# A robust design of an innovative shaped rebar system using a novel uncertainty model

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## Abstract

The current paper has investigated a newly developed re-bar system by implementing uncertainty models to optimise its geometry. The study of the design parameters of this re-bar system has been carried out utilising a novel uncertainty model that has been developed at Swansea University. The importance of this invention comes from the fact that the whole process of optimisation has been automated by linking ANSYS Workbench to MATLAB via the in-house written code, Despite the fact that in the past, ANSYS APDL was linked to MATLAB, however, the APDL was very limited to only simple geometries and boundary conditions unlike the Workbench which can simulate complex features. These shortfalls have been overcome by automating the process of optimisation, identifying the key influential parameters and the possibility to carry out a huge number of trials. Moreover, the tools that have been developed can pave the way for robust optimisation of this proposed structure. The uncertainty in the design parameters of this re-bar system is of a paramount importance in order to optimise the bond strength between the newly developed rebar and the concrete matrix as well as to fully understand the behaviour of the proposed system under pull-out conditions. The interface between the rebar and the concrete matrix was considered as a ‘cohesive zone’ whereby the interfacial area is studied as a function of the bonding strength.

## Keywords

Robust design Rebar systems Bond strength Concrete Uncertainty ANSYS MATLAB## 1 Introduction

The main scope of the current paper is to explore the sensitivity of the various parameters in the newly developed rebar system. That is, the performance of the bonding strength between the rebar system and the surrounding matrix is influenced by many parameters such as the outer radius of the rebar system, the groove radius, the distance between the centre of the rebar shape to the centre of the cutout shape, the radius of the edges alongside the angle from one edge to the successive edge within the same groove. This exercise will employ uncertainty studies using an innovative model that has been developed at Swansea University. In this regard, it has become possible to link ANSYS Workbench via MATLAB using an in-house written code that has been examined and provided precise results. Yet, this paper presents the first publication in a series of future publications utilising this novel method which can open new eras of collaborative research in future. The advantage of this developed routine is that it allows the user to enter as many input parameters as required whereby the code will automatically operate the ANSYS, run the model, get the results and feedback to obtain other results. This means that any parameter, such as the outer radius in the newly developed rebar system, can be chosen as a set of random values within a defined range from which the bond strength, for instance, of the structure can be evaluated. This allows the possibility of choosing the most sensitive parameters alongside the values that provide the optimum bond strength. This will reduce the time, cost and number of trials to choose the optimum design parameters alongside the use of mathematical tools such as the Meta-models in conjunction with this code for efficient uncertainty analysis.

## 2 The employed parameters in the design

A summary of the materials properties utilisied in the current study (Shafaie et al. 2009)

Material Properties | Values (kg/cm |
---|---|

Concrete compressive strength | 300 |

Concrete tensile strength | 30 |

Concrete E modulus | 273,664 |

Concrete Poisson’s coefficient | 0.2 |

Steel E modulus | 2,100,000 |

Steel yield stress | 3000 |

Steel Poisson’s coefficient | 0.3 |

The experimental results carried out elsewhere (Shafaie et al. 2009) on standard steel bars bonded to concrete structure under pull-out conditions were utilised in this paper in order to verify and validate the generated model that will be applied to the Co-tropic rebar system. In the same study of Shafaie et al., a steel bar of 12 mm diameter with a cross sectional area of 113 mm^{2} has been tested. The steel bar was contained within a 90 mm high cylindrical concrete structure of 60 mm diameter with an anchorage depth of 60 mm. The obtained results will be compared with those obtained for the Co-tropic rebar system such that optimisation and uncertainty studies can thereafter be carried out.

## 3 The ANSYS model

_{tangential}is the tangential force between the surfaces,

*K*

_{tangential}is the tangential stiffness between the surfaces and

*Χ*

_{sliding}is the sliding distance as a result of the applied force. The value of

*Χ*

_{sliding}is ideally zero for sticking conditions, however, some slip is allowed in our case. This will require chattering control parameters as well as a maximum allowable elastic slip (ELSI) parameter (i.e. K

_{tangential}), (Doan 2013). In addition to the above boundary conditions, the ‘cohesive zone material’ model has been used to model the delamination process of the interface (i.e. debonding). The adhesion properties of the utilised adhesive were entered via the ‘cohesive zone material model’ with ‘bi-linear’ behaviour mode and these were allocated for the contact elements of the model. To define a bi-linear material’s behaviour of adhesion, the separation distance and the constant properties of the adhesive material, the TBDATA command in ANSYS was used. In this case, the properties of the concrete have been defined by the user (ANSYS 2009; ANSYS 2012).

