# Structural Model for Fibre-Reinforced Precast Concrete Sandwich Panels

Authors: Luis Segura-Castillo; Nicolás García; Iliana Rodríguez Viacava; Gemma Rodríguez de Sensale
Journal: Advances in Civil Engineering
 (2018)

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## Abstract

Fibre-reinforced concrete (FRC) has been used in numerous types of precast elements around the world, as has been shown that reductions in production costs and time can be obtained; however, there is little experience of this material in Uruguay. Therefore, our study analysed the feasibility of its utilisation in this country. This paper reports on the development of a simple analysis model that is useful for the design of FRC precast elements. The model efficiency was evaluated through its application to a practical case study—vertical precast concrete sandwich panel systems tested by bending. Three different types of reinforcement were analysed: synthetic fibres, metal fibres, and steel mesh. With the developed model, the cost-efficiency of different panel geometries and amounts of reinforcement were evaluated. The model allowed consideration of the contribution of the fibres to withstand internal tensile forces of the panels and therefore be able to substitute for the steel mesh in the panel wythes. It was found that it was possible to optimise panel reinforcement and geometry, thereby reducing wythe thickness. Besides the reduction in production time, it was possible to achieve cost savings of up to 10% by replacing steel mesh with fibres and of more than 20% if the geometry was also modified.


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## Body

## 1. Introduction

Fibres have been successfully used in precast elements since the early days of development of fibre-reinforced concrete (FRC) [1]. Numerous types of FRC precast elements can be found around the globe [2, 3], including roof elements [4], tunnel segmental linings [5], and pipes [6]. Besides the reduction in costs and production times associated with the use of FRC, further advantages can be obtained if the use of fibres is combined with a self-compacting concrete (SCC) matrix, obtaining self-compacting fibre-reinforced concrete (SCFRC), which allows the casting of structural elements with complex geometry and/or composed of thin components [7].Although there are several local applications that use FRC in Uruguay, mainly in industrial pavements and small precast elements, and the use of this technique is increasingly common, there are very few cases in which its design and use are based on criteria endorsed by guidelines or recommendations, with only two documented cases where a quality-control program was followed [8, 9]. In most cases, the design is based on recommendations from fibre suppliers, whilst different recommendations and criteria are available; in practice, quality control of such materials is not carried out. Therefore, it is not possible to know whether the specifications meet the requirements for the design.The vast majority of precast elements produced in Uruguay are based on heavy prefabrication, largely using conventional reinforced concrete (CRC), while SCC and FRC are rarely used. Therefore, there is potential to improve the national precast industry with the use of these materials. Precast concrete sandwich panel (PCSP) systems can be mentioned among the elements that could be improved. These panels are composed of two concrete wythes (or concrete layers) separated by a layer of insulation [10]. Different ways of connecting the concrete wythes through the insulation layer include concrete webs, metal or plastic connectors, or a combination of these.Panel prototypes have been tested by Barros et al. [11–13] to assess benefits to the flexural and shear resistance of thin structural systems when CRC wythes are replaced with SCFRC. A smeared multifixed crack model, implemented into a finite-element method-based computer program, was used to simulate panel structural behaviour up to failure. Other research has assessed the use of fibre-reinforced polymer as concrete wythe connectors for this solution [14, 15]. FRC panels have also been used as partial support in the study of a fibre-reinforced slab [16].In line with this, a research project was developed to analyse the feasibility of the utilisation in Uruguay of FRC and SCFRC for precast elements, in particular, in a PCSP system, which was chosen as the case study. An experimental study was first carried out at the material level [17], and then prototypes where cast in order to analyse the feasibility of their production in the concrete plant and tested under bending to assess the main responses [18].The objective of this work was twofold: on one hand, to develop a simple analysis model useful for the design of PCSPs reinforced with FRC, and on the other hand, to optimise the design and analyse the cost-efficiency of the solution.A theoretical model for evaluating the structural behaviour (moment-curvature and load-displacement relationships) of PCSP elements reinforced with concrete with fibres is proposed. The model follows the guidelines established in the Structural Concrete Instructions of Spain (EHE-08, Annex 14) and is based on results obtained at the material level.


## 2. Case Study: Precast Concrete Sandwich Panels

The model efficiency was evaluated through its application in a practical case study: vertical PCSP tested by bending. The panels had a prismatic shape with the dimensions (height × width × thickness) 2.4 × 1.2 × 0.2 m, with a central insulation layer made of expanded polystyrene (EPS) foam sheets (2.1 × 0.9 × 0.1 m), as shown in Figure1. According to the defined geometry, an external-edge stiffening beam, 0.15 m wide, was formed at the panel perimeter.Figure 1
Plan and cross section of the panel.These panels were reinforced with steel trusses formed by three longitudinal bars of 8 mm nominal diameter, two top and one bottom (according to the direction of testing), connected by diagonal bars with a diameter of 4.2 mm (Figure2). In the two external layers of concrete (concrete wythes), 0.05 m thick, the following reinforcement alternatives were evaluated (Figure 2):(i)
P_Mesh: steel mesh with a diameter of 4.2 mm and mesh width of 150 mm (ϕ4.2 mm/150 mm), placed in the middle plane of each concrete layer(ii)
P_FRCM20: 20 kg/m3 steel fibres with hooks made of low-carbon steel, 50 mm long, 1 mm in diameter, with a tensile strength superior to 1100 MPa and specific weight of 7.85 kg/m3 (Wirand FF1)(iii)
P_FRCS6: 6 kg/m3 synthetic fibres, polyolefin macrofibres with corrugated surfaces, 48 mm long, equivalent diameter of 1.37 mm, with a tensile strength superior to 550 MPa and specific weight of 0.92 kg/m3 (Fiber Force PP-48)Figure 2
Details of different types of reinforcement tested: (a) traditional reinforcement with steel bars and (b) synthetic or steel fibre reinforcement.
(a)
(b)Figure3 shows the panel construction process using traditional steel reinforcement bars. The reinforcement assembly and placement stages were eliminated in the case of the panels with fibres.Figure 3
Construction of lightweight concrete panels.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Concrete with a nominal compressive strength offck = 35 MPa and steel with a characteristic strength of fyk = 500 MPa were used. The mechanical parameters of the concrete used in the model were determined experimentally from the specimens made during the concreting of the panels, as described in the following section.The flexural strength test of the panels was carried out on three specimens of each type of reinforcement, applying the standard ASTM E72 (2015) [19]. The panel was tested in a horizontal position and was simply supported by a span between supports of 2.1 m. Two-point loading of equal magnitude was applied, each load at a distance of one-quarter of the span from the supports, as shown in Figure 4. In this way, the central section was subjected to pure bending moment.Figure 4
Two-point load test setup.


