Numerical evaluation of structural stay-in-place formwork in textile reinforced cement composite for concrete shells

Abstract

This article presents the structural evaluation of textile reinforced cement composite elements as formwork and reinforcement for concrete shells. Different shell geometries of 10 m span and a height between 2.5 and 5 m are examined. On one hand, the minimum thickness of the composite element functioning as formwork for these shells is determined when supported with a different number of supports and for different concrete sections. On the other hand, the minimum thickness of the composite functioning as reinforcement is calculated to withstand the occurring maximum bending moment. For shells with a 10 to 15 mm concrete section, a minimum composite formwork of 5–6 mm is needed to carry the load of the cast concrete, while the minimum composite reinforcement needed to withstand the maximum occurring bending moment equals 2–3 mm. Conclusively, this preliminary numerical study demonstrates the structural potential of textile reinforced cement composite stay-in-place formwork for concrete shells and indicates the dominance of the casting stage over the final stage. More specifically, local buckling of the formwork is the determining factor and should drive future work towards this issue.

Keywords

concrete external reinforcement finite element modelling flexible formwork shells stay-in-place formwork textile reinforced cement composite

Introduction

There is an increasing demand from the building society for free-form shell structures with optimal structural behaviour and material use; however, their construction remains a challenge in high-wage economies. The current formwork methods widely used in practice, that is, timber moulds (Weilandt, 2009) and foam blocks (Nedcam, n.d.), are labour intensive and/or material wasting. Therefore, a lot of research is being performed on alternative formwork for curved surfaces: flexible fabrics (Cauberg et al., 2012; Pronk et al., 2007; West, n.d.), pneumatic formwork (Van Hennik and Wagemans, 2004) or hybrid cable-net and fabric formwork (Veenendaal and Block, 2014). The flexibility of these systems allows realizing curved surfaces easily; however, the tested prototypes experienced relatively large deformations under the load of the cast concrete, for example, the fabric formwork of 1.8 m span in Cauberg et al. (2012) deformed 18 mm (L/100) under 35 mm of cast concrete. Appropriate pretension is required to obtain stiff structures which can withstand the (heavy) weight of cast concrete.

A second issue obstructing the construction of concrete shells is the tensile reinforcement; shaping and placing of the rigid traditional steel reinforcement is labour intensive, practically limits the curvature and hence increases the cost. Recently, alternative fibre and textile reinforcement methods are investigated in concrete shells, like demonstrated in Ortlepp et al. (2009), Tysmans et al. (2011), Ehlig et al. (2012) and Scholzen et al. (2015a, 2015b). However, these new reinforcement methods do not solve the formwork problems discussed before.

Anticipating on these issues, we developed an inventive stay-in-place formwork solution, which exploits the properties of a textile reinforced cement composite developed at our university (Wu and Gu, 1997). This composite consists of the matrix inorganic phosphate cement (IPC) in combination with random E-glass fibre mats (Figure 1(a)), further referred to as glass fibre texile reinforced inorganic phosphate cement (GFTR-IPC). As long as the cement is still wet, the used fibre mats are flexible (more than two-dimensional (2D) textiles) as shown in Figure 1(b) and can be shaped onto any (reusable) mould such as foam moulds, flexible formwork moulds, pneumatic formwork (Figure 1(c) and (d)) or prestressed membranes. Very complex curved shapes can thus easily be obtained. The fibre mats are impregnated layer by layer (approximately 0.5 mm/layer) with the IPC matrix until the required thickness is obtained. The low weight of the composite layer compared to the concrete, which is normally directly cast onto these flexible moulds, allows to limit the deformations of these flexible moulds.

Figure 1.

(a) Random glass fibre mats (b) impregnated with IPC are initially flexible and (c) and (d) can be formed on for example (reusable) pneumatic formwork.

After hardening of the cement, the impregnated textile becomes a thin cement composite formwork with sufficient rigidity to cast concrete on it. After hardening of the concrete, the stay-in-place formwork gets an additional structural function and acts as a (partial) tensile reinforcement of the final concrete structure. The structural capacity of stay-in-place formwork in textile reinforced concrete (TRC) was already demonstrated for beam elements in Brameshuber et al. (2004), Papantoniou and Papanicolaou (2008), Papanicolaou and Papantoniou (2010) and at our university for the used composite GFTR-IPC in Verbruggen et al. (2013) and De Sutter et al. (2014). Both materials show similar results as stay-in-place formwork because the material properties are also very similar.

This article focuses on the structural analysis and design of GFTR-IPC elements functioning not only as formwork but also as reinforcement for concrete shells. A methodology is presented to design the formwork and the reinforcement. Varying parameters in the different case studies are the shape of the shells, their thickness and the amount of vertical supports during casting. In Scholzen et al. (2015b), a design method was already proposed to design TRC shells; however, our methodology differs on several levels. First, this article does not present the design of a TRC shell, but the design of GFTR-IPC as formwork and the design of GFTR-IPC as (external) tensile reinforcement for concrete shells. Second, GFTR-IPC uses random glass fibre mats instead of 2D textiles, resulting in the same material properties in all directions and hence strongly affecting the design methodology.

The article is subdivided into two parts. In the first part, five different double curved shells with a span of 10 m and a height between 2.5 and 5 m are studied. Assuming a fixed concrete shell thickness of 50 mm, first the minimum thickness of the GFTR-IPC formwork needed to carry the load of the cast concrete is determined. Second, the minimum thickness for the GFTR-IPC reinforcement needed to withstand the maximum occurring bending moment in the shells section is determined. In the second part, we focus on the two form found shell designs and add the concrete thickness as a varying parameter next to the GFTR-IPC formwork/reinforcement thickness. The minimum thickness of the GFTR-IPC in formwork stage and reinforcement stage are determined separately to see which stage is dominant. Based on these results, conclusions are drawn on the structural feasibility of the new stay-in-place formwork concept for concrete shells, and specific attention points in the design are discussed.

