The Ultrasonic Pulse Velocity and Lagrangian Approaches to Predict the Effective Thickness and Homogeneity of the Sandwich Panel

, 80 mm, and 90 mm, respectively. The test results were analyzed on travel time and wave velocity using a statistical analysis approach including covariance, Kolmogorov-Smirnov, ANOVA, t-test, and Lagrangian. The analysis results show that the mixture's homogeneity can be determined based on the ultrasonic pulse velocity. The proposed Lagrange analysis can reveal the behavior of the propagation speed. Based on the results of the Lagrange approach, the highest speed is obtained at a thickness of 80 with a maximum speed of 2.395 km/s. The results of this study contribute to the non-destructive test procedure, especially in determining homogeneity and the dimensions of the effective thickness of the structural walls (cores and layers) that have been installed in the field quickly, cheaply, accurately, and briefly.


Introduction
Lightweight concrete uses lightweight aggregates with a specific gravity ranging from 600-1600kg/m3 [1]. Using lightweight materials can reduce the structure's weight, which impacts lateral force reduction, especially in earthquakes. Two types of lightweight aggregates are natural (pumice, volcanic ash) and artificial. One of the artificial lightweight aggregates is EPS (expanded polystyrene) [2]. EPS is the most widely used material because it is extremely light [2]. The EPS cutting method in powder form is assumed to replace the fine aggregate in the sandwich wall mixture. EPS is necessary because this material cannot be decomposed and can even become microplastics that can pollute the environment [3].
Furthermore, EPS also has advantages, including that it does not contain chlorofluorocarbons (CFCs) or hydrochlorofluorocarbons (HCFCs) [2] [4], has mechanical properties that can insulate thermals and can be made in such a way that it is easy to use as a composite mixing material on wall panels or floors. EPS has characteristic properties, including good heat conductivity, sound damping, and water absorption. Therefore, it is believed to be an alternative for developing environmentally friendly materials with a percentage greater than fifty percent of the volume of sandwich panel walls.
Presently, the type of EPS testing includes material modification in concrete and mechanical characteristics related to mechanical, chemical, or thermal [2]. Babu et al. [4] propose a combination of EPS and fly ash additives with compositions of 0% up to 66.5% and 50%, respectively, with varying densities between 0.550 g/cm 3 up to 2,200 with compressive strength ranges varying from 0.55MPa to 22MPa. Another researcher, Tabatabaiefar [5], analyzed the distribution of stresses in sandwich panels on flexural loading, the effect of moisture content on the mechanical properties of the panels, and revealed the creep load.
Priya and Sivakumar [6] analyze the material property of EPS as a bearing wall. The focus of the study refers to the mechanical properties (compressive strength and flexural strength) and thermal conductivity due to the role of EPS in the wall mixture. Variations in using EPS are determined in the range of 0% to 100% with an interval of 20. The test results recommend that using EPS at 60% of the wall volume will reduce structural weight and thermal well. Meanwhile, Matsuo [7] reported the other side of the mechanical properties of EPS walls besides compressive strength, namely resistance to infrared radiation. Research continues to be developed towards the use of steel composite materials [8], microstructural investigations with other additives [9], stress and strain behavior, combination with bricks [10], FEM modeling between EPS and concrete block [11], and feasibility studies [12]. The flexural strength test performance was also developed to obtain optimal results regarding the lightweight behavior of foam concrete sandwich panels under axial load. [13], the utilization composite concrete filled circular steel tubes [14], mixed compositions for high-strength concrete using polystyrene beads [15], recommendations for the use of hybrid foamed concrete sandwich panel technology for the precast concrete industry [16], modification of sandwich panel with double shear truss connectors to increase the flexural performance [17], laboratory scale precast concrete sandwich panel for flexural loading test [18], the resistance of lightweight panel during exposed to flame [19], and The new technique expanded polystyrene as an aggregate to prevent segregation during the material mixing process and loss of weight [20].
Although extensive research and modifications have been carried out on EPS, the main problems of lightweight composite materials related to homogeneity and prediction of the most effective thickness have not been discussed. This is important to demonstrate because sandwich walls made of EPS are significantly affected by the mixing process, which can affect quality. It is important to determine the most effective thickness because inappropriate layer thickness can result in rapid wall damage. Therefore, one of the proposed methods to measure material homogeneity is ultrasonic pulse velocity (UPV). The direct method is rightly implemented because it has a relatively high level of accuracy compared to indirect and semi-direct [21] [22].
The level of accuracy of readings is influenced by several factors, including the influence of aggregate dimensions, density, voids in structural components, material homogeneity, transducer distance (transmitter to receiver), and the consistency of the coupling agent as a medium to facilitate the distribution of ultrasonic signals or waves. Surface leveling and initial calibration must be carried out to obtain the material's specific characteristics to minimize interference due to readings.
Based on the initial findings, the test results show instability due to the influence of thickness, core, and layers. The difficulty increases with reading errors because the waves are not in direct contact (transmitter to receiver). As an initial assumption, the waves are predicted not to radiate straight, but there is a bias due to the use of seventy percent polystyrene. Analysis with an ultrasonic pulse velocity approach aims to determine the homogeneity, covariance, and effect of core and layer thickness on velocity. In contrast, the mathematical approach (Lagrange) aims to find new information about the effectiveness of reading and predicting the thickness of the core layer and an adequate layer to strengthen the experimental test results.
This study aims to determine the homogeneity of the material and predict the dimensions of the Expanded Polystyrene (EPS) core and layer. A direct test scheme is applied to obtain the travel time of the data distribution from ninety test points. The reliability of the test was checked through several stages, including the covariance test, homogeneity, Kolmogorov-Smirnov test, and t-test. The results obtained from the analysis are pulse velocity, which will be used as the basis for determining material quality decisions and predicting the dimensions of the core wall and layer. Therefore, this research contributes to the nondestructive test procedure, especially in determining homogeneity and the dimensions of the effective thickness of the structural walls (cores and layers) that have been installed in the field quickly, cheaply, accurately, and briefly.