## 4 The robust design and uncertainty analsyis

Research studies by Bryne and Taguchi and colleagues represent the first efforts in developing robust designs. They have introduced methods to minimise the effect of uncontrollable parameters during the design stage (Bryne 1987; Taguchi 1989). Further studies by Ross and colleagues employed the Taguchi loss function to make the design more tolerable to model variations (Ross 1995). Other researchers proposed methods to reduce the variations in input parameters to obtain designs with lower sensitivities to design parameters (Ramakrishnan 1991). They have suggested a method for robust design with the Taguchi loss function as the object that is subjected to the model constraints. This allows the constant and variable sensitivities from controllable and uncontrollable parameters to be reduced using non-linear analysis. On the other hand, Padulo has investigated two main approaches for robust optimization in which the parameters are stochastic. The purpose of uncertainty in this case was to identify the uncertainties in input and output of a system or simulation tool (Padulo 2008). In structural analysis, it has become essential to determine the relationship between the various parameters with respect to the component geometry, the applied load, the material properties, and the contour conditions. In general, the main sources of uncertainty are associated with the properties of the adhesive, the geometry, material, load direction alongside many other factors (Neto and Rosa 2008). Other scholars have extended the use of uncertainty models on various materials and structures which has facilitated the design and optimisation exercise of structures (Wang and Al 2018a), (Wang and Al 2018b), (Wang and Al 2017a), (Wang and Al 2017b).

In this paper, the shape of the bar has been explored for the analysis. Various parameters have been investigated after which the most sensitive ones have been considered. In this context, 1000 random runs of the chosen parameters have been carried out from the space of input parameters. The utilisation of 1000 trials was based on the fact that beyond this number the results would make no significant improvement as they have already converged at around the 1000th trial. The Monte Carlo simulation approach has been employed to obtain the corresponding 1000 outputs which represent the bonding strength of the structure. Afterwards, the Kernel probability distribution function has been estimated from the sample data using the Kernel Smoothing density function in MATLAB (R2013b). Regions of the acceptable output bonding strength were defined such that the bonding strength in those regions is desired for the design. The new advancement in the current paper is the creation of an automated script file that allows the designer to modify the parameters of complex geometries without the need to work with the ANSYS environment. That is to say, the MATLAB and ANSYS workbench interact with each other and the parameters are modified following this approach. This allows more flexibility to deal with complex geometries since this was, in the past, only restricted to simple designs.The framework for robust design proposed and employed by the same authors on carbon fibre composite materials bonded to aluminium connectors has recently been published elsewhere (Aldoumani et al. 2016). The code can be used for any future collaborative work on any engineering application that involves uncertainties in the design, manufacturing and operating conditions.

## 5 Results and discussion

### 5.1 The validity of the employed model

### 5.2 The investigated parameters in the co-troipc rebar system

### 5.3 The sensitivity study of the parameters

^{2}in comparison to 312.0 mm

^{2}when R1 was taken as 12.0 mm. In other words, it is clear that any increase in R1 will always increase the bonding strength due to the increased outer surface area and hence the interfacial bond. For this reason, the variation of R1 was not considered in the analysis since it only provides basic information that can be easily drawn without the need for any further simulation.

^{2}whereas the black coloured text is related to that calculated when R2 was taken as 2.11 mm with a cross sectional area of 88.87 mm2. In terms of bond strength, the reduction of R2 has resulted in a reduction in the interfacial bond strength which is undesirable. This means that the gripping effect continuously decreases when R2 is decreased. Overall, when looking at the whole graph, it can be seen that the effect of reducing R2 has a minimal effect on the bond strength. In other words, the value of the bond strength is not significantly sensitive to the change in R2 which makes this parameter of less interest in the current analysis. The chosen range between 2.0–3.0 mm ensures that the ‘butterfly’ shape of the rebar remains the same as beyond this range the geometry will be modified which is undesirable.

^{2}(i.e. the black coloured measurement).

^{2}whereas that which has been modified with sharper edges, i.e. smaller R4, is plotted in black. The modification of R4 has caused a slight modification to the overall rebar design; however, the basic geometry of the dovetail is maintained due to the applied constraints.

^{2}. It can be seen that the decrease in R4 leads to a significant improvement in terms of the bond strength reaching a value of 13.8 MPa. Similarly, the increase in R4 starting from the initial geometry has also resulted in an increase of the bond strength reaching a value of 13.5 MPa. This means that this parameter can improve the bond strength without causing an excessive increase/decrease in the cross sectional area as can be seen in the three circled trials in Fig. 14. This makes this parameter worth investigating in terms of uncertainty and robust design as will be shown later in the current study.

*α*’ as shown in Fig. 15 (a). This angle defines the amount by which the groove closes around the concrete. The ‘closing’ effect of the grooves can be understood in Fig. 15 (b) wherein the increase in the groove angle

*α*brings the edges closer to one another, e.g. the black geometry in Fig. 15 (b) has an angle

*α*= 41o when compared to the original geometry in blue of an angle value α = 20

^{o}.