## 3. Structural Analysis

### 3.1. Structural Behaviour Model

The expected structural behaviour can be divided in three stages [20], which are represented in the load-displacement diagram shown in Figure 5. In the first stage (linear elastic), the concrete matrix is in an uncracked phase and, therefore, the element behaves in a linear elastic way. In the second stage (postcracking), the matrix starts cracking. This is a transition stage during which the tensile stresses are transferred from the matrix to the reinforcement. In the third stage (yielding), yielding takes place in at least one of the materials. As there was a low amount of reinforcement used in this study, yielding took place in the reinforcement whilst the concrete was still in the linear elastic stage. Depending on the type and amount of reinforcement, two types of behaviour can take place in the yielding stage [21]: hardening, in which the ultimate strength is greater than the cracking strength of the element, or softening, where the ultimate strength is below the cracking strength, as shown in Figure 5.Figure 5
Load-displacement curves for panels subjected to a bending test.Stage 1: Linear elastic. Transverse deformations of the panel in Stage 1 are shown schematically in Figure 6. The following basic hypothesis on the behaviour of the reinforced concrete was taken into account in the model. The concrete was in the precracked stage; that is, the tensile strength of concrete had not been reached, and therefore, it could develop both tensile and compressive stresses. Also, a linear elastic behaviour of the concrete could be assumed. Finally, the Navier–Bernoulli hypothesis was assumed, meaning that the planar sections would remain planar and perpendicular to the deformed axis. This hypothesis is based on the assumption that the width of the external frame was greater enough to act as a concrete web, efficiently transferring the shear forces.Figure 6
Deformation diagram on the linear elastic section.Under the mentioned hypothesis, a classic formulation of the mechanics of materials is valid. The cracking moment (Mcr) was determined using Equation (1). The reinforcement contribution was neglected for the three types of reinforcements, being negligible in comparison with the forces created by the concrete in tension:(1)Mcr=fct,m,flex.Ich/2,where fct,m,flex is the mean flexural tensile strength of concrete, Ic is the second moment of area (uncracked concrete section), and h is the cross-sectional depth.Also, in Stage 1, the relationship between moment (M) and curvature (χ) is linear up to the cracking moment, which is given by(2)χ=McrEc.Ic,where Ec is the reduced modulus of elasticity of concrete.Given the static configuration of the test, the moment value can be related to that of the applied load (F), according to Equation (3), where L is the span between the supports. Since it is a statically determined setup, this relationship is maintained in all stages of the test:(3)F=8·ML.Finally, the displacements can also be obtained using classic equations. The displacement of the midpoint (δ) is given by the following expression, with the meanings of the variables indicated above:(4)δ=11·F·L3768·Ec·Ic.Stage 2: Postcracking. The postcracking stage is a transitory phase between Stages 1 and 3, in which progressive cracking of the matrix takes place, transferring the tensile stresses from the concrete matrix to the reinforcement (bars or fibres). As the behaviour in this stage reports great complexity and, in turn, does not contribute significant information to the design, this section was not precisely determined. In a practical way, the structural behaviour in this stage was completed by a linear transition from the last point determined in Stage 1 with the first point in Stage 3.Stage 3: Yielding. The moment-curvature curve was determined by discretising the curvature curve (χi) at several points, at which the values of the corresponding moment (Mi) were determined. For each curvature value (χi) analysed, and for a given position of the neutral line (xi in Figure 7), it was possible to calculate the deformations in the entire section through the Navier–Bernoulli hypothesis. By means of the constitutive equation of each material, the tensions were obtained for every position within the section. A triangular stress distribution was considered for concrete under compression, and a multilineal constitutive equation (described below) was used for the FRC in tension. These assumptions are represented in the diagrams shown in Figure 7. The stresses provided by the EPS, as well as the tensile stresses of plain concrete, were considered to be null. Also, a punctual load (Fs) was applied where reinforcement bars were present.Figure 7
Diagrams representing strain (ε) and stress (σ).Thexi value that solves the problem was determined by equilibrium equations. As the beam was under pure bending, the axial force was equated to zero (N=0) by means of an iterative procedure. Finally, for the value of curvature (χi) and xi found, the corresponding moment was calculated. The moment-curvature curve was obtained by joining all pairs of values (Mi,χi) obtained from this analysis. It is important to take into account that it is not possible to adopt curvatures that involve a deformation of the steel in tension greater than 10‰ or of the FRC in tension greater than 20‰, as these values correspond to the ultimate state limits [22]. As indicated above, the relationship between applied load and moment in the central section was determined by Equation (3).A single crack was observed in the fibre-reinforced panels, where the deformation was concentrated. Therefore, to determine the displacements in the panels in the yielding stage, it was considered that each panel behaved as two rigid bodies that rotated on a hinge where the crack was formed, as illustrated in Figure8. To simplify the analysis, it was assumed that the crack opening (ω) was located in the central section of the panel.Figure 8
Displacement in the panel according to the rigid body model.Considering this scheme, it is possible to correlate the value of the deflection at the midpoint (δ) with the crack opening through trigonometric relationships (δ=L·ω/4·h). In turn, the crack opening is related to the strain at any point by the structural characteristic length, lcs, which, in this case, can be assumed to be equal to h [22]. Therefore, for the bottom fibre: ω=ε·lcs=ε·h. Finally, the strain in the bottom fibre of the cross section can be approximately calculated with the curvature (εinf=χ·h). Combining the previous expression, the following equation can be obtained:(5)δ=L·ω4·h=L·εinf·h4·h=L·χ·h·h4·h⇒δ=L·h·χ4.To estimate the displacement in the mesh-reinforced panels, two strategies were used, depending on whether the yield moment (My) of the central section was exceeded. If the maximum moment was below My, Branson’s empirical formula [23] was used to calculate the equivalent moment of inertia (Ie) of the element:(6)Ie=McrMa3·Ib+1−McrMa3·If≯Ib,where Mcr is the cracking moment, Ma is the acting bending moment, Ib is the second moment of area (uncracked concrete section), and If is the moment of inertia of the cracked section.If the maximum moment exceeds the yield moment, it is understood that large deformations take place in the central section. For this reason, the displacement of the central point was calculated by integrating the curvature of the section between loads (χmax, which was assumed to be constant in this section) and neglecting the rest of the deformations. The variation of the angle of rotation from the central section to the point of application of the loads can be computed as(7)θmax=∫0L/4χmaxdx=L4·χmax.And, hence, the displacement of the central point:(8)υmax=∫0L/2θxdx=∫0L/4xL/4θmaxdx+∫L/4L/2θmaxdx=L4·3L8·χmax.