This article is a continuation of the work and ideas presented in Verwimp et al. (2014), yet the results presented in this article are obtained by a more conservative approach regarding buckling and a proposal for the concrete parabola–rectangle diagram is given. The strength of this article compared to that of Verwimp et al. (2014) is the different case studies which are examined, which allows to investigate the influence of shell shape and thickness.

Shell geometries, materials and methods

First, this section describes the examined shell geometries. Second, the used materials and their mechanical properties are presented. Next, the considered load cases and combinations which are taken into account in the design are described. Finally, the proposed methodologies for determining the GFTR-IPC thickness as formwork on one hand and as reinforcement on the other hand are explained.

Geometries and boundary conditions

To have an initial idea of the behaviour of thin GFTR-IPC shells functioning as formwork and as reinforcement for concrete shells, the first part of this article examines five different shells with double curvature (Figure 2). The thickness of the concrete section of these shells is predetermined, that is, 50 mm. First, two spherical domes with a span of 10 m and a height of, respectively, 5 and 2.5 m (Figure 2(a)) are studied. In the formwork stage (meaning the GFTR-IPC is hardened and functions as formwork) the circular base is hinged (Figure 2(b), green line) and the rest of the GFTR-IPC formwork is simply supported by vertical supports during casting (Figure 2(b), red dots). These vertical supports have a cross section of around 200 × 200 mm and are modelled as roller bearings. Their position subdivides the formwork in smaller pieces, preferably with the same geometry, whose dimensions comply with the transportation regulations (Goldstein and Labeeuw, 2001). In the final stage, the concrete domes are only hinged at their base.

Figure 2.

Both synclastic and anticlastic shells are examined.

Hereafter, two shells with an anticlastic curvature, saddle shapes, which cover a 10 m × 10 m area, are examined. The first saddle shape is generated from a circular arch of 5 m high at the edges and a circular arch of 2 m deep in the perpendicular direction (Figure 2(c)). The second saddle shape has similar dimensions (Figure 2(e)), but was generated by form finding with the dynamic relaxation method (Barnes, 1999) in Tysmans et al. (2008) to obtain a saddle shape which is in compression-only under its self-weight. During casting, the edges at ground level of the formwork are hinged (Figure 2(d) and (f), green line) and the rest of the formwork is simply supported by vertical supports (Figure 2(d) and (f), red dots). In the final stage, the edges at ground level are hinged.

Finally, a funicular synclastic shell is studied. This geometry is generated using the force density (Schek, 1974) form finding software Easy. For this study, a funicular (compression-only) shape under its self-weight was generated. The shell (Figure 2(g)) covers an area of 10 m × 10 m, is about 4.8 m high and its corners measure 0.8 m to avoid stress concentrations. During the formwork stage, the formwork is hinged in its four corners (Figure 2(h), green line) and simply supported by vertical supports (Figure 2(h), red dots). In the final stage, the concrete shell is hinged in its four corners.

In the second part of the article, only two of the latter case studies are considered, namely, the form found saddle shape and the form found synclastic shell (Figure 2(e) and (g)). For these case studies, both the concrete section and the GFTR-IPC section are optimized.

Material properties

As already mentioned, the composite GFTR-IPC is used in this study. The textile is actually a random E-glass fibre mat with a density of 300 g/m². The mechanical behaviour of GFTR-IPC is very different in tension and compression (Figure 3(a)) because of the brittle IPC matrix. In compression, GFTR-IPC is assumed to be linear until failure (Cuypers, 2001) and in tension a non-linear behaviour is already reached at low tensile stresses because of the low tensile failure strain of IPC compared to that of the fibres. However, by adding high fibre volume fractions, the composite can obtain a significant post-cracking stiffness and tensile strength (Figure 3(b); Remy, 2011). On a macro-scale, this strain-hardening behaviour is thus analogue to that of traditional TRC, and we can therefore refer to GFTR-IPC as a kind of TRC.

Figure 3.

Tensile test results on plate specimens of 300 mm × 25 mm consisting of five layers of E-glass fibre mats: (a) GFTR-IPC is linear in compression and non-linear in tension and (b) by increasing the fibre volume fraction of the glass fibre mats, the post-cracking stiffness and tensile strength increase (Remy, 2011).

For structural applications, the durability is an important issue in the design. The IPC matrix is neutral, meaning that a chemical attack of the fibres is eliminated and a better durability is obtained compared to other TRCs (Cuypers et al., 2006). Moreover, Cuypers et al. (2006) examined the effect of accelerated ageing, the composite retained 90% of its initial strength after testing. Remy (2011) tested the fatigue failure of GFTR-IPC and concluded that one million cycles or more can be reached when loading the composite up to 50% of its tensile failure load. These properties prove GFTR-IPC is a durable composite. In correspondence with Tysmans et al. (2011), a safety factor of 2 is used on the GFTR-IPC properties, while a safety factor of 1.5 is applied for concrete (European Committee for Standardization, 2004).

It is the aim of this article to explore the structural feasibility of GFTR-IPC as formwork and reinforcement for various cases rather than an in-depth analysis of the material. To this end, the material behaviour of the GFTR-IPC and concrete is assumed to be linear elastic. Considering these assumptions of Hookes law, the design characteristics of the GFTR-IPC (Symbion, n.d.) and the used concrete, that is, normal concrete C25/30 (European Committee for Standardization, 2004), are presented in Table 1.

Table 1.

Properties of glass fibre reinforced inorganic phosphate cement and concrete C25/30.

