Development of statistical design models for concrete sandwich panels with continuous glass-fiber-reinforced polymer shear connectors

Abstract

This article proposes a statistical framework for the development of design models for concrete sandwich panels with glass-fiber-reinforced polymer shear grids. The framework is developed by integrating the Bayesian parameter estimation method and the Eurocode-based capacity reduction factor calibration method. In the first part of the framework, probabilistic and deterministic shear flow prediction models are proposed based on 32 experimental data. It is seen that the contribution of glass-fiber-reinforced polymer grids is dominant, although parameters on the geometrical and material properties of insulation and concrete wythes also contribute. Different constant terms for bias correction of the proposed models are proposed according to the insulation type, and the prediction error of the developed model was reduced. In the second part of the framework, the capacity reduction factor for the proposed deterministic formulas is calculated for design purposes. Statistical calibrations for capacity factors are carried out to meet a target reliability level, and the value is estimated to be approximately 0.75 for all proposed models. Further data collection will improve the applicability of the proposed models and clarify quantification of the contribution of parameters.

Keywords

Bayesian parameter estimation capacity factor calibration concrete sandwich panel glass-fiber-reinforced polymer shear grid insulation effect

Introduction

Introduction to precast concrete sandwich panels

Insulated concrete sandwich panels have been used for more than 50 years as exterior and interior walls for many different structures. They can be easily attached to any type of structural frame, including structural steel, reinforced concrete (RC), pre-engineered metal, and precast/pre-stressed concrete (Losch et al., 2011). They provide high thermal resistance and structural performance by inserting the most vulnerable insulation component between two concrete wythes (layers). Insulated concrete sandwich panels are available in three types, non-composite, composite, and partially composite in terms of strength, which differ in the composite shear resistance action of the concrete wythes and the existence of full shear transfer between the wythes. When insulated concrete sandwich panels are used as structural components, the shear connectors should be designed to provide full shear transfer between the two concrete wythes in order to take full advantage of the strength of the two wythes and to prevent individual wythe buckling (Benayoune et al., 2006).

Insulated concrete sandwich panels are manufactured using various materials including structural concrete, reinforcing bars, welded-wire reinforcement, steel embedment, and pre-stressing strands. The unique components of insulated concrete sandwich panels are various types of insulation and a variety of wythe connectors (Losch et al., 2011). Expanded polystyrene (EPS) and extruded polystyrene (XPS) rigid foams are commonly used types and exhibit significant, but not easily predictable, adhesion bonds to concrete.

The shear strength of insulated concrete sandwich wall panels depends on the bond between insulation and concrete wythes, the shear resistance of insulation, and the shear flow strength of shear connectors (Salmon et al., 1997), but shear connectors provide the dominant contribution to the shear strength of insulated concrete sandwich panels according to their arrangement, spacing, and materials (Benayoune et al., 2008). Conventional shear connectors include C-tie, Z-tie, M-tie, steel trusses, bent wire, and solid zones of concrete penetrating the foam core (Rizkalla et al., 2009). Since these traditional connector types have a notable thermal bridge effect between the two concrete wythes, which greatly reduces the thermal efficiency of insulated concrete sandwich panels, many researchers have recently tried to find alternative types of shear connectors to achieve better thermal efficiency and composite action between concrete wythes (Ekenel, 2014). At the present time, one of the most commonly used materials is glass-fiber-reinforced polymer (GFRP), which has very low thermal conductivity and thus does not compromise the thermal performance of insulation (Gleich, 2007; Rizkalla et al., 2009). Continuous shear connectors such as GFRP grids also provide an effective shear transfer mechanism between the inner and outer concrete wythes, achieving high structural resistive capacity while maintaining the thermal efficiency of the panel.

Many experimental studies have been carried out to investigate the structural performance of concrete sandwich panels reinforced with GFRP shear connectors, and their results are reported in the literature as follows. Pantelides et al. (2008) conducted four-point bending tests to investigate the flexural strength of shell-type GFRP connectors designed based on a shear flow approach. Morcous et al. (2010) carried out experiments to optimize the distribution of truss-type shear connectors and their effects on the flexural strength and stiffness of concrete sandwich panels. Naito et al. (2012) conducted push-off shear strength tests and uniform load flexural tests of concrete sandwich panels considering 14 different shear ties made of GFRP, carbon-fiber-reinforced polymer (CFRP), and carbon-steel. Woltman et al. (2013) conducted experiments on the shear strength of concrete sandwich panels with three kinds of stud-type GFRP connectors for 50 specimens, in which the adhesion bond between concrete and insulation was found to be significant but showed large variation. Lameiras et al. (2013a, 2013b) carried out experimental and numerical studies on steel-fiber-reinforced self-compacting concrete sandwich panels with GFRP plates and I-shaped GFRP connectors. Oh et al. (2013a, 2013b) performed push-off tests to measure the shear strength of concrete sandwich panels with truss-type GFRP shear connectors using different types of insulation and surface conditions.

However, further to these experimental investigations, there is no legally adopted building code for the design of concrete sandwich panels, particularly with GFRP grid connectors. The Precast/Prestressed Concrete Institute (PCI) document “State of the art of precast/pre-stressed concrete sandwich wall panels” (Losch et al., 2011) provides guidelines for the design of sandwich wall panels, but it does not include guidelines for determining the capacity of composite shear connectors and relies only on data obtained from shear connector manufacturers. AC422 (ICC Evaluation Service, 2010) provides acceptance criteria for the construction of concrete sandwich panels with fiber-reinforced polymer (FRP) grids, but it is entirely dependent on standardized performance tests to determine the shear design strength of the panels, which does not consider the effect of the number of tests, and provides capacity reduction factors that are simply adopted from ACI 318 without reliability analysis. There is no “standardized” guidance that provides capacity determination for sandwich panels with GFRP connectors. A formulation of design assumptions and capacity reduction factors based on the experimental data gathered using the standardized testing provided by AC422 is currently not available (Ekenel, 2014).

