3.1. Influential Variables and Dataset
In this research, both conventional and PT-reinforced concrete flat slabs were designed based on the provisions of the ACI 318-18 using the same design parameters. In total, 68 PT and 9 conventional RC flat slab models for residential buildings with different geometry and span lengths were developed.
Figure 2 illustrates the area and geometry of all investigated design cases. The slab models were initially designed in SAFE 2016, the academic version [
30], and the outcomes were verified and confirmed by industrial partner specialists in PT floor design [
31]. In each design case, the CO
2 and cost were compared and comprehensively analyzed in relation to the main design parameters and variations in material composition and governing constraints of the derived designs.
The compressive strength of concrete () and slab thickness () were considered as the design variables related to concrete, while the yield strength () and diameter of the steel reinforcement () were the design variables related to steel. The tendon pattern, either distributed or bonded, was set to be a tendon variable. Tendons are normally made of seven wires made of high-strength steel, wound together and placed inside a plastic duct. At each end, a PT anchor is installed, and these are fitted in pockets embedded into the slab edge. When the strands are stressed, the wires will stretch—about 100 mm for a 15 m strand—to apply about 146 kN of load. After stressing, the tendon is cut off and the pocket in which the anchors are located is filled with grout for corrosion protection.
Various slab design cases were produced by combining these design variables. The ranges of the design variables were selected in compliance with the practical values recommended in industrial practice.
Table 1 illustrates the details and ranges of the above-mentioned design variables.
The current strength-based approaches for RC flat slabs attempt to maximize economic effectiveness through satisfying the strength requirement recommended by design codes using a minimal size of the structural member and thus a minimal quantity of materials. The calculation of embodied CO
2 emissions concerned in the current research for each design case was characterized by the major processes associated with raw material extraction and material production stages in compliance with ICE [
32]. The data for embodied CO
2 emissions and cost were based on the results of an input–output life cycle assessment (LCA) analysis and the data frequently used in the existing literature that reflect the accepted practice in the construction industry [
33,
34,
35]. The following equations were considered to estimate the embodied CO
2 emissions and cost for each flat slab design case.
where
,
,
, and
denote the volume, density, unit embodied CO
2 emissions, and cost, respectively, of the structural materials in the slabs. Subscripts
c,
s, and
t represent concrete, steel, and tendon, respectively. In Equations (1) and (2), the
,
, and
indicate the amounts of concrete, steel, and tendon, which are determined by the design variables
,
, and tendon pattern distribution, as outlined in
Table 1, for the various derived design cases. In this study, a typical value for
was taken as
, while
was set according to the density of the steel reinforcement size (
), which ranges from 10 to 16 mm. The prestressing tendons were mono strands (100 mm
2 in cross-section) where the average rupture stress (
) was 1897 MPa and the stress at 0.1% residual strain was 1689 MPa.
Table 2 shows the unit of embodied CO
2 emissions and cost for different strengths of steel and concrete.
It was not always possible to determine complete boundary conditions for the embodied energy and carbon data in the original studies. A common example was the energy that was not traced completely back to the earth or electricity that was not traced all the way upstream. The ICE database has cradle-to-gate boundaries, but a robust assessment of the carbon released would consider whole-life implications, including operation and end of life, i.e., cradle-to-grave. For materials with high embodied energy and high density such as concrete, the difference between cradle-to-gate and cradle-to-grave could be considered reasonably negligible. In contrast, the difference is significant for materials with a very low cradle-to-gate embodied energy per kilogram, such as fine and coarse aggregates.
The product stage of concrete was divided into the raw material stage, the transportation stage, and the manufacturing stage. The raw material stage refers to CO2 emission during the production of the major components of concrete such as cement, aggregate, and water. For the transportation stage, CO2 emissions occur during the hauling of raw materials to the ready-mixed concrete manufacturing plant. In the manufacturing stage, CO2 emissions emanate from the electricity and oil used in concrete batch plants. The transport distance and concrete technology were the same for all the mixtures in the ICE database. The only difference was the concrete grade.
The embodied carbon or embodied CO
2 comes from the embodied energy consumed to extract, refine, process, transport, and fabricate the material or product (including buildings). It is often measured from cradle to (factory) gate, cradle to site (of use), or cradle to grave (end of life). Therefore, the embodied carbon footprint is the amount of carbon (CO
2 or CO
2 emission) to produce a material. The embodied CO
2 unit in
Table 2 was estimated based on the life cycle inventory database constructed by ICE and some other references [
33,
36,
37], where the embodied CO
2 was estimated from the embodied CO
2 emission factors of all energy sources for the given materials’ production, from cradle to (factory) gate. The carbon footprint of concrete varies because of the required compressive strength. Nevertheless, the aggregate properties and operation standards remain constant for all case studies.
According to Kajaste and Hurme [
38], geographical differences in CO
2 emissions from cement and steel production are the reason for developing regional datasets. They analyzed regional CO
2 emissions of the cement industry by applying a climate impact management matrix on a cradle-to-gate basis, which resulted in datasets to estimate regional emission data covering over 77% of the world’s cement production. However, the ICE data for carbon and energy would have been sourced from within the British Isles. Nevertheless, for most materials, this was not feasible, and embodied energy data from international sources had to be adopted using, for instance, European and worldwide averages. Considering the datasets developed by Kajaste and Hurme, which indicated national differences in fuel sources and electricity generation, preference should be given to embodied carbon data from UK sources.