*α*is shown in Fig. 16 by plotting the bond strength against this parameter. It is apparent that with increasing the angle

*α*, the gripping effect also increases as indicated by the trend line, i.e. dashed line, leading to an increase in the bond strength. This effect takes place whilst the cross sectional area remains almost constant which is a great advantage to maintain the same geometry. Yet, the limitation of this parameter is that the central part of the bar becomes significantly thinner when increasing the value of

*α*which will result in an increased stress at the central part of the rebar. This means that the bar itself might fail at a much lower stress than anticipated. Despite the fact that this might be the case, the scope of this paper is to investigate the overall bond strength and not the mechanical properties of the rebar which might be another topic for future studies. The change in

*α*has resulted in a significant improvement of the bond strength and therefore, this parameter will be considered for further exploration in the current work.

### 5.4 The uncertainty and optimisation of the chosen parameters

^{2}. The range of variations for the uncertain parameters, i.e. R4 and

*α*, utilised in the current analysis is shown in Fig. 14 and Fig. 16 . From the probability distribution, Fig. 17 it is clear that more than one third of the examined samples have shown a bonding strength of about 12 MPa exceeding that found in the literature for the conventional rebar systems of 10 MPa (Shafaie et al. 2009). This means that the employed model in this report agrees well with that of the conventional systems and provides satisfactory results. On the other hand, the change in

*R4*and

*α*have also provided very strong bond strength reaching a value of around 15 MPa which is very desirable. For the purpose of uncertainty and optimization, the acceptable level in the current investigation was taken above 15 MPa. This will reduce the number of samples that provide an acceptable level of bonding strength as well as provide the best bond performance which is higher than that obtained by standard rebar systems.

^{o}.

^{o}. From a materials point of view in terms of cost saving, the value of R4 between 1.0–1.2 mm has resulted in a reduced cross sectional area of 105.15 mm

^{2}when compared to the original area of 113mm

^{2}or to the area 123.54mm

^{2}when R4 was 0.1–0.4 mm. This means that the values of R4 that will be considered will be those between 1.0–1.2 mm. This reduces the problem of optimisation and makes it simple to run the robust analysis as it will be discussed later.

### 5.5 The meta-model design optimisation

The Meta-model-based design optimisation is becoming increasingly popular in the industrial practice for optimisation of complex engineering problems, especially to reduce the burden of computationally expensive simulations. The idea behind the Meta-model-based design optimisation is to build a surrogate model (or a meta-model) from a reduced number of simulation runs and subsequently use the model for optimisation purposes (Gano et al. 2006). The surrogate model, i.e. y = f(x_{1}, x_{2}…, x_{n}), approximates the relationship between the design variables, i.e. x_{1}, x_{2}…, x_{n}, and the output variable, y. This method can speed up the design optimisation process since the function evaluations of the surrogate model are less expensive to execute when compared to deterministic simulations. The simplest type of ‘Response Surface’ is a linear model in which the functional relationship f(x_{1}, x_{2}…, x_{n}) is assumed to be a linear function of the design variables. Linear models can be extended to polynomial response surface models wherein the response surface is a polynomial function of the design variables. In either way, the linear or higher order response surface (polynomials) can be obtained using the ‘Ordinary Least Squared’ approach by minimising the sum of the squared distances of a given data points from the surface. In this case, the surrogate modelling utilising the least squared approach assumes that all errors are normally distributed with given mean and variance. This assumption is often too stringent in real-world problems.

One of the most popular methods that falls under the Meta-model approaches is the Kriging (or Gaussian process interpolation). This is considered a surrogate model that is able to approximate the deterministic noise-free data and has proven to provide high level optimisation results alongside design space exploration, visualisation, prototyping and sensitivity analyses (Booker et al. 1999). The Kriging approach is an interpolation technique which differs from the conventional least squares approach as its model goes through each calculated point. With the Kriging method, it is possible to describe the uncertainty of the interpolation outside the given points (Ulaganathan et al. 2015). The widespread adoption of the Kriging method is due to the ability to approximate complex response functions (Martin and Simpson 2005) and less restrictive assumptions, compared to the least square method, on distribution of residual errors.

#### 5.5.1 The ordinary kriging model

_{i}(x) = b

_{1}(x), b

_{2}(x),…, b

_{p}(x), are the basis functions (e.g. the power base for a polynomial) and α

_{i}= (α

_{1,}α

_{2}… α

_{p}) denote the coefficients. The idea is that the regression function captures the largest variance in the data (the general trend) and then the Gaussian Process interpolates the residuals. In fact, the regression function f(x) is actually the mean of the broader Gaussian Process Y.