To extend existing experimental observations on the shear capacity of concrete sandwich panels with GFRP shear connectors to a design practice, this article proposes a statistical framework to develop a probabilistic shear flow model of a concrete sandwich panel with shear grids, and the corresponding design strength model for its practical use. This framework utilizes the two probabilistic methods, the Bayesian parameter estimation method and the capacity reduction factor calibration method. The framework uses 32 recently acquired experimental data on the ultimate shear strength of concrete sandwich panels with shear grids to estimate the importance of the parameters and the bias and scatter of the modeling error in the developed probabilistic shear strength models. The probabilistic model is developed through the identification of important design parameters and an equation simplification process with systematic removal of non-informative terms in the Bayesian parameter estimation method. The safety margin for the model is then calibrated based on the estimated parametric and modeling uncertainties. The proposed framework can be updated upon collection of new experimental data, thus extending the applicability of the framework to a wider range of input design parameters.

Database on shear strength of concrete sandwich panels reinforced with GFRP shear grids

Experimental test program on pure shear tests

In this study, we use the push-off test results for the shear strength of concrete sandwich panels reinforced with GFRP grids reported in You et al. (2011). A sandwich panel should be designed to withstand applied wind load through its flexural and shear capacities, but a full-scale test is often difficult to carry out because of its large size. In this report, unit-scale five-layer concrete sandwich panels (600 mm × 1050 mm) were tested to obtain their pure shear strengths. Table 1 provides the geometric and material properties of the test specimens, including the tensile strength of an individual strand in a GFRP shear grid, the embedment depths of grids, the types and thicknesses of insulation layers, and the compressive strengths of concrete wythes. The specimen labels are named according to the following order of properties: insulation type, insulation width, embedment depth, and shear grid type. As an example, the label “EPS-100-EL40T1” means a sandwich panel with EPS-type insulation, the insulation width of 100 mm, embedment depth/length (EL) of 40 mm, and type 1 shear grids. The following three types of insulation are considered in this study: XPS, EPS, and a special type of XPS with surface treatment for improving its roughness XPS(eXtruded PolyStyrene) with Surface treatment (XPSS). Both XPS and EPS foams are comprised of polystyrene, but different manufacturing processes are applied to them. XPS is manufactured in a continuous extrusion process that produces a homogeneous closed-cell cross section, whereas EPS is manufactured by expanding spherical beads in a mold, using heat and pressure to fuse the beads together. XPSS foam is a special type of XPS with surface treatment to improve its roughness. Their photos are shown in Figure 1. A photo of a GFRP shear grid used in the test is shown in Figure 2, and the statistics of the tensile strength test results of individual strands for five types of grids are reported in Table 2. All the strands in both the machine and cross-machine directions in the shear grids were manufactured to have the same tensile strength by carefully adjusting the amount of fiber or changing the design of their intersections.

Table 1.

Test specimen matrix.

	Specimen label	GFRP shear grid		Insulation		Concrete	No. of specimens
	Specimen label	Strand tensile strength (kN)	Embedment depth (mm)	Type	Width (mm)	Compressive strength (MPa)	No. of specimens
No shear grid	EPS-100			EPS		31	2
	EPS2-100			Gray EPS		28	3
	XPS-100			XPS		31	1
	XPSS-110			XPS with roughened surface		26	2
With shear grid	EPS-100-EL40T3	4.75	40	EPS	100	35	2
	XPS-100-EL40T3	4.75	40	XPS	100	35	2
	XPS-100-EL40T1	3.22	40	XPS	100	35	2
	EPS-100-EL40T4	5.03	40	EPS	100	35	2
	EPS-140-EL40T4	5.03	40	EPS	140	35	3
	XPS-80-EL40T4	5.03	40	XPS	80	35	3
	XPS-100-EL20T4	5.03	20	XPS	100	31	3
	XPS-100-EL30T4	5.03	30	XPS	100	31	3
	XPS-100-EL40T4	5.03	40	XPS	100	31	3
	XPSS-110-EL40T4	5.03	40	XPS with roughened surface	110	26	3
	XPS-100-EL40T2	4.12	40	XPS	100	35	3
	EPS2-100-EL40T5	6.4	40	Gray EPS	100	28	3
	Total number						40

GFRP: glass-fiber-reinforced polymer; EPS: expanded polystyrene; XPS: extruded polystyrene.

Figure 1.

Types of insulation for sandwich panels: (a) XPS foam, (b) XPS foam with surface treatment, and (c) EPS.

Figure 2.

GFRP shear grid (unit: mm).

Table 2.

Tensile strengths of GFRP strands (unit: kN).

	Type 1	Type 2	Type 3	Type 4	Type 5
Mean	3.22	4.12	4.75	5.03	6.4
Standard deviation	0.13	0.89	0.31	0.29	0.24

GFRP: glass-fiber-reinforced polymer.

Figure 3 provides dimensions of a typical specimen. Figure 3(a) shows the front and side views of a specimen without GFRP shear grids. These specimens were tested to examine the contribution of bond strength between insulation and concrete layers to the overall shear strength of the concrete sandwich panels. Figure 3(b) shows the front and side views of a specimen with GFRP shear grids. The GFRP shear grids are arranged at a uniform interval of 30 cm.

Figure 3.

Dimensions of a typical sandwich panel specimen: (a) test specimen without shear grids and (b) test specimen with shear grids.

Figure 4 shows a schematic illustration and a photograph of the push-off test setup. Each panel consists of three concrete layers and two insulation foams. Vertical loading is applied to the top of the middle concrete layer, and the other two outside layers are supported on the bottom. These three concrete layers were used to avoid the occurrence of any eccentric load.