### 3.2. Ultimate Moment Approximation

Besides the previously described procedure, it is possible to establish approximated formulas that allow the calculation, by hand, of the ultimate moment and ultimate curvature of the element. For calculation of the simplified ultimate moment, it is assumed that the whole compressive force is concentrated in the top fibre of the section.In the case of fibre-reinforced panels, concrete tensile stresses are assumed to have a uniform value (fctR,d = 0.33fR,3,d [22], see section 3.3) and are applied only to the lower layer of the sandwich. Therefore, tensile stresses in the upper layer, and in the lateral beams, are neglected. Strain and stress diagrams representing the assumptions are shown in Figure 9.Figure 9
Strain (ε) and stress (σ) diagrams for the approximation of the ultimate moment.With the assumed simplifications, the curvature can be determined as(9)χ=εsh−c,where εs is the deformation of the reinforcement bar (which, in the ultimate state, is taken to be 10‰), h is the panel depth, and c is the reinforcement mechanical cover.On contrary, the resultant tensile force (T) is(10)T=fctR,d⋅b⋅h_w+fyd⋅Aswhere fctR,d is the residual resistance of the fibres, b is the panel width, h_w is the thickness of the concrete wythes, fyd is the design yield strength of the reinforcement steel, and As is the cross-sectional area of the reinforcement steel.The approximate ultimate moment of the cross section of the element can then be determined by multiplying the lever arm of the internal forcesz=h−c by the resultant tensile force (T):(11)MU=T.z=fctR,d⋅b⋅h_w+fyd⋅As.h−c.


### 3.3. Determination of Stress-Strain (σ−ε) Law for FRC by means of the Beam Test

Load-displacement curves for the FRC were experimentally obtained by a four-point bending test, according to UNE 83510 [24], carried out over three samples cast during the production of each of the panel types [17].From the average of these values, and applying the four-stage criteria indicated below, the multilinear stress-strain constitutive equation (Figure10), as indicated by EHE-08, was obtained. The average values of these parameters are detailed in Table 1 for the synthetic (P_FRCS6) and steel (P_FRCM20) fibres used in the panels.Figure 10
Multilinealσ−ε law obtained experimentally.
(a)
(b)Table 1
Parameters for FRC constitutive equation in tension.
Stage 1: Loads (kN), ASTM C1609
Stage 2: Strengths (MPa), ASTM C1609
Stage 3: Residual strengths (MPa), EN 14651
Stage 4: Constitutive equation FRC
P_FRCS6
F
1
31,65
f
1
4,22
f
ct,m
2,53 MPa
ɛ
0
0,075‰
F
0.75
18,18
f
600
2,42
f
R,1
2,07
f
ct,R1,m
0,93 MPa
ɛ
1
0,175‰
F
3.00
21,07
f
150
2,81
f
R,3
3,51
f
ct,R3,m
1,34 MPa
ɛ
2
12,50‰
f
ct,lim
1,59 MPa
ɛ
lim
20,00‰
P_FRCM20
F
1
33,12
f
1
4,42
f
ct,m
2,65 MPa
ɛ
0
0,077‰
F
0.75
19,94
f
600
2,66
f
R,1
2,27
f
ct,R1,m
1,02 MPa
ɛ
1
0,177‰
F
3.00
13,23
f
150
1,76
f
R,3
2,21
f
ct,R3,m
0,65 MPa
ɛ
2
12,50‰
f
ct,lim
0,42 MPa
ɛ
lim
20,00‰Stage 1. ASTM C1609 [25] and UNE 83510 [24] standards, both strongly based on the Japanese standard JSCE-SF4 [26], are considered to be equivalent. Therefore, from the load-displacement curve of the material obtained from the test, the following values, considered by the ASTM standard, were obtained:(a)
First-peak load (F1) and first-peak deflection(b)
Residual load (F0.75) corresponding to a net deflection of L/600, equivalent to 0.75 mm(c)
Residual load (F3.00) corresponding to a net deflection of L/150, equivalent to 3.00 mmStage 2. The first-peak strength (f1) and the residual strength corresponding to the deflections of 0.75 mm (f600) and 3 mm (f150) were calculated according to ASTM C1609, based on the following equation:(12)f=F·Lb·h2,where f is the strength in N/mm2, F is the load in N, L is the distance between supports in mm, b is the average width of the specimen in mm, and h the average edge of the specimen in mm.Stage 3. Values of fR,1 and fR,3, according to the three-point bending test EN 14651 [27], can be calculated from the ASTM C1609 [25] four-point bending test results, using the correlations indicated by Conforti et al. [28] Equation (13):(13)f600MPa=1.17·fR,1MPaR2=0.94,f150MPa=0.80·fR,3MPaR2=0.90.Stage 4. Tensile strength (fct) and its corresponding deflection (ε0), the tensile residual strengths (fct,R1 and fct,R3) associated with deflections ε1 and ε2, respectively, in the post-peak regime, and the value of εlim were determined by applying the equations indicated in point 39.5 of Annex 14 of the Spanish Instructions for Structural Concrete EHE-08 [22], which is based on EN 14651 test results.