Quantity	GFTR-IPC		Concrete
Density	1750	kg/m²	2400	kg/m²
Compressive characteristic strength	80	MPa	25	MPa
Compressive design strength	40	MPa	14.2	MPa
Compressive design strain	2200	µm/m	–
Tensile characteristic strength	40	MPa	0	MPa
Tensile design strength	20	MPa	0	MPa
Tensile design strain	1100	µm/m	–
Young’s modulus	18	GPa	31.5	GPa
Fibre volume fraction	18	%	–
Poisson ratio	0.3		0.2

Finite element modelling

The shells are structurally analysed in the finite element software Abaqus version 6.12. Both for the analysis of the formwork and the reinforcement, the geometries are modelled as a shell model with a continuous shell section. Two types of quadratic elements are used, namely, the S8R and STRI65 elements, which stand for 8-node shell elements with reduced integration and a 6-node triangular shell elements, respectively. Mostly, the S8R elements are used, but if locally the geometry of the shell does not allow them, STRI65 elements are used instead. Three integration points over one material layer are used through Simpson’s integration. Mesh convergence was checked for the studied shells and an element size of 0.2 m is used.

Load cases and combinations

Eurocode 1 (European Committee for Standardization, 2002) describes the serviceability limit state (SLS) and ultimate limit state (ULS) which need to be fulfilled. For the design of the concrete shells (final stage) in SLS, Eurocode 2 (European Committee for Standardization, 2004) allows a maximum deflection of span/300. For the formwork design in SLS, the strictest criteria of span/300 and upper limit of 20 mm are considered (Remy, 2011). For the design in ULS, the shells, both in formwork and final stage, must not exceed the maximum tensile and compressive material design strength criteria (Table 1) and stability of the structure must be ensured (no buckling).

To design the GFTR-IPC formwork to cast the concrete shells on, the ULS load combination 1.35 self-weight GFTR-IPC + 1.50 weight of cast concrete and the SLS load combination 1.00 self-weight GFTR-IPC + 1.00 weight of cast concrete are used.

For the design of GFTR-IPC as tensile reinforcement for the concrete shells, all load cases, which must be taken into account according to Eurocode 1 (European Committee for Standardization, 2002), are considered, namely self-weight, (asymmetric) wind, (symmetric and asymmetric) snow and service loads. The shells represent a roof of a closed building whose facades act as separate structures and therefore transfer or exert no forces on the shells. The dominant ULS and SLS load combinations in the analyses for the reinforcement stage are mentioned in the results as they differ for each concrete shell.

Methodology to design GFTR-IPC formwork

This section describes the methodology to design the GFTR-IPC formwork onto which the concrete shell is cast. The minimum thickness of the GFTR-IPC formwork is determined by the requirements of strain (and thus stress), deformations and buckling under the load combinations described in section ‘Load cases and combinations’. Both the strains, deformations and buckling load are calculated in Abaqus. This model was already validated by an experimental analysis on a 2 m span GFTR-IPC shell in Verwimp et al. (2015).

For the verification of buckling, the approximated method proposed by Medwadowski (2004) is used. First, the linear critical buckling load of the shell is calculated, using the eigenvalue buckling prediction in Abaqus (Abaqus 6.12-1 Documentation, 2012), disregarding any imperfections and assuming a linear elastic material. Following Medwadowski (2004), this value must be modified to take into account imperfections, creep, plasticity, cracking and the role of the reinforcement. Imperfections can reduce significantly the buckling resistance of shells (Reitinger and Ramm, 1995) leading to enormous reduction factors which are applied in shell design. Moreover, higher safety factors are applied to shells sensitive to imperfection. The safety factor and imperfection sensitivity factor are the most critical factors in the design and therefore taken into account in this study. The most conservative safety factor γ according to Medwadowski (2004) equals 3.5 and is used in this study. The imperfection sensitivity factor α_imp is obtained from the graphs generated in Tomás and Tovar (2012) and this for a geometrical imperfection size equals the formwork thickness. The imperfection sensitivity factor α_imp depends on the geometry of the considered shell; therefore, its value is given in the results for every geometry. The critical buckling load is represented in the results by a buckling factor λ_cr, which is obtained by multiplying the eigenvalue λ calculated in Abaqus by α_imp and dividing it by γ. The buckling factor λ_cr must be higher than 1 to assure the formwork does not buckle under the concrete load.

Methodology to design GFTR-IPC reinforcement for concrete shells

To determine the GFTR-IPC thickness, needed to resist the tensile stresses in the concrete shell, we developed an iterative method based on the bending moments acting in the shell’s section. Under all acting loads, tensile stresses are present at the top and bottom surface of the (thin) concrete shell. Therefore, reinforcement is needed on both sides of the shell. In this study, the reinforcement needed at the top is realized by placing a GFTR-IPC layer both at the bottom and top surface of the section, that is, creating a symmetrical section. The steps of the iterative method are described below:

For a shell with a predefined concrete section, the maximum occurring bending moment M_D (Nm/m) is determined in Abaqus under every load combination;

The equilibrium equations for a (1 m wide) cross section are solved in a MATLAB routine and give a first value of the constant thickness t (mm/m) of the GFTR-IPC needed to provide sufficient tensile capacity to withstand M_D;

M_D is recalculated in Abaqus but now for the hybrid GFTR-IPC-concrete section, using the previously determined GFTR-IPC thickness t;

The equilibrium equations for this new GFTR-IPC-concrete section are solved in the MATLAB routine and give the new thickness t of the GFTR-IPC;

Steps 3 and 4 are repeated until convergence is reached, the GFTR-IPC thickness t is rounded up to the minimum thickness of one GFTR-IPC layer, that is, 0.5 mm;

The deformations and linear buckling load of the shell structure are checked under all load combinations using the finite element (FE) model built in Abaqus. The applied reduction factor for buckling is as described in Tomás and Tovar (2012). Stresses are calculated using the equilibrium equations.