Figure 4.

Test setup and measurement installation.

Test result summary

Table 3 shows a summary of experimental results including maximum loads (kN), average load, and average shear flow. The first four rows of the table show the test results of specimens without shear grids (EPS-100, EPS2-100, XPS-100, and XPSS-110). The average shear flows of EPS and EPS2 are 33 and 38 kN/m, respectively, which are greater than those for XPS and XPSS. This means that the bond between concrete and EPS insulation is stronger than that in the other types of insulation, and accordingly the shear deformation of EPS insulation is relatively large, as shown in Figure 5(a). XPS insulation showed sudden separation along the interface between the insulation and concrete at a very low average shear flow of 1.5 kN/m. The shear flow of the surface-roughened XPS (XPSS) specimen increased to 29 kN/m compared to 1.5 kN/m for XPS because of the improved mechanical bond between concrete and insulation.

Table 3.

Experimental test results.

Specimen label	Maximum load (kN)			Average load (kN)	Standard deviation of load (kN)	Average shear flow (kN/m)
Specimen label	1	2	3	Average load (kN)	Standard deviation of load (kN)	Average shear flow (kN/m)
EPS-100	127.2	135.0		131.1	5.5	32.8
EPS2-100	154.7	139.3	159.1	151.0	10.4	37.8
XPS-100	5.8	−	−	5.8	−	1.5
XPSS-110	117.4	116.0	−	116.7	1.0	29.2
EPS-100-EL40T3	221.2	229.9	−	225.6	6.2	56.4
XPS-100-EL40T3	176.5	205.3	−	190.9	20.4	47.7
XPS-100-EL40T1	137.9	125.8	−	131.9	8.6	33.0
EPS-100-EL40T4	251.6	252.9	−	252.3	0.9	63.1
EPS-140-EL40T4	218.0	215.5	228.2	220.6	6.7	55.1
XPS-80-EL40T4	302.3	281.3	292.9	292.2	10.5	73.0
XPS-100-EL20T4	245.2	255.2	249.6	250.0	5.0	62.5
XPS-100-EL30T4	252.4	267.5	263.4	261.1	7.8	65.3
XPS-100-EL40T4	251.2	247.5	245.3	248.0	3.0	62.0
XPSS-110-EL40T4	318.4	313.8	319.3	317.2	3.0	79.3
XPS-100-EL40T2	167.1	166.5	169.1	167.6	1.4	41.9
EPS2-100-EL40T5	315.9	281.8	279.8	292.5	20.3	73.1

EPS: expanded polystyrene; XPS: extruded polystyrene.

Figure 5.

Observed failure modes for different insulation types: (a) EPS without shear girds, (b) EPS with shear grids, (c) XPS with shear grids, and (d) XPSS with shear grids.

The test results of specimens containing shear grids are shown from the fifth row down of Table 3. The specimens with EPS insulation showed average shear flows within the range of 56–73 kN/m according to the tensile strength of a shear grid strand and the thickness of insulation. Shear failure as a result of the high bond effect of EPS insulation was observed as approximately 45° insulation tearing, as shown in Figure 5(b). The specimens with XPS insulation showed average shear flows within the range of 33–73 kN/m depending on the tensile strength of a shear grid strand and thickness of insulation. The specimens with XPSS insulation showed an average shear flow of 79 kN/m because of the improved bond effect. Figure 5(c) and (d) shows the interface bond failure mode of XPS and XPSS specimens.

In addition to the type of insulation, the following key parameters were identified from the experimental results: the tensile strength of a shear grid strand, the thickness of insulation, and the compressive strength of concrete wythes. First, it is seen that the shear strength of the overall sandwich panel is directly proportional to the tensile strength of shear grid strands. Second, the shear strength of the overall sandwich panel decreases with the thickness of insulation because insulation is the weakest component with respect to shear stress, and the effective number of strands embedded in both concrete wythes also decreases. This is clearly shown in Table 3; a comparison of EPS-100-EL40T4 and EPS-140-EL40T4 shows that they have insulation thicknesses of 100 and 140 mm and average shear strengths of 252.3 and 220.6 kN, respectively. Third, the concrete compressive strength can indirectly affect the pull-out strength of shear grid strands, but no pull-out was observed during the tests because the shear grids have a sufficient embedment depth of greater than 20 mm to avoid pull-out failure of the shear grid from concrete wythes. In this study, 32 out of 40 test results excluding the test results with no shear grid are used in the capacity prediction and design model development in the following sections.

Probabilistic methodology for capacity prediction and design models

The shear capacity of a typical sandwich panel can be determined based on possible mechanisms of shear failure including: (1) mechanical connectors, such as shear grids between the insulation and concrete wythes; (2) the bond between the concrete wythes and the insulation; (3) the shear resistance of insulation; and (4) the shear capacity of concrete wythes, although the fourth mechanism does not generally govern the shear failure of a sandwich panel as it is much stronger than the shear capacity of insulation. It is, however, difficult to mechanically investigate the individual effects of mechanisms (2) and (3) and their contribution to the shear capacity of the whole sandwich panel. Therefore, AC422 recommends experimentally obtaining the shear capacity of a sandwich panel when using such a type of panel by carrying out multiple push-off tests. However, it may not be easy to carry out experiments every time to predict the capacity of a sandwich panel, and the safety of this design method has not been quantified or verified based on reliability analysis in the literature.

Therefore, it is necessary to develop an analytical prediction model and determine its safety margin for the convenient design and practical use of a sandwich panel. Based on available experimental results, this study proposes a framework for developing a simple prediction model for the shear capacity of a sandwich panel, which is expected to be predominantly affected by the three shear failure mechanisms of the shear connectors, bond, and insulation. The proposed framework synthesizes the following two methods: (1) development of a probabilistic shear capacity prediction model using the Bayesian parameter estimation approach (Gardoni, 2002; Song et al., 2010) and (2) determination of capacity reduction factor based on the statistical partial safety factor calibration method provided in EN 1990 Annex D.8 (European Committee for Standardization, 2002). Both methods are based on the same test database.