### 3.4. Elastic Modulus

Table2 shows the values of the elastic modulus used, which were established as a function of the compressive strength, determined by the standard test UNIT-NM 101 [29], from the following formula indicated in the fib Model Code (2010) [21]:(14)Ect=Ec0·αE·fcm101/3.Table 2
Elastic modulus values used.
P_FRCS6
P_FRCM20
f
cm (MPa)
38,13
40,52
E
cm (GPa)
33,59
34,28It should be noted that, taking into account Tables 5.1–6 of the Model Code, for aggregates of quartzite origin, it is considered thatEc0·αE=21500MPa.


## 3.1. Structural Behaviour Model

The expected structural behaviour can be divided in three stages [20], which are represented in the load-displacement diagram shown in Figure 5. In the first stage (linear elastic), the concrete matrix is in an uncracked phase and, therefore, the element behaves in a linear elastic way. In the second stage (postcracking), the matrix starts cracking. This is a transition stage during which the tensile stresses are transferred from the matrix to the reinforcement. In the third stage (yielding), yielding takes place in at least one of the materials. As there was a low amount of reinforcement used in this study, yielding took place in the reinforcement whilst the concrete was still in the linear elastic stage. Depending on the type and amount of reinforcement, two types of behaviour can take place in the yielding stage [21]: hardening, in which the ultimate strength is greater than the cracking strength of the element, or softening, where the ultimate strength is below the cracking strength, as shown in Figure 5.Figure 5
Load-displacement curves for panels subjected to a bending test.Stage 1: Linear elastic. Transverse deformations of the panel in Stage 1 are shown schematically in Figure 6. The following basic hypothesis on the behaviour of the reinforced concrete was taken into account in the model. The concrete was in the precracked stage; that is, the tensile strength of concrete had not been reached, and therefore, it could develop both tensile and compressive stresses. Also, a linear elastic behaviour of the concrete could be assumed. Finally, the Navier–Bernoulli hypothesis was assumed, meaning that the planar sections would remain planar and perpendicular to the deformed axis. This hypothesis is based on the assumption that the width of the external frame was greater enough to act as a concrete web, efficiently transferring the shear forces.Figure 6
Deformation diagram on the linear elastic section.Under the mentioned hypothesis, a classic formulation of the mechanics of materials is valid. The cracking moment (Mcr) was determined using Equation (1). The reinforcement contribution was neglected for the three types of reinforcements, being negligible in comparison with the forces created by the concrete in tension:(1)Mcr=fct,m,flex.Ich/2,where fct,m,flex is the mean flexural tensile strength of concrete, Ic is the second moment of area (uncracked concrete section), and h is the cross-sectional depth.Also, in Stage 1, the relationship between moment (M) and curvature (χ) is linear up to the cracking moment, which is given by(2)χ=McrEc.Ic,where Ec is the reduced modulus of elasticity of concrete.Given the static configuration of the test, the moment value can be related to that of the applied load (F), according to Equation (3), where L is the span between the supports. Since it is a statically determined setup, this relationship is maintained in all stages of the test:(3)F=8·ML.Finally, the displacements can also be obtained using classic equations. The displacement of the midpoint (δ) is given by the following expression, with the meanings of the variables indicated above:(4)δ=11·F·L3768·Ec·Ic.Stage 2: Postcracking. The postcracking stage is a transitory phase between Stages 1 and 3, in which progressive cracking of the matrix takes place, transferring the tensile stresses from the concrete matrix to the reinforcement (bars or fibres). As the behaviour in this stage reports great complexity and, in turn, does not contribute significant information to the design, this section was not precisely determined. In a practical way, the structural behaviour in this stage was completed by a linear transition from the last point determined in Stage 1 with the first point in Stage 3.Stage 3: Yielding. The moment-curvature curve was determined by discretising the curvature curve (χi) at several points, at which the values of the corresponding moment (Mi) were determined. For each curvature value (χi) analysed, and for a given position of the neutral line (xi in Figure 7), it was possible to calculate the deformations in the entire section through the Navier–Bernoulli hypothesis. By means of the constitutive equation of each material, the tensions were obtained for every position within the section. A triangular stress distribution was considered for concrete under compression, and a multilineal constitutive equation (described below) was used for the FRC in tension. These assumptions are represented in the diagrams shown in Figure 7. The stresses provided by the EPS, as well as the tensile stresses of plain concrete, were considered to be null. Also, a punctual load (Fs) was applied where reinforcement bars were present.Figure 7
Diagrams representing strain (ε) and stress (σ).Thexi value that solves the problem was determined by equilibrium equations. As the beam was under pure bending, the axial force was equated to zero (N=0) by means of an iterative procedure. Finally, for the value of curvature (χi) and xi found, the corresponding moment was calculated. The moment-curvature curve was obtained by joining all pairs of values (Mi,χi) obtained from this analysis. It is important to take into account that it is not possible to adopt curvatures that involve a deformation of the steel in tension greater than 10‰ or of the FRC in tension greater than 20‰, as these values correspond to the ultimate state limits [22]. As indicated above, the relationship between applied load and moment in the central section was determined by Equation (3).A single crack was observed in the fibre-reinforced panels, where the deformation was concentrated. Therefore, to determine the displacements in the panels in the yielding stage, it was considered that each panel behaved as two rigid bodies that rotated on a hinge where the crack was formed, as illustrated in Figure8. To simplify the analysis, it was assumed that the crack opening (ω) was located in the central section of the panel.Figure 8
Displacement in the panel according to the rigid body model.Considering this scheme, it is possible to correlate the value of the deflection at the midpoint (δ) with the crack opening through trigonometric relationships (δ=L·ω/4·h). In turn, the crack opening is related to the strain at any point by the structural characteristic length, lcs, which, in this case, can be assumed to be equal to h [22]. Therefore, for the bottom fibre: ω=ε·lcs=ε·h. Finally, the strain in the bottom fibre of the cross section can be approximately calculated with the curvature (εinf=χ·h). Combining the previous expression, the following equation can be obtained:(5)δ=L·ω4·h=L·εinf·h4·h=L·χ·h·h4·h⇒δ=L·h·χ4.To estimate the displacement in the mesh-reinforced panels, two strategies were used, depending on whether the yield moment (My) of the central section was exceeded. If the maximum moment was below My, Branson’s empirical formula [23] was used to calculate the equivalent moment of inertia (Ie) of the element:(6)Ie=McrMa3·Ib+1−McrMa3·If≯Ib,where Mcr is the cracking moment, Ma is the acting bending moment, Ib is the second moment of area (uncracked concrete section), and If is the moment of inertia of the cracked section.If the maximum moment exceeds the yield moment, it is understood that large deformations take place in the central section. For this reason, the displacement of the central point was calculated by integrating the curvature of the section between loads (χmax, which was assumed to be constant in this section) and neglecting the rest of the deformations. The variation of the angle of rotation from the central section to the point of application of the loads can be computed as(7)θmax=∫0L/4χmaxdx=L4·χmax.And, hence, the displacement of the central point:(8)υmax=∫0L/2θxdx=∫0L/4xL/4θmaxdx+∫L/4L/2θmaxdx=L4·3L8·χmax.


