to that vibration mode, as suggested in
[2]. Because this thesis involves a general assessment of several slabs in different conditions, it is
difficult to estimate what the minimum frequency of such a movement should be. Nevertheless, it is
very likely that frequencies as low as 2 to 3 Hz can cause excessive vertical accelerations.
The range of frequencies of the slabs studied in this thesis is presented in the following graphs. Note
that the graphs depict the variance with the span of the slab (x axis) as well as with the live load.
f (Hz)
Group A
6,0
5,0
4,0
3,0
2,0
1,0
0,0
10
12
14
f (Hz)
16
18
L (m) 20
18
L (m) 20
Group B
6,0
5,0
4,0
3,0
2,0
1,0
0,0
10
12
14
16
7
f (Hz)
Group C
6,0
5,0
4,0
3,0
2,0
1,0
0,0
1,6
LWAC Class 2,0
1,8
f (Hz)
Group D
6,0
5,0
4,0
3,0
2,0
1,0
0,0
3
4
5
6
7
8
9
10
11
12
13
14
LL
15
(kN/m2)
Graph 1 - Frequencies associated with the vertical movement of the different types of slabs
3.3. Ultimate Limit States
3.3.1. Bending
Figure 4 illustrates the resistance mechanism of a slab when subject to bending and compression, as
well as the equations from which the area of ordinary reinforcement is calculated from.
8
(1)
(
)
(2)
(3)
Bonded:
Note
Unbonded:
Equilibrium of moments:
(4)
Equilibrium of forces :
(5)
Figure 4 - Resistance mechanism of a slab subject to bending and compression
Prior to the calculation of As, it is necessary to guarantee that the tension in the reinforcement and
prestressing steel reach fyd and fpyd, respectively.
3.3.2. Punching Shear
The punching shear is a main concern in what regards to the design of flat slabs, particularly the ones
with such long spans. However, this issue is not approached in this paper, since this verification
would only lead to adopting drop panels in the slab, or local punching shear reinforcement, a small
concern once the main goal of this thesis is to make a comparative analysis of the different solutions
mentioned in the introductory chapter.
4. Comparative analyses
4.1. Balanced Spans vs. Unbalanced Spans
As the following graphs show, unbalanced spans require in average higher thicknesses than balanced
spans. This is mainly caused by the fact that unbalanced spans are more flexible, thus requiring more
prestress in the exterior panels than in the interior ones. However, this additional prestress would
generate high compression stresses in the concrete, leading to poor serviceability behavior. In order
to decrease those stresses to tolerable values, an increase in the thickness of the slab is required. As
9
depicted in graph 2, there is an increase of about 9% in the average thickness of the unbalanced span
slab when compared to the balanced one.
Band Thickness
Average Thickness
0,80
0,45
0,70
0,40
0,63
0,60
0,36
0,35
h (m)
h (m)
0,50
0,47
0,40
0,33
16x16x16
0,30
0,30
16x16x16
12x16x12
0,20
12x16x12
0,25
0,10
0,00
0,20
3
4
5
6
7
8
9
10 11 12 13 14 15
3
4
5
6
7
8
9
10 11 12 13 14 15
LL (kN/m2)
LL (kN/m2)
Graph 2 - Thicknesses of balanced vs. unbalanced span slabs
Because of the fact that increasing the thickness of the slab is more efficient than solely increasing
the quantity of prestressing steel, there is not a significant difference between the quantity of
prestress in balanced or unbalanced span slabs. In fact, for high live loads, a balanced span slab