Probabilistic capacity prediction models using a Bayesian approach

To develop probabilistic shear capacity prediction models for sandwich panels based on the failure test data, we adopted Bayesian methodology, which was originally developed for constructing probabilistic models for the capacities of RC columns (Gardoni, 2002) and the seismic demands of RC bridges (Gardoni et al., 2003). The probabilistic shear capacity prediction models predict the shear flow strength of a sandwich panel, C, as follows

C (x, Θ) = c_{d} (x) + γ (x, θ) + σ ε

(1)

where x is the vector of mean measured input parameters during sandwich panel shear tests (e.g. the tensile strength of a strand of a GFRP grid, strand spacing, and the compressive strength of concrete); $Θ = (θ, σ)$ denotes a set of unknown parameters introduced to fit the proposed model into the test results; $c_{d} (x)$ is an existing deterministic capacity prediction model; $γ (x, θ)$ is a correction term introduced to minimize the bias and scatter of the deterministic capacity prediction model, which includes the input parameters x and uncertain model parameters $θ = [θ_{1}, θ_{2}, \dots, θ_{p}]^{T}$ ; $ε$ is a standard normal random variable representing the remaining modeling error after a bias correction using $γ (x, θ)$ ; and σ represents the magnitude of the remaining modeling error.

This model is valid based on the following two assumptions: (1) the homoscedasticity assumption, that is, the model variance σ is constant for all given values of input parameters x; and (2) the normality assumption: the error term ε follows a standard normal distribution.

In this study, we assume that the bias-correction function $γ (x, θ)$ is expressed as the summation of a suitable set of p“explanatory” functions $h_{i} (x)$ , $i = 1, \dots, p$ , as follows

γ (x, θ) = \sum_{i = 1}^{p} θ_{i} h_{i} (x)

(2)

To make the whole capacity prediction model a product form, we modify this equation using the natural logarithms satisfying the homoscedasticity assumption as follows (Song et al., 2010)

\ln [C (x, Θ)] = \ln [c_{d} (x)] + \sum_{i = 1}^{p} θ_{i} h_{i} (x) + σ ε

(3)

where the explanatory functions $h_{i} (x)$ are defined as the lognormal functions of input parameters. The model is fully constructed by finding the model parameters $Θ = (θ, σ)$ that best fit the model to the test results. For this model parameter estimation, we use the following Bayesian parameter estimation method to estimate the posterior distribution $f (Θ)$ of the unknown model parameters based on the test results. The well-known Bayesian updating rule (Box and Tiao, 1992) is

f (Θ) = κ L (Θ) p (Θ)

(4)

where $p (Θ)$ is the prior function of $Θ$ , $L (Θ)$ is the likelihood function representing the likelihood of the test results, and $κ = [\int L (Θ) p (Θ) d Θ]^{- 1}$ is the normalizing factor.

In the formulation of equation (4), if there is no available information for the prior function $p (Θ)$ , a non-informative prior can be used that has little influence on the posterior distribution. Gardoni (2002) showed that the non-informative prior distribution of $θ$ is locally uniform such that $p (Θ) ≅ p (σ)$ . The non-informative prior of $σ$ then takes the form

p (σ) \propto \frac{1}{σ}

(5)

The likelihood function $L (Θ)$ in equation (4) is defined as a function proportional to the conditional probability of the observations for given input parameters x. If we only use failure data and there are no upper or lower bound data, the likelihood function is simply defined as follows (Gardoni, 2002; Song et al., 2010)

L (Θ) \propto \underset{failure data}{Π} {\frac{1}{σ} φ [\frac{\ln [C_{i}] - \ln [c_{d} (x_{i})] - γ (x_{i}, θ))}{σ}]}

(6)

where $φ (\cdot)$ denotes the probabilistic density function (PDF) of the standard normal distribution. This likelihood function represents the proportionality of the probability distribution of the difference between the measured data and the unbiased prediction. This form is adopted due to it incorporating the prediction error that is measured based on experimental data.

As it is not straightforward to compute the multifold integrals to obtain the normalizing factor $κ$ in equation (4), the posterior mean vector $M_{Θ}$ , and the covariance matrix $Σ_{Θ Θ} = \int Θ Θ^{T} f (Θ) d Θ - M_{Θ} M_{Θ}^{T}$ , we adopt an important sampling method employing the sampling density function centered at the maximum likelihood point to expedite the convergence of posterior distribution estimation (Gardoni, 2002).

The Bayesian approach provides a stepwise equation construction procedure by removing insignificant explanatory terms. When any of the input parameters has a very large posterior coefficient of variation (COV) after Bayesian updating, its corresponding explanatory function is considered non-informative and can then be dropped from the bias-correction function $γ (x, θ)$ , because the function does not significantly contribute to the accuracy of the constructed equation. After dropping the non-informative term, Bayesian updating is performed again to check that the posterior mean of the magnitude of the modeling error σ does not increase greatly. This process is repeated until a removal increases the posterior mean of $σ$ by an unacceptable amount. This removal process of the Bayesian methodology allows researchers not only to try all explanatory functions they consider important but also to identify informative terms in a systematic manner without performing numerous regression analyses with all possible combinations of explanatory terms.

Safety margin calibration procedure using a Eurocode method based on test observations

In order to use a probabilistic shear capacity prediction model in a practical design, an appropriate safety margin considering the uncertainties included in the formula should be provided. In this section, all of the uncertainties created during the formula development processes in the previous sections are identified and incorporated into the determination of safety margins. In this framework, we modify the EN 1990 method (European Committee for Standardization, 2002), which provides procedures based on experimental data to estimate a single capacity reduction factor as the ratio of the design resistance to the nominal resistance. The modification involves direct use of the quantified modeling error of unbiased models developed using the Bayesian parameter estimation method in the Eurocode framework.