## 3.2. Ultimate Moment Approximation

Besides the previously described procedure, it is possible to establish approximated formulas that allow the calculation, by hand, of the ultimate moment and ultimate curvature of the element. For calculation of the simplified ultimate moment, it is assumed that the whole compressive force is concentrated in the top fibre of the section.In the case of fibre-reinforced panels, concrete tensile stresses are assumed to have a uniform value (fctR,d = 0.33fR,3,d [22], see section 3.3) and are applied only to the lower layer of the sandwich. Therefore, tensile stresses in the upper layer, and in the lateral beams, are neglected. Strain and stress diagrams representing the assumptions are shown in Figure 9.Figure 9
Strain (ε) and stress (σ) diagrams for the approximation of the ultimate moment.With the assumed simplifications, the curvature can be determined as(9)χ=εsh−c,where εs is the deformation of the reinforcement bar (which, in the ultimate state, is taken to be 10‰), h is the panel depth, and c is the reinforcement mechanical cover.On contrary, the resultant tensile force (T) is(10)T=fctR,d⋅b⋅h_w+fyd⋅Aswhere fctR,d is the residual resistance of the fibres, b is the panel width, h_w is the thickness of the concrete wythes, fyd is the design yield strength of the reinforcement steel, and As is the cross-sectional area of the reinforcement steel.The approximate ultimate moment of the cross section of the element can then be determined by multiplying the lever arm of the internal forcesz=h−c by the resultant tensile force (T):(11)MU=T.z=fctR,d⋅b⋅h_w+fyd⋅As.h−c.