requires more prestress than an unbalanced one.
As shown in graph 4, the difference in cost between a balanced and an unbalanced slab is in average
11%. Bear in mind however that this difference is closely related to the relative area of the balanced
and unbalanced spans in the total area. Consequently, the bigger the total area, the lesser the
difference in cost between those two types of slabs. This way, in a real commercial or industrial
structure where the total area of slab is expected to be higher than the one studied in this paper, the
difference between these types of slabs is likely to be less than 11%.
Prestressing Steel Index
10,0
Ordinary Steel Index
60,0
9,0
50,0
8,0
7,3
6,8
6,0
5,0
16x16x16
4,0
12x16x12
45,1
40,0
(Kg/m2)
(Kg/m2)
7,0
38,9
30,0
20,0
3,0
2,0
16x16x16
12x16x12
10,0
1,0
0,0
0,0
3
4
5
6
7
8
9
10
11
12
13
14
15
3
4
5
6
7
8
9
LL (kN/m2)
10 11 12 13 14 15
LL (kN/m2)
Graph 3 - Reinforcement steel in balanced and unbalanced span slabs
10
+ 11%
Average cost per m2 of slab
140,0
16x16x16
120,0
12x16x12
100,0
+ 16%
60,0
+ 11%
+ 8%
40,0
+ 3%
€/m2
80,0
20,0
0,0
Total
Ordinary Reinf.
Prestress
Concrete
Formwork
2
Graph 4 - Average cost per m of balanced and unbalanced slabs
4.2. Normal Concrete vs. Lightweight Concrete
A slab composed of lightweight concrete requires in general lower band thicknesses, as shown in the
graphs below. However, in order to maintain an adequate serviceability behavior (low deformation
and cracking), the thickness of the slab between bands needs to be greater than the one in normal
concrete, especially for heavy live loads, resulting in higher mean thicknesses of the slabs composed
of LWAC.
Band Thickness
Mean Thickness
0,85
0,50
16x16x16
16x16x16
LWAC 1.6
0,75
LWAC 1.8
0,70
LWAC 2.0
LWAC 1.6
0,45
LWAC 1.8
LWAC 2.0
0,63
0,65
0,62
0,60
h (m)
h (m)
0,80
0,37
0,37
0,61
0,55
0,38
0,40
0,35
0,36
0,57
0,50
0,30
0,45
0,40
0,25
3
4
5
6
7
8
9
10
11
12
13
14
15
3
4
5
6
7
8
LL (kN/m2)
9
10
11
12
13
14
15
LL (kN/m2)
Graph 5 - Thicknesses of slabs in normal concrete vs. LWAC
The main benefit when adopting LWAC is shown in the following graphs:
11
Prestressing steel index
Reinforcement steel index
8,5
60,0
8,0
7,5
LWAC 1.6
50,0
7,2
7,0
6,8
6,5
(Kg/m2)
(Kg/m2)
16x16x16
55,0
7,3
6,4
6,0
16x16x16
5,5
LWAC 1.6
5,0
LWAC 1.8
45,0
45,1
LWAC 2.0
41,7
40,0
38,1
35,0
37,3
30,0
LWAC 1.8
4,5
25,0
LWAC 2.0
4,0
20,0
3
4
5
6
7
8
9
10 11 12 13 14 15
3
4
5
6
7
8
9
10 11 12 13 14 15
sc (kN/m2)
sc (kN/m2)
Graph 6 - Reinforcement steel in normal concrete vs. LWAC slabs
These graphs show that when adopting LWAC there is a significant decrease in the quantity of
prestressing steel and, more important, in the reinforcement steel.
All things considered, as graph 6 shows, in spite of the reduction in the quantities of steel, the high
price of LWAC makes these solutions up to 20% more expensive than those in normal concrete.
Further analyses of how the price of LWAC influences the total cost of the structure showed that for
the LWAC solutions to be as expensive as the normal concrete ones, their prices had to be reduced as
the following table shows:
Cost (€/m3)
Actual
Break-Even
Class 1.6
180
Class 1.8
166
Class 2.0
131
122
118
111
↓ 32%
↓ 29%
↓ 15%