The reasons for choosing the Eurocode method among many other methods are as follows. (1) This method is based on the lognormal distribution models for resistance with the lower limit at zero, instead of a normal distribution, and this corresponds to reality (Gulvanessian and Holický, 2005); the corresponding target reliability should be selected based on the same models and assumptions. (2) This study develops shear capacity prediction models defined as the products of design parameters, which are well represented by a lognormal distribution rather than a normal distribution. (3) This method can calibrate capacity reduction factors separately from the load effects using the first-order reliability method (FORM) sensitivity factors. For these reasons, when calibrating capacity reduction factors, we can fully utilize the test database and the developed unbiased shear capacity prediction models.

The first step of the capacity reduction factor calibration procedure is to evaluate the modeling error of a capacity prediction model. Applying exponential functions to equation (3), the probabilistic model becomes the following form

\begin{matrix} C (x) = c_{d} (x) \cdot \exp [\sum_{i = 1}^{n} μ_{θ_{i}} h_{i} (x)] \cdot \exp (μ_{σ} ε) \\ = {\tilde{c}}_{d} (x) \cdot \exp (μ_{σ} ε) \end{matrix}

(7)

where n denotes the number of the explanatory terms that survive after the stepwise removal process, and ${\tilde{c}}_{d} (x)$ is the shear capacity prediction function developed using the stepwise calibration approach introduced in the previous section. If we replace the model parameters in $Θ$ and plug in the mean measured values into the input parameters x, the normal random variable $ε$ becomes the only random variable in the model. Then, the model follows a lognormal distribution as an exponential function of a normal random variable is a lognormal random variable. In this case, the mean and COV of the shear strength prediction are derived as ${\tilde{c}}_{d} (x) \cdot \exp (μ_{σ}^{2} / 2)$ and $[\exp (μ_{σ}^{2}) - 1]^{1 / 2}$ , respectively (Ang and Tang, 2007). In the case that $μ_{σ} << 1$ , the mean and COV can be approximated as ${\tilde{c}}_{d} (x)$ and $μ_{σ}$ , respectively. The COV of the exponential term $\exp (μ_{σ} ε)$ represents the inaccuracy of the prediction model.

In addition to this modeling error, there is one more source of uncertainty—the COV of the resistance function ${\tilde{c}}_{d} (x)$ based on the uncertainties in the input parameters x—as there should be variation around the mean measured values of x. We define the COV of ${\tilde{c}}_{d} (x)$ as the parametric variance $(V_{par})$ .

Since $C (x)$ in equation (7) follows a lognormal distribution as we assumed that $\ln [C (x)]$ follows a normal distribution, the COV of $C (x)$ is estimated as follows (European Committee for Standardization, 2002)

V_{C} ≅ \sqrt{(μ_{σ}^{2} + V_{par}^{2})}

(8)

where $μ_{σ}$ is the the COV of the shear strength prediction and V_par is the COV of $C (x)$ due to the uncertainties in the input parameters x. V_par can be estimated for each test result using a Monte Carlo simulation or the first-order approximation of moments (Ang and Tang, 2007). Under the lognormal assumption, the standard deviation of $\ln [C (σ_{\ln C})]$ is calculated as follows

σ_{\ln C} = \sqrt{\ln (1 + V_{C}^{2})}

(9)

This standard deviation is used to calculate the target design value of the resistance (C_D) for a target reliability index β as follows

C_{D} = {\tilde{c}}_{d} (x) \exp (- k σ_{\ln C} - 0.5 σ_{\ln C}^{2})

(10)

where

k = \frac{(k_{d} μ_{σ}^{2} + β V_{par}^{2})}{V_{C}^{2}}

(11)

and k_d is the fractile factor corresponding to β at the 75% confidence level, determined for a number of test data N from a non-central t-distribution. Note that the target reliability index β is determined by considering the resistance only and ignoring the effect of loads. In this case, according to ISO 2394:1998 (International Organization for Standardization, 1998), β can be empirically estimated as α_R × β_t, where β_t is the target reliability considering both resistance and load effects and α_R is the FORM sensitivity factor for resistance, which is taken as 0.8 as recommended in EN 1990 (European Committee for Standardization, 2002).

Finally, the strength reduction factor is calculated as the ratio of the design resistance in equation (10) and nominal capacity $C_{N}$ as follows

ϕ = \frac{C_{D}}{C_{N}}

(12)

where $C_{N}$ is calculated by plugging in the nominal values of parameters into the resistance prediction model ${\tilde{c}}_{d} (x_{N})$ , where $x_{N}$ is nominal values of the input parameters. When the experimental data do not provide information about the nominal values of the input parameters, the characteristic values based on the 5% fractal value of 1.64 can alternatively be used to replace the nominal input parameters (European Committee for Standardization, 2002; Gulvanessian and Holický, 1996). For example, the characteristic tensile strength of a GFRP strand $(T_{gk})$ is defined as follows

T_{gk} = {\bar{T}}_{g} \exp (- 1.64 σ_{\ln T_{g}} - 0.5 σ_{\ln T_{g}}^{2})

(13)

where

σ_{\ln T_{g}} = \sqrt{\ln (1 + V_{T_{g}}^{2})}

(14)

and ${\bar{T}}_{g}$ and $V_{T_{g}}$ are the mean measured value and the COV of the tensile strength of a GFRP strand, respectively.