Solving the equilibrium equations of the section (in step 2 and 4), some hypotheses are made (European Committee for Standardization, 2004). First, the hypothesis of Bernoulli (plane cross sections remain plane) is used and the principal directions in the shell are considered (the GFTR-IPC has isotropic behaviour due to its random fibre mats). Second, the bond between concrete and GFTR-IPC is assumed to be perfect, so the strains of both materials are equal. Even though, this is not entirely the case in reality, tests on beams with this combined section (De Sutter et al., 2015; Verbruggen et al., 2010) have shown this is a valid approximation in the early design stage. Third, the compressive stresses of concrete are determined from a proposed parabola–rectangle stress–strain diagram, assuming the maximum compressive strain cannot exceed the ultimate strain of GFTR-IPC, 2200 µm/m (see Table 1). This results in a different value for the filling coefficient ψ and the gravity coefficient δ_G (Figure 4(e)) than prescribed in Eurocode 2 (European Committee for Standardization, 2004). Finally, the tensile strength of concrete is neglected.

Figure 4.

(a) Schematic representation of a GFTR-IPC reinforced concrete section, (b) strain diagram with GFTR-IPC in tension in ultimate limit strain $\in_{IPC},_{C}$ , (c) force equilibrium in the cross section, (d) bending moment working on the cross section and (e) proposed parabola–rectangle diagram of concrete for a maximum compressive strain of 2200 µm/m.

The equilibrium equations are based on Figure 4(a) to (d). A strain diagram is proposed, where the tensile GFTR-IPC layer is assumed to be in ultimate state, that is, 1100 µm/m, because this is the most stringent limit tensile strain of all materials. The design of the reinforcement is thus determined by the tensile strength of GFTR-IPC. The equilibrium equations of the cross section as a function of the unknown parameters, that is, the GFTR-IPC thickness and the position of the neutral axis, are derived in the next paragraphs.

The translational equilibrium in the hybrid section is written as

F_{C} + F_{IPC, C} = F_{IPC, T}

(1)

in which

F_{C} = ψ \cdot (σ_{c} \cdot x)

(2)

where F_C is the compressive force in the concrete section, F_IPC,_C is the compressive force in the GFTR-IPC section, F_IPC,_T is the tensile force in the GFTR-IPC section, σ_c is the concrete compressive stress, x is the position neutral axis and ψ is the filling coefficient of the rectangle in the diagram σ_c·x.

Substituting equation (2) in equation (1), expressing forces as a function of strains and areas, and assuming a linear relation between stress and strain gives

ψ \cdot (\in_{c} \cdot E_{cm} \cdot x) + \in_{IPC},_{C} \cdot E_{IPC} \cdot t = f_{IPC},_{T} \cdot t

(3)

where $\in_{c}$ is the strain in the concrete section, E_cm is the mean secant modulus of concrete, $\in_{IPC},_{C}$ is the strain in the compressive GFTR-IPC section, E_IPC is Young’s modulus of GFTR-IPC and f_IPC is the tensile design strength of GFTR-IPC.

Expressing the strains in equation (3) as a function of the position of the neutral axis and the tensile design strength of GFTR-IPC (1100 µm/m see Table 1) gives

\frac{\in_{c}}{x} = \frac{\in_{IPC, T, U}}{h - x + t} \Rightarrow \in_{c} = \frac{\in_{IPC, T, U} \cdot x}{h - x + t}

(4)

\begin{matrix} \frac{\in_{IPC, C}}{x + t} = \frac{\in_{IPC, T, U}}{h - x + t} \\ \Rightarrow \in_{IPC, C} = \frac{\in_{IPC, T, U} \cdot (x + t)}{h - x + t} \end{matrix}

(5)

where h is the height of the concrete section.

Substituting equations (4) and (5) in equation (3) gives the translational equilibrium as a function of unknowns x and t

\begin{matrix} ψ \cdot \frac{\in_{IPC, T, U} \cdot x}{h - x + t} \cdot E_{cm} \cdot t \\ + \frac{\in_{IPC, T, U} \cdot (x + t)}{h - x + t} \cdot E_{IPC} \cdot t = f_{IPC, T} \cdot t \end{matrix}

(6)

The rotational equilibrium per unit width in the hybrid section is written as

\begin{matrix} M_{D} = F_{C} \cdot z_{C} + {F_{IPC}}_{, C} \cdot z_{IPC} \\ = ψ \cdot (σ_{c} \cdot x) \cdot z_{C} + σ_{IPC, C} \cdot t \cdot z_{IPC} \end{matrix}

(7)

In which

z_{C} = h - δ_{G} \cdot x + \frac{t}{2}

(8)

z_{IPC} = h + t

(9)

where δ_G is the gravity coefficient for the concrete force.

\begin{matrix} M_{D} = ψ \cdot (σ_{c} \cdot x) \cdot (h - δ_{G} \cdot x + \frac{t}{2}) \\ + σ_{IPC, C} \cdot t \cdot (h + t) \end{matrix}

(10)

Substituting equations (4) and (5) in equation (10) gives the rotational equilibrium in function of unknowns x and t

\begin{matrix} M_{D} = ψ \cdot \frac{\in_{IPC, T, U} \cdot x}{h - x + t} \cdot E_{cm} \cdot t (h - δ_{G} \cdot x + \frac{t}{2}) \\ + \frac{\in_{IPC, T, U} \cdot (x + t)}{h - x + t} \cdot E_{IPC} \cdot t \cdot (h + t) \end{matrix}

(11)

Solving the translational (6) and rotational equations (11) gives the position of the neutral axis x and the GFTR-IPC thickness t needed to withstand the tensile stresses. With this thickness t, the bending moments are updated by recalculating the reinforced shell in Abaqus, following the proposed iterative procedure.