3.2. Design Constraint for Safety and Serviceability
The final design of PT RC slabs should satisfy the criteria of safety and serviceability recommended by ACI 318-18 regarding the span-to-depth ratio, minimum slab reinforcement, punching shear strength, and heel drop vibration. These criteria were reflected as constraints in the design of each PT slab case.
Figure 3 depicts the proposed design flowchart for the PT floor slabs in compliance with ACI 318-18.
According to the pertinent ACI 318-18 design provisions, the span-to-depth ratio for PT flat slabs with a total imposed load of 5 kN/m
2 is recommended to be 36. In negative moment areas at the column support, the constraint for the minimum area of bonded reinforcement at the top of the PT slab in each direction is expressed using Equation (3), where
is the larger gross cross-sectional area of the slab-beam strip in the two orthogonal equivalent frames intersecting the column in a two-way slab system.
In addition, once the spacing of tendons exceeds 1.37 m, additional bonded shrinkage and temperature reinforcement is required between the tendons at slab edges, extending from the slab edge for a distance equal to the tendon spacing.
Figure 4 depicts the steel reinforcement distribution for a typical slab with a 6 m bay.
The punching shear strength of flat RC slabs is typically controlled by the concrete strength and geometric parameters of the slab and columns. In a flat RC slab without shear reinforcement subjected to a centered axial load by an isolated column, two categories of orthogonal bending moments,
Mφ producing tangential stresses and
Mr producing radial stresses, are developed due to the bidirectional flexural action of the slab. According to ACI 318-18, the concrete punching shear stress capacity of a two-way prestressed section can be taken as:
where
is the factor used to compute
in the post-tensioned slab;
is the perimeter of the critical section;
is the average value of
for the two directions ≤ 3.5 MPa and ≥0.9 MPa;
is the factored vertical component of all prestressed forces crossing the critical section; and
is the scale factor based on the location of the critical section (40 for interior columns). In this study, all PT slab cases were designed without shear reinforcement, and therefore the applied punching shear loads were lower than the allowable punching shear stress of Equation (4).
The vibration serviceability limit state is an important design consideration for two-way PT slabs. However, it is not always adequately addressed by several international design standards. As shown in
Figure 3, the serviceability vibration check is the most important step in the PT floor design process and should be controlled once the load, slab geometry, and material properties have been defined. The frequency tuning and response calculation methods are the most adopted approaches for evaluating the vibration serviceability limit state in PT floors. The frequency tuning method sets the natural frequency of the PT floor above frequencies sensitive to resonating once excited by the lower harmonic of foot drop heel forces. The response calculation method is a performance-based approach concerned with determining the vibration response of a PT floor under the foot drop heel forces. Aalami [
39] proposed a simplified procedure based on the response calculation method in which the parameters governing the vibration response of a PT floor system are its modulus of elasticity, mass, damping, degree of cracking, if any, and post-tensioning. In this simplified model, the floor system and its superimposed load are considered for the mass calculation. Since the elastic modulus for vibration analysis is larger than corresponding static values, the recommended value is 25% higher than the static modulus. The damping also has high variability that is difficult to estimate before the PT floor is placed in service. According to Allen et al. [
40], the damping varies at 2–3% for bare concrete PT floors and 5–8% in the case of full-height partition walls. In concrete slabs, cracking decreases the floor stiffness and, therefore, decreases its natural frequency. Nevertheless, for post-tensioned floors designed according to ACI 318-18, allowable tensile stresses are relatively low; thus, such a decrease in stiffness is not essential.
The perception of floor vibration and whether it is exasperating is highly subjective and differs from reference to reference. The Canadian Steel Design Code [
41] provides the perceptibility of foot drop vibration for several floor-damping levels. The Applied Technology Council [
42] addresses the same issue and recommends a threshold of human sensitivity to vertical vibration, as depicted in
Figure 5 [
43]. Generally, there is a relationship between the floor response acceleration with different natural frequencies and perceptibility levels. It is widely accepted that humans are most sensitive to vibration for frequencies ranging between 4 and 8 Hz, while more significant acceleration can be tolerated at lower or higher frequencies.
To determine the vibration characteristics of PT floors, the natural frequency and its associated peak acceleration should be identified. Apart from commercial programs, several analytical models are available to estimate the natural frequency of RC PT floors. The simplified procedure proposed by Aalami [
44] recommends the following empirical equation to estimate the first natural frequency (
) of PT slabs with rectangular panels and uniform thickness:
is the first natural frequency [Hz],
a is the span length in the X-direction,
E is the dynamic modulus of elasticity 1.25 static E; in [MPa],
h is the slab thickness [mm],
is Poisson’s ratio [0.2],
is the gravitational acceleration [9810 mm/s
2],
is the weight per unit surface area of the slab, and
represents the boundary conditions of the slab, for a central panel bound by similar panels on each side
where
is the
x to
y span length ratio.
Using Equation (6), ATC [
42] and AISC/CISC [
45] recommend the following relationship for calculating the peak acceleration of the RC PT floor due to the foot drop heel:
where
is the peak acceleration, g is the gravitational acceleration [32.2 ft/s
2; 9.81 m/s
2],
is the constant force representing the walking force,
is the modal damping ratio,
is the effective weight of the panel and the superimposed dead load, and
is the first natural frequency [Hz].