Proposed shear capacity prediction models and their safety margins based on pure shear test database

Shear capacity prediction model construction using Bayesian method

Using the Bayesian method together with the explanatory function form in equation (3), we construct a shear capacity prediction equation for a sandwich panel. Since we construct an equation from scratch, there is no deterministic equation, that is, $c_{d} (x)$ is zero. The bias-correction function $γ (x, θ)$ is formulated as follows

γ (x, θ) = \ln [2^{θ_{1}} \cdot T_{fs, t}^{θ_{2}} \cdot {(\frac{t}{t_{s}})}^{θ_{3}} \cdot {f'}_{c}^{θ_{4}}]

(15)

where 2 is a constant term (any number except 1 can be used; 1 always results in 1^θ1 = 1), t is the thickness of rigid insulation (mm), t_s is the thickness of the representative sandwich panel (mm), ${f'}_{c}$ is the compressive strength of concrete (MPa), and T_fs,t is the theoretical shear flow strength of shear grids (kN/m). The terms in equation (15) are selected due to the following rationales. (1) It is obvious that shear grid is the most dominant component contributing to the whole shear strength of a sandwich panel, and the term T_fs,t is selected. (2) To represent the effect of the gap between concrete panels that is represented by the thickness ratio of insulation to the whole sandwich panel, the term t/t_s is selected. This is a unitless term. (3) To represent a potential effect of the concrete strength to the shear strength of the whole sandwich panel, the term ${f'}_{c}$ is selected. It is expected that the shear grid has the most dominant contribution to the shear capacity of a sandwich panel, and we propose to estimate it using the following deterministic equation when statistics on the grid tensile strength per unit length are available

T_{fs, t} = \frac{1 - (t_{sg} / L)}{s \sqrt{2}} \times T_{g} \times \cos (45 \circ)

(16)

where t_sg is the width of shear grids (mm), T_g is the tensile strength of a GFRP strand (kN), s is the spacing of GFRP strands in a shear grid (mm), and L is the total length of shear grids (herein assumed to be 1000 mm).

By conducting a Bayesian parameter estimation, the posterior means and COVs of the unknown θ_i parameters in equation (15) are estimated as shown in Table 4. From this table, we can observe that the COV of θ₁, which corresponds to the explanatory term ln(2), has the greatest value compared to other θ_i values. This means that this explanatory term is the least informative and does not significantly affect the improvement in accuracy of the constructed equation. Second, we also observe that the power term for the thickness of insulation, θ₄, has a negative mean value, which shows a reciprocal relationship between the shear strength of a sandwich panel and the thickness of insulation.

Table 4.

Posterior means and COV of θ_i after Bayesian parameter estimation.

	θ ₁	θ ₂	θ ₃	θ ₄	σ
Corresponding explanatory term	ln(2)	ln(T_fs,t)	ln(t/t_s)	$\ln ({f'}_{c})$	ln(ε)
Explanation	Constant term	Grid shear flow strength	Insulation thickness ratio	Concrete compressive strength	Model error
Posterior mean	2.3375	1.111	−0.47697	−0.65493	0.11975
Posterior COV	0.9281	0.15313	0.76697	0.37395	0.13399

COV: coefficient of variation.

The experiments in this study consider three different insulation types, EPS, XPS, and XPSS, which have different contributions to the shear strength of a sandwich panel as a result of different mechanical characteristics such as shear and bonding strengths. However, it is difficult to fully investigate their mechanical properties based only on the available database because of missing information about their shear strength, systematic bonding effect between the insulation and shear grid, and the quantified roughness of insulation. Therefore, in this study, this information is accounted for in a statistical way, and we propose respective constant factors for these three different insulation types using the following explanatory function modified from equation (15)

γ (x, θ) = \ln [2^{I (θ_{1}, 0, θ_{2})} \cdot T_{fs, t}^{θ_{3}} \cdot {(\frac{t}{t_{s}})}^{θ_{4}} \cdot {f'}_{c}^{θ_{5}}]

(17)

where the function I(θ₁, 0, θ₂) is defined such that it returns the value of θ₁ for the insulation type EPS, 0 for XPS, and θ₂ for XPSS. Since XPS is expected to have the lowest shear contribution due to its lowest bonding effect compared to EPS and XPSS, the power term for XPS is fixed at 0 such that the constant term of XPS is 2⁰ = 1, in order to make XPS the reference insulation type with no additional constant. Then, the calculated θ₁ and θ₃ will work as amplification/diminishing factors for EPS and XPSS in the equation for XPS. Note that using the function I(θ₁, 0, θ₂), we can obtain separate constant terms for three different insulation types, but the other remaining parameters are fitted together regardless of the insulation type.

The Bayesian parameter estimation results for this explanatory function are provided in Table 5. We first see that the mean value of σ has been reduced from 0.11975 to 0.10001, which means that the modeling error has been reduced after introducing different constant terms for different insulation types, that is, EPS, XPS, and XPSS. These different constant terms represent the different bonding and shear capacities of the three types of insulation in a statistical manner. It is also seen that the COV of θ₁ is very high (=3.2982) compared with that of other parameters and the mean value is 0.023853, which is close to 0. This means that the constant term for the EPS insulation type can be taken as 1 either because it is the least informative term or it is close to 1 based on its mean value, that is, 2^0.023853 = 1.0167.

Table 5.

Posterior means and COV of θ_i using equation (17).

	θ ₁	θ ₂	θ ₃	θ ₄	θ ₅	σ
Corresponding explanatory term	ln(2) for EPS	ln(2) for XPSS	ln(T_fs,t)	ln(t/t_s)	$\ln ({f'}_{c})$	ln(ε)
Explanation	Constant term	Constant term	Grid shear flow strength	Insulation thickness ratio	Concrete compressive strength	Model error
Posterior mean	0.023853	0.39151	1.1308	−0.64561	−0.25999	0.10001
Posterior COV	3.2982	0.27919	0.077166	0.57554	0.43025	0.13498

EPS: expanded polystyrene; COV: coefficient of variation.