When designing a concrete section, there is a minimum of tensile reinforcement needed in the section. In this article, the recommendations according to the Eurocode 2 (European Committee for Standardization, 2004) for the minimum steel reinforcement are adapted to determine the minimum GFTR-IPC area A_s,min per mean width of the tension zone by replacing the strength of steel by the GFTR-IPC tensile strength

A_{s, \min} = 0.26 \cdot \frac{f_{ctm}}{f_{IPC, t}} \cdot d

(12)

where f_ctm = 0.30 $f_{ck}^{2 / 3}$ for strength classes ≤C50/60, f_ck = characteristic compressive concrete strength and d = the effective depth.

Results of the study on shells with a50 mm concrete section

This section describes the results of the structural analysis for the five shell geometries consisting of a GFTR-IPC layer with variable thickness (to be determined) and a concrete layer with a fixed concrete thickness of 50 mm. It describes the determination of the minimum thickness of the GFTR-IPC (1) as formwork in the casting stage and (2) as the reinforcement of the concrete shell.

Analysis of GFTR-IPC formwork in casting stage

Table 2 shows the results of the analysis of the GFTR-IPC formwork under the load of 50 mm wet concrete for every shell geometry and for a different number of vertical supports. In all cases, the minimum thickness of the formwork (between 6 and 15 mm) is determined by local buckling of the shells (Figure 5).

Table 2.

The formwork thickness is determined by local buckling of the GFTR-IPC shell under the 50 mm wet concrete load.


No. of supports	25	31	17	20	25	30	19	30	42	21	32
Span between supports (m)	2	1.5	2	2	2	1.5	2	1.5	1.5	2.5	2
GFTR-IPC thickness (mm)	7	6.5	6	15	13	13	15	15	13.5	14	12
Compressive strain, E_c (µm/m)	−482	−203	−182	−736	−784	−545	−154	−217	−23	−353	−398
Tensile strain, E_t (µm/m)	535	236	166	789	943	911	118	193	28	391	398
Deformation, U (mm)	0.44	0.50	0.38	2.44	2.58	2.78	1.46	1.37	1.47	3.93	3.66
Imperfection factor, α_imp	0.3	0.3	0.4	0.78	0.78	0.78	0.78	0.78	0.78	0.9	0.9
Buckling factor, λ_cr	1.14	1.00	1.05	1.08	1.02	1.10	1.01	1.01	1.04	1.02	1.05

Figure 5.

Local buckling is determining in the design of the GFTR-IPC formwork for (a) the 5 m high dome, (c) the form found saddle shape and (d) the funicular synclastic shell. For (b) the saddle shape with circular arched elevation, global buckling is determining.

For the domes, local buckling just above the hinged base is determining (Figure 5(a)). A GFTR-IPC formwork of 7 mm is needed for the 5 m high dome when supported by 25 supports during casting and 6.5 mm when supported by 31 supports. Precisely because of local buckling, adding more supports has practically no influence. For the 2.5 m high dome, the local buckling effect is lower than for the 5 m high dome, and only 6 mm of GFTR-IPC formwork is needed for less supports (17). For both spherical domes, the maximum strains (535 µm/m < 1100 µm/m) and displacements (0.50 mm < 20 mm) in the formwork are very small because the domes are very stiff structures, that is, strongly double curved and hinged over the whole base.

For the saddle shape with circular arched elevation, 15 mm of GFTR-IPC formwork is needed during casting when supported by 20 supports. For both 25 and 30 supports, 13 mm of GFTR-IPC formwork is needed because buckling is determining. The maximum strain is relatively large (943 µm/m < 1100 µm/m) while deformations are small (6.16 mm < 20 mm). Buckling, now global, is again the deciding factor.

For the form found saddle shape local buckling in the four corners is determining (Figure 5(c)), as a result of which an increase in the number of supports for the formwork from 19 to 30 does not decrease the GFTR-IPC formwork thickness (15 mm). Increasing the number of supports to 42 slightly decreases the GFTR-IPC thickness to 13.5 mm, because the supports are closer to the critical corners in this composition. However, the gain in thickness (10%) is little compared to the increase in number of supports (40%). The behaviour of this form found saddle shape is clearly different compared to the saddle shape with circular arched elevation; generally, a thicker formwork is needed for the form found saddle shape and the shell buckles locally instead of globally. Thickening the formwork in the corners could offer a solution for this local buckling phenomenon and decrease the GFTR-IPC thickness in the rest of the formwork.

Finally, for the funicular synclastic shell, local buckling of the free edges determines the thickness of the GFTR-IPC formwork (Figure 5(d)). Increasing the number of supports from 21 to 32 (54.2%) and thus also the supports at these free edges, decreases the formwork thickness only with 14.3% (14–12 mm). Continuously supporting the free edges could offer a solution and decrease the formwork thickness more significantly.

As these results show, local buckling is the most important factor in the design of the GFTR-IPC formwork. To carry the weight of a 50 mm concrete section, a rather thick GFTR-IPC formwork (between 6 and 15 mm for the studied cases) is needed to prevent buckling. Because local buckling determines the thickness of the formwork, the design of the TRC formwork should be considered more complex than only positioning the supports. Special attention should be paid to the zones sensitive to local buckling and extra supports or thickening the formwork locally should be examined.