#### 5.5.2 The blind kriging model

_{1},…,β

_{t}). The estimation of β provides a relevance score of the candidate features. A frequentist estimation of β (e.g., least-squares solution) would be a straightforward approach to rank the features (e.g. the least-squares solution) would be a straightforward approach to rank the features.

#### 5.5.3 The co-kriging model

The Co-Kriging, a special case of multi-task or multi-output Gaussian Processes, exploits the correlation between fine and coarse model data to enhance the predictive accuracy (Kennedy & O’Hagan, 2000). Generally, creating a Co-Kriging model can be interpreted as constructing two Kriging models in sequence: a first Kriging model of 100 samples (the coarse data) followed by a second Kriging model constructed on the residuals of the 1000 samples (fine and coarse data). This is a useful technique since it uses a small set of samples to predict the long-term properties. This will be useful to save time and cost in relation to the required data since it is able to predict the overall behaviour within and outside the given range of properties.

These methods, i.e. the Ordinary, Blind and Co- Kriging will be employed in the current study to provide a comparative study of all techniques so as to capture the optimum regions of bond strength in the given case study. The purpose is to validate the obtained data as well as to provide the optimum and robust design of the parameters. These methods can be used to find the surrogate model that approximates the solution to the problem since it employs less stringent assumptions about the residual errors and they are able to model complex systems. These models were built using the Design and Analysis of Computer Experiments (DACE) software toolbox (Couckuyt et al., 2013). The model DACE software package is a freely available toolbox which preforms both calculations of the Kriging function and parameters optimisation. The Meta-model found using the various Kriging methods can then be used to identify the optimal regions of the bond strength. Subsequently, a global optimum zone can be found by applying the same methodology to the optimal regions.

### 5.6 The results of the various kriging applied to the re-bar system

Figure 22 (a), indicates a similar trend and provides similar optimum regions with groove angles between 50 and 70o and R4 values of 1–1.2 and 0–0.3 mm, Fig. 22 (b), it was unable to capture all the actual data set when compared to the Blind Kriging approach. The Blind Kriging provided a wider range for the optimum bond strength of 14 MPa when compared to the Ordinary Kriging. Moreover, the variance plot shows higher variance values especially at the corners, Fig. 22 (c), where the two optimum regions are located. This shows the inferior predictability as well as the low robustness obtained when this method was employed. When the Co-Kriging method was utilised, the poorest quality of the fitting was observed as shown in the response surface, Fig. 23 (a). It can be clearly seen that most of the points were not captured by the model along with a less robustness, i.e. range of the highest bond strength, obtained, Fig. 23 (b). Moreover, a huge amount of variance is observed when the variance contour plot was generated, Fig. 23 (c), almost in all regions

The associated errors for the Ordinary, Blind and the Co-Kriging methods using the Mean Squared error analysis of the Leave-Out Cross Validation

A Summary of the Associated Errors in all Methods | ||
---|---|---|

Ordinary Kriging Method | Blind Kriging Method | Co-Kriging Method |

200.0034 × 10 | 0.0039 × 10 | 6430.4812 × 10 |

### 5.7 The optimum and robust Design of the Rebar System

^{o}. The variance is mostly negligible as can be seen in the variance contour plot, Fig. 24 (c).

^{o}. The variance contours, Fig. 25 (c), show a minimal amount of variance for the fit which is desirable for the robust design.

## 6 Conclusions

The Co-tropic rebar system has been investigated in terms of the optimum and robust design of the various parameters involved in its geometry. The sensitivity studies of the system have shown that the system is most sensitive to two parameters, namely: the groove angle α and the edge radius R4. These two parameters were thoroughly studied using uncertainty models and robust design analyses. The new advancement in the current study is the employment of a novel approach that links the ANSYS Workbench with MATLAB to generate thousands of data that have a normal distribution (using the Latin hypercube method). The best combination of α and R4 that has provided the maximum bond strength of 14 MPa with the concrete was identified. In order to better understand and validate the obtained results, the Kriging method was utilised to create the response surface. This approach has provided the robust design in 3D combination between the bond strength, α and R4. It has been observed that the best groove angle α lies in the region between 60^{o} and 70^{o} with edge radii of either 0.2–0.25 or 1.0–1.1 mm. This analysis was a very useful exercise that has employed the various uncertainty and robust design technique in order to optimise such a structure. This approach can be applied to any engineering application in order to save time and cost associated with such simulations. The predictive capability of the methods were assessed and compared against each other to increase the level of confidence in the obtained results.

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## Notes

### Acknowledgements

The funding provided by the ASTUTE 2020 project (Advanced Sustainable Manufacturing Technologies) based at Swansea University, the European Regional Development Fund through the Welsh Government and the Welsh European Funding Office (WEFO) is highly appreciated.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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