Based on the results of equation (17), we propose the following simplified deterministic formula to predict the shear flow capacity of a sandwich panel including shear grid layers by adopting the posterior mean values of the parameters θ_i

q_{max} = C_{IT} \cdot T_{fs, t}^{1.131} \cdot {(\frac{t}{t_{s}})}^{- 0.646} \cdot {f'}_{c}^{- 0.260}

(18)

where C_IT = 1.000 for EPS, 1.000 for XPS, and 1.312 for XPSS.

In this equation, the power term of T_fs,t has a mean value greater than 1.000 because there is no consideration of possible strength additions such as the resistance of compression cords until buckling. Also, this power term includes a scaling factor for the remaining terms. The power terms of (t/t_s) and ${f'}_{c}$ with negative values do not have significant physical meaning and merely contribute statistically to the reduction of fitting error. This is shown in their values of COV, which are much greater than the power term of T_fs,t. It is seen that the C_IT value for EPS is close to 1.000, which is somewhat greater than that of XPS although the difference is negligible. One explanation for this is that although EPS has stronger bond resistance than XPS due to the adhesion between EPS and wythes by absorbing moisture, the strength of EPS insulation is generally lower than that of XPS, and EPS insulation does not grip shear grids as strongly as XPS. Therefore, the strength gain from the bond effect is canceled out by the lower strength of EPS insulation. This is consistent with the observation in the literature that although EPS has lower shear strength than XPS, EPS usually has a stronger adhesive bond between insulation and concrete than XPS (Woltman et al., 2013). In fact, XPS has almost no bond strength as evidenced by the bond test data in section “Database on shear strength of concrete sandwich panels reinforced with GFRP shear grids” that show the shear strength test results of sandwich panels without shear grids. However, there is still a slight gain in the power term of T_fs,t due to the shear strength of insulation, which adds to the strength of the shear grid by gripping it and limiting deformation. If there is no shear grid, the XPS panel will fail mainly due to bond failure; however, a shear grid will protect XPS insulation from bond failure and the insulation shear capacity will also contribute to the shear capacity of the entire panel. In XPSS, both the gripping effect and bond resistance are applied together, and both make a significant contribution to the increase in shear strength as shown in the corresponding C_IT value of 1.312. The bond capacity between these concrete wythes and the capacity of insulation are not easily quantifiable because of difficulties in experimental measurement of their individual contributions and a limited amount of experimental data.

We can further simplify equation (17) by removing some uninformative terms, such as t/t_s and ${f'}_{c}$ . Although they have a greater contribution than the constant term for EPS, it is still worthwhile quantifying their actual contribution to the accuracy of the equation by monitoring the change of the mean of σ after removing them as follows

γ (x, θ) = \ln [2^{I (θ_{1}, 0, θ_{2})} \cdot T_{fs, t}^{θ_{3}}]

(19)

Bayesian parameter estimation is repeated using equation (19), and the results are reported in Table 6. We first observe that the mean of σ in Table 6 does not increase greatly compared to the value in Table 5 (from 0.10001 to 0.10928), although the form of the equation is much more simple. This means that dropping the explanatory terms t/t_s and ${f'}_{c}$ does not considerably sacrifice the accuracy of the equation and these terms can therefore be dropped for further simplification. Based on equation (19), for practical use, we propose the following simplified deterministic formula to predict the shear flow capacity of a sandwich panel including shear grid by adopting the posterior mean values of the parameters θ_i

q_{max} = C_{IT} \cdot T_{fs, t}^{1.044}

(20)

where C_IT = 1.000 for EPS, 1.000 for XPS, and 1.339 for XPSS. Equations (18) and (20) need to be carefully used when they have parameter values that significantly exceed the range of the parameter values in the test database used in this study.

Table 6.

Posterior means and COV of θ_i using equation (19).

	θ ₁	θ ₂	θ ₃	σ
Corresponding explanatory term	ln(2) for EPS	ln(2) for XPSS	ln(T_fs,t)	ln(ε)
Explanation	Constant term	Constant term	Grid shear flow strength	Model error
Posterior mean	−0.024782	0.42084	1.0443	0.10928
Posterior COV	2.4598	0.22721	0.004641	0.13429

EPS: expanded polystyrene; COV: coefficient of variation.

Calibration of safety margins for proposed shear capacity models

The proposed formula provides improved accuracy compared to considering only the capacity of shear grids, but there are still uncertainties in terms of modeling error due to the inaccuracy of the proposed formula, parametric error due to the uncertainties contained in the nominal values of the design parameters, and statistical error due to the insufficient number of data. Consideration of these uncertainties and determination of a proper safety margin are essential for the application of the proposed formula to design practices.

Using the calibration procedure summarized in section “Safety margin calibration procedure using a Eurocode method based on test observations,” the capacity reduction factors for equations (18) and (20) are calculated for varying target reliability indices (β) from 1.5 to 4.5 in Figure 6(a) and (b), respectively. In this analysis, the values of the COVs of input parameters are as follows: 5% for the tensile strength of a strand (Neocleous et al., 2001), 15% for the compressive strength of concrete, and 1% for all geometry-related parameters (JCSS, 2011). The plotted values are the average of the capacity reduction factors for all test data. Note that this study is undertaken based on the assumption that only the resistance effect is separately considered and the uncertainties in the load effects and the corresponding load factors are ignored. Therefore, the target reliability index β is chosen for the partial effect of resistance, which is represented as the product of the FORM sensitivity factor α_R = 0.8 and the target reliability index for both resistance and load effects (β_t). Therefore, the range of β = 1.5–4.5 covers that of β_t = 1.875–5.625. The calibration is repeated for the following two different assumptions for nominal shear strengths of a sandwich panel: (1) the nominal strength of a sandwich panel is calculated by subtracting three times the standard deviation of maximum shear strength measured from the test database from the mean predicted shear strengths, similar to the AC422 (ICC Evaluation Service, 2010) and (2) the nominal unit tensile strength of a shear grid is calculated by the 5% fractile of its probabilistic distribution where the COV of a GFRP strand is taken as 0.05. The 5% fractile, which has a value close to the nominal strength, is often used to estimate the characteristic strength of materials such as steel and concrete. Figure 6(a) and (b) shows results for these two different assumptions for the nominal shear strength; in both figures, the capacity reduction factors decrease with the target reliability index because more safety needs to be provided to meet the higher target reliability level. The auxiliary lines shown in the figures represent the capacity reduction factor for the target reliability index β = 3.04 (=0.8 β_t), which corresponds to the value of β_t = 3.8 used to calibrate the capacity reduction factors of structural design equations according to ISO 2394 for ultimate limit state design.