Analysis of the GFTR-IPC layer as reinforcement for concrete shells

This section presents the structural analysis results of the concrete shell geometries reinforced by a GFTR-IPC stay-in-place formwork layer and according to the limit states defined in section ‘Load cases and combinations’. The thickness of the GFTR-IPC reinforcement, which is needed to withstand the maximum occurring bending moment M_D occurring under all load combinations of wind, snow and so on, is determined assuming a fixed 50 mm concrete section and based on the tensile design strength of GFTR-IPC (20 MPa). The GFTR-IPC thickness for the reinforcement which is presented in the next paragraphs is the minimal GFTR-IPC thickness to withstand M_D and not the GFTR-IPC thickness determined in the formwork stage. In this way, it is evaluated which stage is the determining one for the thickness of the GFTR-IPC layer.

Table 3 shows results of the analysis of the GFTR-IPC reinforcement of the concrete shells with a 50 mm concrete section. The 5 m high spherical dome is subjected to a maximum bending moment of −359 Nm/m under the dominant load combination 1.35 self-weight + 1.50 point load on the centre top. With this maximum moment M_D, the thickness of the GFTR-IPC reinforcement is iteratively determined following the procedure previously defined in section ‘Methodology to design GFTR-IPC reinforcement for concrete shells’. After iterating, 0.5 mm of GFTR-IPC reinforcement is needed on top and bottom surface of the shell to carry this bending moment. The small increase in the 50 mm concrete section by reinforcing it with the GFTR-IPC hardly increases the maximum bending moment (to −361 Nm/m). Buckling is far from occurring, that is, eigenvalue of 152.4. The compressive stresses of concrete and GFTR-IPC, 1.34 and 1.58 MPa, respectively, are small. The deformations under the dominant SLS load combination 1.00 self-weight + 1.00 point load on the centre top are extremely small, that is, only 0.061 mm (<40 mm) because of the geometrical stiffness of the spherical dome.

Table 3.

Analysis of the GFTR-IPC reinforced concrete shells under the dominant load combinations.


Bending moment M_D under the dominant load combination (Nm/m)	−359	−363.9	1167	−729	447.5
Concrete thickness (mm)	50	50	50	50	50
GFTR-IPC reinforcement, t (mm)	0.5	0.5	1.5	1.0	0.5
New bending moment, M_D (Nm/m)	−361	−368.8	1197	−733.2	450.3
Deformation, U (mm)	0.061	0.066	2.91	0.67	0.72
Imperfection factor, α_imp	0.5	0.55	0.9	0.9	0.9
Buckling factor, λ_cr	152.4	126.5	8.8	18.00	12.31
Compressive stress concrete (MPa)	1.34	1.34	2.23	1.86	1.34
Compressive stress GFTR-IPC (MPa)	1.98	1.98	3.66	2.91	1.98
Tensile stress GFTR-IPC (MPa)	20	20	20	20	20

The results for the other shell geometries are analogue, that is, for the determined mixed GFTR-IPC-concrete sections buckling is never occurring, the compressive stresses and deformations are incessantly smaller than the limit values and the difference in the initial and final maximum bending moment is always small. The 2.5 m high dome needs a GFTR-IPC reinforcement of 0.5 mm to withstand the maximum bending moment under its dominant load combination (1.35 self-weight + 1.50 point load between top and edge). While the circular saddle shape needs a GFTR-IPC reinforcement of 1.5 mm to withstand the maximum bending moment under its dominant load combination (1.35 self-weight + 1.50 upward wind load + 1.50 0.5 triangular snow load), the form found saddle shape needs 1.0 mm (under dominant ULS load combination 1.35 self-weight + 1.50 point load on the edge). Finally, the funicular synclastic shell needs 0.5 mm of GFTR-IPC reinforcement to withstand the maximum bending moment (under 1.35 self-weight + 1.50 point load on the edge).

After determining the GFTR-IPC thickness with the equilibrium equations, the minimum required reinforcement has to be checked using equation (12). For 50 mm concrete sections, minimally 0.85 mm or thus practically 1.0 mm of reinforcement is needed. The GFTR-IPC reinforcement thickness of the spherical domes and funicular synclastic shell should thus be increased from 0.5 to 1 mm. The deformations, buckling and stresses in this adapted section still fulfil all limit states.

As these results show, the GFTR-IPC thickness needed for the reinforcement of the concrete shells is a lot lower than the thickness needed in the formwork stage (between 92.9% and 83.3% thinner). Moreover, for the examined 50 mm thick concrete shells, deformations and stresses are very small and buckling is far from determining when the GFTR-IPC reinforcement is designed for the tensile design strength of GFTR-IPC. Hence, the shells might be executed a lot thinner. Reducing the concrete section would also be an additional advantage in the stage of casting concrete on the GFTR-IPC formwork. A less thick concrete section means less load on the formwork, so the formwork can also be a lot thinner, which can lead to a better synergy/compromise between the thickness of the stay-in-place formwork required for casting and for the hardened stage. This issue is further investigated by two case studies in section ‘Structural design of GFTR-IPC formwork and reinforcement for two case studies’.

Structural design of GFTR-IPC formwork and reinforcement for two case studies

As seen in section ‘Results of the study on shells with a 50 mm concrete section’, the 50 mm thick concrete shells are actually over-dimensioned and might be executed with a decreased thickness. This section presents the determination of the minimum concrete thickness and the investigation of structural adaptations to reduce the local buckling effect. Afterwards, it is concluded if the formwork stage is still dominant over the final stage.

Determining the concrete shell thickness

This section focuses on the two form found case studies introduced in section ‘Geometries and boundary conditions’ (Figure 2(e) and (g)). To determine their minimum concrete section, the concrete thickness was iteratively reduced and subjected to all the load combinations – previously defined in section ‘Load cases and combinations’– and the deformations and buckling load of the plain concrete shells were checked with the limit values.