## 3.3. Determination of Stress-Strain (σ−ε) Law for FRC by means of the Beam Test

Load-displacement curves for the FRC were experimentally obtained by a four-point bending test, according to UNE 83510 [24], carried out over three samples cast during the production of each of the panel types [17].From the average of these values, and applying the four-stage criteria indicated below, the multilinear stress-strain constitutive equation (Figure10), as indicated by EHE-08, was obtained. The average values of these parameters are detailed in Table 1 for the synthetic (P_FRCS6) and steel (P_FRCM20) fibres used in the panels.Figure 10
Multilinealσ−ε law obtained experimentally.
(a)
(b)Table 1
Parameters for FRC constitutive equation in tension.
Stage 1: Loads (kN), ASTM C1609
Stage 2: Strengths (MPa), ASTM C1609
Stage 3: Residual strengths (MPa), EN 14651
Stage 4: Constitutive equation FRC
P_FRCS6
F
1
31,65
f
1
4,22
f
ct,m
2,53 MPa
ɛ
0
0,075‰
F
0.75
18,18
f
600
2,42
f
R,1
2,07
f
ct,R1,m
0,93 MPa
ɛ
1
0,175‰
F
3.00
21,07
f
150
2,81
f
R,3
3,51
f
ct,R3,m
1,34 MPa
ɛ
2
12,50‰
f
ct,lim
1,59 MPa
ɛ
lim
20,00‰
P_FRCM20
F
1
33,12
f
1
4,42
f
ct,m
2,65 MPa
ɛ
0
0,077‰
F
0.75
19,94
f
600
2,66
f
R,1
2,27
f
ct,R1,m
1,02 MPa
ɛ
1
0,177‰
F
3.00
13,23
f
150
1,76
f
R,3
2,21
f
ct,R3,m
0,65 MPa
ɛ
2
12,50‰
f
ct,lim
0,42 MPa
ɛ
lim
20,00‰Stage 1. ASTM C1609 [25] and UNE 83510 [24] standards, both strongly based on the Japanese standard JSCE-SF4 [26], are considered to be equivalent. Therefore, from the load-displacement curve of the material obtained from the test, the following values, considered by the ASTM standard, were obtained:(a)
First-peak load (F1) and first-peak deflection(b)
Residual load (F0.75) corresponding to a net deflection of L/600, equivalent to 0.75 mm(c)
Residual load (F3.00) corresponding to a net deflection of L/150, equivalent to 3.00 mmStage 2. The first-peak strength (f1) and the residual strength corresponding to the deflections of 0.75 mm (f600) and 3 mm (f150) were calculated according to ASTM C1609, based on the following equation:(12)f=F·Lb·h2,where f is the strength in N/mm2, F is the load in N, L is the distance between supports in mm, b is the average width of the specimen in mm, and h the average edge of the specimen in mm.Stage 3. Values of fR,1 and fR,3, according to the three-point bending test EN 14651 [27], can be calculated from the ASTM C1609 [25] four-point bending test results, using the correlations indicated by Conforti et al. [28] Equation (13):(13)f600MPa=1.17·fR,1MPaR2=0.94,f150MPa=0.80·fR,3MPaR2=0.90.Stage 4. Tensile strength (fct) and its corresponding deflection (ε0), the tensile residual strengths (fct,R1 and fct,R3) associated with deflections ε1 and ε2, respectively, in the post-peak regime, and the value of εlim were determined by applying the equations indicated in point 39.5 of Annex 14 of the Spanish Instructions for Structural Concrete EHE-08 [22], which is based on EN 14651 test results.


## 3.4. Elastic Modulus

Table2 shows the values of the elastic modulus used, which were established as a function of the compressive strength, determined by the standard test UNIT-NM 101 [29], from the following formula indicated in the fib Model Code (2010) [21]:(14)Ect=Ec0·αE·fcm101/3.Table 2
Elastic modulus values used.
P_FRCS6
P_FRCM20
f
cm (MPa)
38,13
40,52
E
cm (GPa)
33,59
34,28It should be noted that, taking into account Tables 5.1–6 of the Model Code, for aggregates of quartzite origin, it is considered thatEc0·αE=21500MPa.


## 4. Result Analysis

### 4.1. Qualitative Behaviour

The numerical results for the three analysed panels are shown in Figure11, including the moment-curvature (Figure 11(a)) and load-displacement (Figure 11(b)) curves. In both figures, the solid points represent the complete model, and open points are the simplified model. It can be seen that, in the complete model, the three stages expected (linear elastic, postcracking, and yielding) are represented. Also, due to the statically determinate setup, the structural behaviour reflects the sectional behaviour to a large extent.Figure 11
Structural results for the three types of reinforcement analysed: (a) moment-curvature and (b) load-displacement curves.
(a)
(b)The simplified calculation of the ultimate moment matches, with great accuracy, the complete analysis for the bar-reinforced panels. For the FRC panels, a conservative value was obtained (around 17% smaller), due to the simplifications described above.A usually accepted design criterion requires, to avoid brittle failure under bending, that the ultimate bending capacity (MU) be larger than the flexural cracking moment (Mcr) [30, 31]. It can be seen that this criterion is approximately met for the three types of reinforcement, with a small drop for large curvatures in the case of steel fibres.


### 4.2. Influence of the Amount of Reinforcement and Fibres

For the three types of reinforcement analysed, the influence of the amount of reinforcement in the structural response was analysed. The load-displacement curves are shown in Figure12 for the three types and different amounts of reinforcement (20, 40, and 60 kg/m3 for steel fibres (Figure 12(a)); 4, 6, and 8 kg/m3 for synthetic fibres (Figure 12(b)); and ϕ4.2/15, ϕ6/15, and ϕ8/15 for mesh reinforcement (Figure 12(c))).Figure 12
Influence of the amount of reinforcement in (a) steel fibres, (b) synthetic fibres, and (c) steel mesh.
(a)
(b)
(c)It can be seen that the initial linear stage is the same in all cases because it is mainly controlled by the concrete matrix. On the contrary, a clear difference in behaviour can be seen among the different types of reinforcement after cracking takes place, and the tensile loads are transferred to the reinforcement. Also, for each type, the ultimate moment is higher when the amount of reinforcement increases.Finally, it can be seen that, for the amounts of reinforcement analysed, which are in the usual ranges for conventional FRC, the ultimate loads obtained are in a range of between 80 and 180 kN. A similar range was observed for the rebar-reinforced panels; however, for this type of reinforcement, the amount of reinforcement could be considered to be in the low range, regarding the bending moments. It can be said that, regarding the flexural resistance, fibres are capable of substituting for rebar reinforcement at low to medium structural capacity.