Table 5 - Cost of the different classes of LWAC (LC30/33) to match the cost of normal concrete slabs
The cost reduction is significant in every class, especially for classes 1.6 and 1.8.
Finally, if the LWAC prices were the same as the ones for normal concrete, the economy of this
solution would not surpass 7% of the total cost for normal concrete. This clearly shows that even if
the price of LWAC were not an issue, its adoption in order to obtain more economical solutions is not
significant.
12
€/m2
Average cost per m2 of slab (Current Prices)
160,0
16x16x16
140,0
LWAC 1.6
120,0
LWAC 1.8
100,0
LWAC 2.0
80,0
60,0
40,0
20,0
0,0
Total
Ordinary Reinf.
Prestress
Concrete
Formwork
Average cost per m2 of slab (Break-Even)
€/m2
16x16x16
140,0
LWAC 1.6
120,0
LWAC 1.8
100,0
LWAC 2.0
80,0
60,0
40,0
20,0
0,0
Total
Ordinary Reinf.
Prestress
Concrete
Formwork
€/m2
Average cost per m2 of slab (LWAC price = 100 €/m3)
140,0
16x16x16
120,0
LWAC 1.6
100,0
LWAC 1.8
LWAC 2.0
80,0
60,0
40,0
20,0
0,0
Total
Ordinary Reinf.
Prestress
13
Concrete
Formwork
4.3. Post-tensioned Concrete vs. Reinforced Concrete
It is widely known that for spans as big as the ones analyzed in this paper, reinforced concrete
solutions are never economically competitive when compared to post-tensioned concrete. The
reasons for such a fact are clearly demonstrated in the following graphs, starting with the overall
thickness of the slab:
Slab thickness at the support
Average thickness of the slab
1,40
1,40
12x16x12 (RC)
1,20
12x16x12 (RC)
1,20
12x16x12 (PC)
0,97
0,80
12x16x12 (PC)
1,00
h (m)
h (m)
1,00
0,60
0,80
0,60
0,50
0,47
0,40
0,40
0,33
0,20
0,20
0,00
0,00
3
4
5
6
7
8
9
10
11
12
13
LL
14
15
3
4
5
6
7
8
9
10
11
(kN/m2)
12
13
14
15
LL (kN/m2)
Table 6 - Thicknesses of reinforced vs. prestressed concrete slabs
The thickness of slabs in reinforced concrete (RC) is always higher than the one in prestressed
concrete (PC), especially for higher live loads. In average, the thickness of RC slabs can be as much as
50% higher than PC slabs.
While the increase in concrete is significant, the main difference between RC and PC lies on the
quantity of reinforced concrete. Even though RC solutions don’t have the cost of prestress, its
absence leads to a whopping increase of 160% in the quantity of ordinary reinforcement.
Ordinary Reinforcement Index
140,0
120,0
(Kg/m2)
100,0
101,7
80,0
12x16x12 (RC)
60,0
12x16x12 (PC)
40,0
38,9
20,0
0,0
3
4
5
6
7
8
9
10
11
12
13
14
15
LL (kN/m2)
Table 7 - Reinforcement steel of reinforced vs. prestressed concrete slabs
14
All the above considerations lead to the conclusion that a reinforced concrete balanced 16m long
span slab is in average 52% more expensive than one in prestressed concrete. This clearly shows that
there is no match for prestressed concrete for this range of spans.
Total cost per m2
12x16x12 (BAP)
12x16x12 (BA)
200,0
180,0
160,0
120,0
100,0
80,0
60,0
40,0
20,0
0,0
3
4
5
6
7
8
9
10
11
12
13
14
15
LL (kN/m2)
180,0
+ 52%
Average Cost per m2 of slab
PC
160,0
RC
140,0
+ 161%
120,0
100,0
60,0
40,0
+ 15%
+ 51%
80,0
- 100%
€/m2
€/m2
140,0
20,0
0,0
Total
Ordinary Reinf.
Prestress
Concrete
Formwork
Table 8 - Average cost of reinforced vs. prestressed concrete slabs
15
5. Conclusions and further developments
The main conclusions of this paper are summarized as follows:

The total cost of either unbalanced or balanced span slabs is not directly proportional to the
span. In fact, a span increase of 100% (10m to 20m) leads to an increase in the total cost of
about 85% only.

An unbalanced span slab is in average 11% more expensive than a balanced one. This
increase is mainly due to the higher quantity of ordinary reinforcement that the first type of
slabs requires, due to the fact that thicker bands demand more minimum reinforcement.

The current high prices of lightweight concrete in Portugal discards any economical benefit
by adopting such a material. Unless its price reduces by about 30%, LWAC slabs will still
remain a more expensive solution, when compared to normal concrete. Moreover, a further
analysis has led to the conclusion that even if LWAC had the same price as normal concrete,
the maximum reduction in cost would be about 7% (for LWAC of class 1.8), which does not
inspire a great deal of potential reduction in the global cost of the solution. This material
should only continue to serve its original purpose: weight reduction.

For a slab with balanced 16m long spans, a solution in reinforced concrete will be about 40%
more expensive that one with prestressed concrete, apart from the fact that it will be a
solution of poorer quality. This price increase is due to the fact that the absence of prestress
leads to a rise of about 161% in the cost of ordinary reinforcement. Moreover, a reinforced
concrete solution requires about 50% more concrete, due to the size of the drop panels that
must be adopted in a slab this long. It is also important to mention the poorer quality of such
a solution: on the one hand, the maximum deflection permitted had to be restricted to
span/250, since enforcing span/500 would lead to economically unfeasible solutions (as an
example, requiring a total deflection of span/500 in a balanced 16m long span with a live
load of 3 kN/m2 would imply a thickness of 1,10m over the columns and 0,40m at mid span!);
on the other hand, and for the same reasons, it is almost impossible to guarantee that the
slab will not crack over the quasi-permanent loads. This cracking leads to a significant
increase of the time dependent deformation, as well as making the slab more vulnerable to
weather conditions.
16
Future developments of this paper might include:

In-depth analysis of more cases of balanced span slabs;

Analyze more classes of LWAC;

Extend the study of reinforced concrete slabs to try to find the span for which both
reinforced and prestressed concrete solutions cost the same;

Compare the results of this paper to composite concrete and steel solutions;

Analyze the design difference between adopting bonded or unbonded prestressing steel
6. Acknowledgements
I want to express once again my deepest gratitude to Professor João Almeida for his kindness,
support and permanent availability, which greatly helped me to complete this thesis. He has played a
big part in motivating me in the start of my professional career and I see him as a role model. I was
very fortunate to have him as my tutor.
I would also like to thank Dr. António Cabrita for his prompt and eager availability to review my
English grammar and vocabulary. A person with his kindness is very rare.
7. References
[1] Comité Européen de Normalisation (CEN), Eurocode 2: Design of concrete structures - Part 1-1:
General rules and rules for buildings. Bruxelas, Dezembro 2004.
[2] Comité Euro-International du Béton (CEB), Vibration Problems in Structures - Practical
Guidelines. Lausanne, 1991.
[3] Fédération Internationale de la Précontrainte (FIP), Recommendations for the design of posttensioned slabs and foundation rafts., 1998.
[4] VSL International Ltd., Post-Tensioning in Buildings. Berna, Suiça, 1992.
[5] VSL International Ltd., Post-Tensioned Slabs. Berna, Suiça, Janeiro 1985.
17
[6] REBAP - Regulamento de Estruturas de Betão Armado e Pré-Esforçado. Imprensa Nacional da
Casa da Moeda, 1983.
[7] Flambó, Tiago M. A., Vibrações em Pavimentos de Edifícios - Verificação dos Critérios de
Conforto. Lisboa: Instituto Superior Técnico, 2010.
[8] Júlio Appleton and Carla Marchão, Módulo 1 - Pré-Esforço (Folhas de Apoio às Aulas). Lisboa:
Instituto Superior Técnico, 2008.
[9] Fédération International du Béton (FIB), Post-Tensioning in Buildings (Bulletin 31)., 2005.
[10] João Almeida, Pré-Esforço em Edifícios. Lisboa: Instituto Superior Técnico, 2009.
[11] Miguel Pereira, Sofia Barros, and Joana Teixeira, Centro Comercial em Matosinhos. BE2008 Encontro Nacional Betão Estrutural, Guimarães, 2008.
[12] Júlio Appleton and Carla Marchão, Módulo 2 - Lajes de Betão Armado (Folhas de Apoio às Aulas).
Lisboa: Instituto Superior Técnico, 2008.
[13] Comité Euro-International du Béton (CEB), Cracking and Deformations. Lausanne, Suiça, 1985.
[14] Augusto Gomes and João Vinagre, Betão Armado e Pré-Esforçado I: Tabelas de Cálculo (Volume
III). Lisboa, Portugal: Instituto Superior Técnico, 1997.
[15] Júlio Appleton and Carla Marchão, Módulo 3 - Verificação da Segurança aos Estados Limites de
Utilização (Folhas de Apoio às Aulas). Lisboa: Instituto Superior Técnico, 2007.
[16] Fédération International du Béton (FIB), Design of Post-Tensioned Slabs and Foundations.
Londres, 1998.
18