For the form found saddle shape, both concrete sections of 10 and 15 mm did not encounter problems regarding the deformations and buckling load. Both sections are therefore examined in the next paragraphs. While for the funicular synclastic shell a concrete shell with a 10 mm section was found to buckle under almost every load combination, a 15 mm section did not encounter any problems as deformations were small (9.08 mm under 1.35 self-weight + 1.50 point load on the edge) and stresses relatively low.

In the following paragraphs, the minimum GFTR-IPC thickness required to carry the 10 and 15 mm of concrete load in casting stage and the reinforcement of the 10 and 15 mm concrete shells in final stage is determined.

Structural design of GFTR-IPC formwork for two case studies

For the analysis of the formwork of the case studies, the same methodology is used as defined in section ‘Methodology to design GFTR-IPC formwork’ and the necessary GFTR-IPC thickness to act as formwork is determined.

Table 4 shows the results of the GFTR-IPC formwork for the form found saddle shape with different load of wet concrete (10 or 15 mm) and different number of supports. As seen in section ‘Results of the study on shells with a 50 mm concrete section’, the design of the form found saddle formwork is determined by local buckling in the four corners (Figure 5(c)). For this reason, the formwork is locally thickened in the corners over an area of 3 m². When the formwork is supported by 19 supports, 6.5 mm formwork thickness overall and 11 mm local thickening in the corners is needed to carry 10 mm of wet concrete; 7.5 mm formwork thickness overall and 13 mm local thickening in the corners is minimally needed to carry 15 mm concrete. Largely increasing the number of supports from 19 to 42, decreases the overall formwork thickness only with 23.1% (from 6.5 to 5 mm) when 10 mm of concrete is cast and 20% (from 7.5 to 6 mm) when 15 mm of concrete is cast. However, comparing these results to those of formwork for the 50 mm concrete load (section ‘Analysis of GFTR-IPC formwork in casting stage’), a decrease in the formwork thickness of 63% and 55.6%, when 10 and 15 mm of wet concrete, respectively, has to be carried, is noticed.

Table 4.

Local buckling is again determining the GFTR-IPC formwork thickness under wet concrete load.


No. of supports	19	30	30	42	19	19	30	42
Span between supports (m)	2	1.5	1.5	1.5	2	2	1.5	1.5
Concrete thickness (mm)	10				15
GFTR-IPC overall (mm)	6.5	6	5.5	5	7.5	8	6.5	6
GFTR-IPC corners (mm)	11	10	11	8.5	13	12	12	10.5
Compressive strain, E_c (µm/m)	−77	−122	−128	−107	−95	−91	−147	−131
Tensile strain, E_t (µm/m)	73	131	138	100	87	87	159	111
Deformation, U (mm)	0.89	0.92	0.95	1.06	0.93	0.95	1.03	1.08
Imperfection factor, α_imp	0.78	0.78	0.78	0.78	0.78	0.78	0.78	0.78
Buckling factor, λ_cr	1.04	1.02	1.06	1.01	1.08	1.09	1.05	1.07

Table 5 shows the results of the analysis of the formwork for the form found synclastic shape. As the results in section ‘Analysis of GFTR-IPC formwork in casting stage’ showed, the formwork is determined by local buckling of the free edges. In this respect, it would be interesting to increase the number of supports only at the edges, or even support the entire edge with a beam. First, five supports are placed at the edge and the influence of decreasing the number of supports in the rest of the shell’s formwork (from a total of 33 supports to 25) is examined. For both 33 and 25 supports, 7.5 mm of GFTR-IPC formwork is needed because it is again local buckling of the free edges which defines this thickness and both cases have the same amount of supports at their edge. The number of supports in the rest of the formwork only has a small influence on the strains and deformations. When the whole edge is supported, the formwork thickness is strongly reduced by 33.3% to 5 mm; however, local buckling – now in the corners – is still the determining factor.

Table 5.

Increasing the supports at the edges decreases the GFTR-IPC thickness.


No. of total supports	33	25	5
No. of supports per edge	5	5	Beam
Span between supports (m)	1.5	2	2
Concrete thickness (mm)	15
GFTR-IPC thickness (mm)	7.5	7.5	5
Compressive strain, E_c (µm/m)	−591	−762	−774
Tensile strain, E_t (µm/m)	279	235	228
Deformation, U (mm)	2.30	2.89	0.93
Imperfection factor, α_imp	0.9	0.9	0.6
Buckling factor, λ_cr	1.17	1.17	1.18

Conclusively, when designing GFTR-IPC stay-in-place formwork for shells, special attention should be paid to the zones sensitive to local buckling, and extra supports or thickening the formwork locally should be examined. Logically, when using less concrete to be poured on the formwork, the needed GFTR-IPC thickness is less than when casting 50 mm concrete section. This proves the importance of optimizing the concrete section of the considered shell along with the GFTR-IPC formwork/reinforcement layer.

Structural design of GFTR-IPC reinforcement for two case studies

This section presents the structural analysis results of the two form found case studies reinforced by a stay-in-place GFTR-IPC formwork layer and according to the limit states defined in section ‘Load cases and combinations’. The minimum thickness of the GFTR-IPC reinforcement, which is needed to withstand the maximum bending moment M_D occurring under all load combinations of wind, snow and so on, is determined for each concrete section and based on the tensile design strength of GFTR-IPC. Afterwards, stresses, deformations and buckling are checked. The concrete layer is thinner (10–15 mm) compared to the 50 mm concrete shells of section ‘Results of the study on shells with a 50 mm concrete section’, which will result in another concrete/GFTR-IPC ratio of the shells section.