### 4.3. Influence of Wythe Thickness

As there was no rebar reinforcement in the concrete wythes of the fibre-reinforced panels, a minimum reinforcement cover was not needed, and therefore, it was possible to reduce the wythe thickness to optimise the panel geometry. Total cross-sectional thickness, which usually meets insulation requirements and architectural aspects, was kept constant (h = 200 mm) for the analysis. Wythe thickness (h_w) was chosen as the study variable. Three thicknesses (h_w = 50, 40, and 30 mm) were analysed, which corresponded to EPS layer thicknesses of 100, 120, and 140 mm, respectively.The load-displacement results for the wythe thickness analysis are shown in Figure13. As expected, for both types of fibre, the bending moments in the postcracking stage increased with wythe thickness; however, the variation was smaller than that observed for the different amounts of reinforcement studied. Therefore, it seems plausible that the resistance drop produced by a wythe thickness reduction could be restored by an increase in the amount of fibres. This would allow a reduction in the use of concrete, whilst obtaining lighter panels with the same structural capacity.Figure 13
Influence of concrete layer thickness on (a) steel fibres and (b) synthetic fibres.
(a)
(b)


### 4.4. Panel Optimisation

Based on the model developed, which takes into account the structural collaboration of the fibres, a parametric study was carried out in order to optimise the geometry and reinforcement of the panels at a sectional level. The aim of the analysis was to obtain the most economic combination of geometry and reinforcement for six different objective design moments (5, 10, 15, 20, 30, and 50 kNm). For the calculation of each panel cost, the unitary costs shown in Table3, in US dollars (USD), were considered.Table 3
Unitary costs of materials and labour.
Steel fibres
Synthetic fibres
Steel mesh
Steel bars
Poured concrete
Foreman
Worker
(USD/kg)
(USD/kg)
(USD/m2)
(USD/kg)
(USD/m3)
(USD/hour)
(USD/hour)
3.3
11.2
5.7
1.2
210.5
10.5
6.0Regarding the panel geometry, the wythe thickness was chosen as the study variable, with the three options described in the previous section (h_w = 50, 40, and 30 mm), maintaining constant total thickness (0.2 m), as well as the panel exterior width (1.2 m), the EPS width (0.9 m), and the external concrete web width (0.15 m).The reinforcement design strategy consisted of, first of all, setting the minimum reinforcement to be placed in the wythes to control shrinkage and thermal effects. Then, the concrete web reinforcement was determined to cover, together with the wythes, the objective ultimate bending moment. Reinforcing steel bars, with the diameters of 8, 10, 12, 16, and 20 mm, were considered.A comparative analysis of the optimisation results is shown in Figure14. The mesh-reinforced panel for the 15 kNm objective moment was chosen as the reference panel. The figure shows the total cost of the optimised panel in relation to the reference panel for each objective moment, each wythe thickness and type of reinforcement. The lower objective moments (5, 10, and 15 kNm) fell below the flexural cracking moment (Mcr), and therefore, the brittle failure criterion (MU ≥ Mcr) controls the design.Figure 14
Variation in costs of the optimised panels.It can be seen that, for each type of reinforcement, the larger the objective moment, the greater the cost. This increase is mainly due to the amount of reinforcement needed to withstand the acting moment. For a specific value of objective moment and wythe thickness, there is a small reduction in the panel cost when fibres are used as reinforcement (up to 10%, in the case of synthetic fibres).On the contrary, it can be seen that, for a specific objective moment, costs are smaller in the panels with reduced wythe thickness, with reductions greater than 20% for both synthetic and steel fibres. This solution is viable only in the fibre-reinforced panels, where a minimum steel bar cover is not needed. Wythe thickness then becomes conditional mainly upon the concrete execution capacity, which should remain within the specified tolerance range and with acceptable surface finishing.


## 4.1. Qualitative Behaviour

The numerical results for the three analysed panels are shown in Figure11, including the moment-curvature (Figure 11(a)) and load-displacement (Figure 11(b)) curves. In both figures, the solid points represent the complete model, and open points are the simplified model. It can be seen that, in the complete model, the three stages expected (linear elastic, postcracking, and yielding) are represented. Also, due to the statically determinate setup, the structural behaviour reflects the sectional behaviour to a large extent.Figure 11
Structural results for the three types of reinforcement analysed: (a) moment-curvature and (b) load-displacement curves.
(a)
(b)The simplified calculation of the ultimate moment matches, with great accuracy, the complete analysis for the bar-reinforced panels. For the FRC panels, a conservative value was obtained (around 17% smaller), due to the simplifications described above.A usually accepted design criterion requires, to avoid brittle failure under bending, that the ultimate bending capacity (MU) be larger than the flexural cracking moment (Mcr) [30, 31]. It can be seen that this criterion is approximately met for the three types of reinforcement, with a small drop for large curvatures in the case of steel fibres.