Figure 6.

Capacity reduction factor for two different nominal shear strength assumptions: (a) equation (18) and (b) equation (20).

For these two assumptions for nominal shear flow strengths of a sandwich panel, the capacity reduction factors are calibrated as 0.75205 and 0.75709 in Figure 6(a) and 0.74987 and 0.74944 in Figure 6(b), for the target reliability index β = 3.04. These values show that the capacity reduction factors for both equations (18) and (20) are very similar even though the modeling error of equation (18) is smaller than that of equation (20). This is because equation (18) has a more complex form than equation (20) because of two additional input parameters, and the additional parametric uncertainties of these two input parameters offset the accuracy of the model.

This is clearly shown in Figure 7(a) and (b), where the capacity reduction factors of equations (18) and (20) for the target reliability index β = 3.04 are plotted for each of the sandwich panels in the test database. Figure 7(b) shows very mild fluctuations compared to Figure 7(a) because it has smaller parametric uncertainties. Since the modeling error is a constant in this study for all test data and only the parametric error changes for each test data, the fluctuations are affected only by parametric errors and a larger fluctuation means a larger parametric error.

Figure 7.

Capacity reduction factor for each test result: (a) equation (18) and (b) equation (20).

It is also seen that the capacity reduction factors calculated in both equations (18) and (20) are less conservative than the suggested values in AC422, where the suggested total capacity reduction factor is 0.75 (capacity reduction factor) × 0.85 (additional capacity reduction factor) = 0.64, which is used when we rely purely on the test data without shear capacity prediction models.

Figure 8(a) and (b) shows the predicted shear strengths of the sandwich panels using equations (18) and (20) in the test database, which confirm the performances of the developed probabilistic model. The 32 test results are rearranged in ascending order of the mean predictions by the probabilistic models. The mean curve is plotted with a shaded area representing mean ± 1 standard deviation interval based on the modeling errors, where the standard deviation is calculated considering both modeling and parametric errors. This interval covers approximately 68% of the probability distribution of the strength. The mean curves of the developed probabilistic models successfully represent the central tendencies of the observed shear strengths. This means that the both probabilistic models successfully achieve unbiased predictions. It is also seen that the majority of the test data fall within the mean ± 1 standard deviation intervals for both models, although Figure 8(a) shows a slightly smaller number of test data outside the shaded area compared to Figure 8(b) due to a slightly smaller modeling error.

Figure 8.

Unbiased shear strength model and design model: (a) equation (18) and (b) equation (20).

In addition, based on the calculated capacity reduction factors for both models with extra safety to consider the uncertainties in test data and the proposed framework, a capacity reduction factor of 0.7 is constantly applied to both models and the corresponding safe design predictions are represented by a dotted line. These dotted lines show conservative design prediction because they are far below the mean curve. As these lines target at a very small failure probability of 0.0012 with the corresponding reliability index (β) of 3.04, none of the 32 test results fall below the design curve.

Conclusion and discussion

This article proposed a statistical framework for the development of a design model for a concrete sandwich panel with GFRP shear grids. The framework was established by integrating the Bayesian parameter estimation method and the Eurocode-based capacity reduction factor calibration method. Prediction and design models for the shear flow strength of concrete sandwich panels with GFRP shear grids were developed based on 32 experimental data from shear strength tests. From the database, three parameters potentially contributing to the shear flow strength of sandwich panels were selected: the tensile strength of GFRP grid strands, the thickness of insulation, and the compressive strength of concrete. This framework enabled accurate quantification of the importance of the selected parameters in terms of their posterior coefficients of variation. From the developed prediction model, the GFRP shear grids were shown to have a dominant contribution to the shear flow strength of sandwich panels as expected, but the other parameters related to insulation and concrete properties also contributed. By introducing different constant terms for three different insulation types—EPS, XPS, and XPSS—the prediction error of the developed model was reduced. The posterior mean values for the constant terms were 1.017 for EPS, 1.000 for XPS, and 1.312 for XPSS. XPSS had the greatest value because of its enhanced bond effect and high insulation shear resistance. Further simplified probabilistic and deterministic equations were developed using the same framework by dropping relatively less informative parameters that did not significantly sacrifice the model accuracy. The capacity reduction factors for the proposed deterministic formulas were also calculated for design purposes. For the target reliability index β = 3.04 when ignoring the load effect, the capacity reduction factor was determined to be approximately 0.75 for both models, but a lower value of 0.7 can be used by considering extra safety. This value is less conservative than that suggested in AC422, which purely relies on the test data without shear capacity prediction models. With further collection of data, this framework could be used to improve the proposed models in order to expand their applicability to wider parameter ranges and to clearly quantify the contribution of parameters. The proposed framework can be extensively applied to sandwich panels using other types of materials or shear connectors, but test data, material property, and identification of parameters included in the bias-correction function need to be properly addressed.

Footnotes

Appendix 1 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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Australian Research Council (ARC) under its Linkage project (Project No. LP140100030) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2014-11-0869).

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