Table 6 shows the results of the analysis of the mixed GFTR-IPC-concrete form found shells. The form found saddle shape with a 10 mm concrete section is subjected to a maximum bending moment of −718.9 Nm/m under the dominant load combination 1.35 self-weight + 1.50 point load on the edge. After iterating, 3 mm of GFTR-IPC reinforcement is needed on the top and bottom surface of the shell. Reinforcing the 10 mm concrete section with 3 mm of GFTR-IPC decreases the bending moment to −588.5 Nm/m. The compressive stresses of concrete and GFTR-IPC equal 4.46 MPa (<14.2 MPa) and 12.32 MPa (<40 MPa), respectively, and are allowable. Buckling is not determining, but is clearly of more importance now that the section is a lot thinner (eigenvalue 1.19). The maximum deformation under the dominant SLS combination 1.00 self-weight + 1.00 triangular snow load + 1.00 0.6 downward wind load equals 13.01 mm and does not exceed the limit deformation of 40 mm.

Table 6.

Analysis of the optimized GFTR-IPC reinforced concrete shells under the dominant load combination.


Bending moment M_D under the dominant load combination (Nm/m)	−718.9	−768.9	−656.8
Concrete thickness (mm)	10	15	15
GFTR-IPC reinforcement, t (mm)	3	2	2.5
New bending moment, M_D (Nm/m)	−588.5	−611.9	690.4
Deformation, U (mm)	13.01	7.67	5.91
Imperfection factor, α_imp	0.85	0.85	0.9
Buckling factor, λ_cr	1.19	1.97	1.33
Compressive stress concrete (MPa)	4.46	3.88	4.11
Compressive stress GFTR-IPC (MPa)	12.32	8.35	9.41
Tensile stress GFTR-IPC (MPa)	20	20	20

The results for the 15 mm thick form found saddle shape and the 15 mm thick form found synclastic shape are similar, that is, buckling is never occurring, the compressive stresses and deformations are incessantly smaller than the limit values and the difference in the initial and final maximum bending moment is always minimal. If the form found saddle shape has a concrete section of 15 mm, 2 mm of GFTR-IPC reinforcement is needed to withstand the maximum bending moment under its dominant load combination (1.35 self-weight + 1.50 point load between top and edge). The form found synclastic shell, with a concrete section of 15 mm, needs a minimum GFTR-IPC reinforcement of 2.5 mm under its dominant load combination (1.35 self-weight + 1.50 point load on the edge).

As already mentioned, a minimum of tensile reinforcement is needed in the concrete section. Following formula (12) for both 10- and 15 mm concrete sections, at least 0.5 mm GFTR-IPC reinforcement is needed. For all case studies, this requirement is met. As seen in these results, only a few millimetres of reinforcement is needed on top and bottom surface, which indicates promising capacities of GFTR-IPC as tensile reinforcement. When the GFTR-IPC reinforcement is designed for the tensile design strength of GFTR-IPC, the stresses and deformations of the shells remain small, but clearly buckling gains importance now that a thinner section is used (eigenvalues between 1.97 and 1.19 lie closer to 1).

Discussion and conclusion

As seen in the results of the analysis on the shell geometries with a 50 mm concrete section, the GFTR-IPC thickness needed for the reinforcement of the concrete shells is 83.3%–92.9% thinner than needed in the formwork stage, where local buckling was determining. In the final stage, all deformations and stresses were extremely small and buckling was far from occurring. The first part of the article clearly designates the importance to optimize the concrete section along with the GFTR-IPC formwork/reinforcement. In this way, the materials are not only used better in the final stage, but this is also beneficial for the formwork stage, as studied in the second part of the article.

In the second part, two 10 m × 10 m form found shells with reduced concrete section are analysed to check if the formwork stage remains the determining stage. The form found saddle shape requires a 5 mm thick GFTR-IPC formwork to support a 10 mm concrete shell. For the form found synclastic shell also 5 mm of GFTR-IPC formwork (8.5 mm thick in the corners) is needed when the free edges are supported during casting of 15 mm concrete. The GFTR-IPC formwork is always determined by local buckling of the GFTR-IPC formwork. For this reason, buckling of GFTR-IPC shells must be more profoundly examined and it should be questioned if the same safety factors apply as those for concrete shells. The formwork stage is not permanent and will not return; after a few days, the concrete is already structurally contributing. The minimum GFTR-IPC reinforcement for the shells with reduced concrete section varies between 2 and 3 mm, which means 40%–60% thinner than the GFTR-IPC formwork, that is, the casting stage is determining for the design of the structural stay-in-place formwork.

As shown in these results, all shell shapes need a GFTR-IPC reinforcement less than 3 mm, which is very promising and proving the structural contribution of the GFTR-IPC formwork. These results should be interpreted with care, since the performed study is limited to a purely numerical analysis which may not capture all structural effects happening in a true shell. Due to the assumptions made in the modelling and the considered failure mechanisms of buckling and exceeding of material strength, the study should be considered as a preliminary indication of the feasibility of the new concept, which was the aim of this article.

The contribution to the tensile reinforcement is an additional advantage of the flexible formwork method because of two reasons. First, the placing of the rigid traditional reinforcement is labour intensive and practically limits the curvature of the shells. Second, the minimum thickness of steel reinforced concrete surfaces depends on the concrete cover needed to protect the steel reinforcement against corrosion. GFTR-IPC composites are non-corroding so the shell thickness is not determined by the concrete cover. The use of the GFTR-IPC formwork as an exclusive reinforcement opens up the way to very slender, strongly curved concrete surfaces for small to large spans. Before implementation of this promising concept, future work should however still be performed, particularly on the level of experimental validation of the structural behaviour. A first mechanical test on a GFTR-IPC formwork dome has already been performed by Verwimp et al. (2015), yet more tests should be performed, both on a larger scale and on the final stage concrete shells reinforced by GFTR-IPC formwork.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

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