## 4.2. Influence of the Amount of Reinforcement and Fibres

For the three types of reinforcement analysed, the influence of the amount of reinforcement in the structural response was analysed. The load-displacement curves are shown in Figure12 for the three types and different amounts of reinforcement (20, 40, and 60 kg/m3 for steel fibres (Figure 12(a)); 4, 6, and 8 kg/m3 for synthetic fibres (Figure 12(b)); and ϕ4.2/15, ϕ6/15, and ϕ8/15 for mesh reinforcement (Figure 12(c))).Figure 12
Influence of the amount of reinforcement in (a) steel fibres, (b) synthetic fibres, and (c) steel mesh.
(a)
(b)
(c)It can be seen that the initial linear stage is the same in all cases because it is mainly controlled by the concrete matrix. On the contrary, a clear difference in behaviour can be seen among the different types of reinforcement after cracking takes place, and the tensile loads are transferred to the reinforcement. Also, for each type, the ultimate moment is higher when the amount of reinforcement increases.Finally, it can be seen that, for the amounts of reinforcement analysed, which are in the usual ranges for conventional FRC, the ultimate loads obtained are in a range of between 80 and 180 kN. A similar range was observed for the rebar-reinforced panels; however, for this type of reinforcement, the amount of reinforcement could be considered to be in the low range, regarding the bending moments. It can be said that, regarding the flexural resistance, fibres are capable of substituting for rebar reinforcement at low to medium structural capacity.


## 4.3. Influence of Wythe Thickness

As there was no rebar reinforcement in the concrete wythes of the fibre-reinforced panels, a minimum reinforcement cover was not needed, and therefore, it was possible to reduce the wythe thickness to optimise the panel geometry. Total cross-sectional thickness, which usually meets insulation requirements and architectural aspects, was kept constant (h = 200 mm) for the analysis. Wythe thickness (h_w) was chosen as the study variable. Three thicknesses (h_w = 50, 40, and 30 mm) were analysed, which corresponded to EPS layer thicknesses of 100, 120, and 140 mm, respectively.The load-displacement results for the wythe thickness analysis are shown in Figure13. As expected, for both types of fibre, the bending moments in the postcracking stage increased with wythe thickness; however, the variation was smaller than that observed for the different amounts of reinforcement studied. Therefore, it seems plausible that the resistance drop produced by a wythe thickness reduction could be restored by an increase in the amount of fibres. This would allow a reduction in the use of concrete, whilst obtaining lighter panels with the same structural capacity.Figure 13
Influence of concrete layer thickness on (a) steel fibres and (b) synthetic fibres.
(a)
(b)


## 4.4. Panel Optimisation

Based on the model developed, which takes into account the structural collaboration of the fibres, a parametric study was carried out in order to optimise the geometry and reinforcement of the panels at a sectional level. The aim of the analysis was to obtain the most economic combination of geometry and reinforcement for six different objective design moments (5, 10, 15, 20, 30, and 50 kNm). For the calculation of each panel cost, the unitary costs shown in Table3, in US dollars (USD), were considered.Table 3
Unitary costs of materials and labour.
Steel fibres
Synthetic fibres
Steel mesh
Steel bars
Poured concrete
Foreman
Worker
(USD/kg)
(USD/kg)
(USD/m2)
(USD/kg)
(USD/m3)
(USD/hour)
(USD/hour)
3.3
11.2
5.7
1.2
210.5
10.5
6.0Regarding the panel geometry, the wythe thickness was chosen as the study variable, with the three options described in the previous section (h_w = 50, 40, and 30 mm), maintaining constant total thickness (0.2 m), as well as the panel exterior width (1.2 m), the EPS width (0.9 m), and the external concrete web width (0.15 m).The reinforcement design strategy consisted of, first of all, setting the minimum reinforcement to be placed in the wythes to control shrinkage and thermal effects. Then, the concrete web reinforcement was determined to cover, together with the wythes, the objective ultimate bending moment. Reinforcing steel bars, with the diameters of 8, 10, 12, 16, and 20 mm, were considered.A comparative analysis of the optimisation results is shown in Figure14. The mesh-reinforced panel for the 15 kNm objective moment was chosen as the reference panel. The figure shows the total cost of the optimised panel in relation to the reference panel for each objective moment, each wythe thickness and type of reinforcement. The lower objective moments (5, 10, and 15 kNm) fell below the flexural cracking moment (Mcr), and therefore, the brittle failure criterion (MU ≥ Mcr) controls the design.Figure 14
Variation in costs of the optimised panels.It can be seen that, for each type of reinforcement, the larger the objective moment, the greater the cost. This increase is mainly due to the amount of reinforcement needed to withstand the acting moment. For a specific value of objective moment and wythe thickness, there is a small reduction in the panel cost when fibres are used as reinforcement (up to 10%, in the case of synthetic fibres).On the contrary, it can be seen that, for a specific objective moment, costs are smaller in the panels with reduced wythe thickness, with reductions greater than 20% for both synthetic and steel fibres. This solution is viable only in the fibre-reinforced panels, where a minimum steel bar cover is not needed. Wythe thickness then becomes conditional mainly upon the concrete execution capacity, which should remain within the specified tolerance range and with acceptable surface finishing.


## 5. Conclusions

A theoretical model, mainly based on the Spanish Concrete Instruction Guidelines (EHE-08, Annex 14) and capable of evaluating the structural behaviour of FRC panels, was integrated. The results showed the expected behaviour of the panels under study. The model allows consideration of the contribution of fibres to withstand internal tensile forces of the panels and, therefore, be substituted for the steel mesh in the panel wythes. Likewise, as the FRC tensile mechanical properties were defined based on normalised tests, it was possible to establish a quality-control program to evaluate panel production.Finally, it was possible to optimise panel geometry and reinforcement, reducing wythe thickness and adjusting the different types of reinforcement to comply with the design conditions. It was found that it was possible to achieve a cost reduction of up to 10% if the reinforcement was modified and of more than 20% when the geometry was also modified. This constitutes a significant saving that is worthy of further exploration. In addition, besides the economic advantage, a reduction in production time was also achieved when FRC was used.



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*Source: 3235012-2018-11-22.xml*