Research Method
This study uses an experimental method with quantitative descriptive data analysis.
The research was conducted at the structure and materials laboratory, Faculty of Engineering, Universitas Negeri Yogyakarta. The research phases start from (1)

UPV Methods
The method used in this test uses a direct approach, with the test formation in Figure   2. In contrast, the ultrasonic pulse velocity test refers to [23]. Based on European standards EN 12504-4, BS EN 12504-4 [24], two transmission methods (transducer and receiver) is applied with a frequency transmission system of 54kHz. UPV testing methods are divided into three: (1) direct, (2) indirect, and (3)

Statistic test and Lagrangian
Statistical testing consists of covariance, Kolmogorov-Smirnov, ANOVA, and the t- to support the UPV test and analyze an optimization function for the velocity at each stage of the direct test. Therefore, the presence of EPS-interfering media will be transformed into a function without constraints, and the concern of thickness prediction can be decrypted.

Pulse Velocity
The result of the ultrasonic test is the travel time. These values are analyzed to produce the wave propagation (pulse velocity), the analysis results are presented in Table 3. In this study, velocity is a representation of density and homogeneity. Based on Table   3, SP70A,B SP80A,B and SP90A,B the average are 1.065 km/s, 2.018 km/s, and 2.015 km/s, respectively. Pulse velocity decreases using 70 mm and 25 mm thick core and layer layers. The wave speed will be high if it passes through solid media [23]. However, based on the test with a larger layer thickness on the SP70A,B specimen compared to other test objects, there is a disturbance in the speed of propagation which means that there is a variety of wave refraction patterns. This can be caused by the density of the test object and wave refraction due to the influence of Styrofoam (EPS) material.
In the exact mechanism, increasing the core dimensions to 80 mm and 90 mm can increase the speed by 25.74% and 25.55%, respectively. This indicates that the core dimensions significantly affect the ultrasonic wave propagation speed value. Attempts to thicken the layers will refract the waves and reduce the velocity value, depending on the predetermined compressive strength. Therefore, several possible causes for the decreasing velocity value include the layer mass density, which has a linear effect on the strength of the material.
Efforts to increase the core thickness do not change the quality of the wave propagation speed. The decrease in velocity indicates that the optimal values are obtained for the core dimensions and layer thicknesses of 80 mm and 20 mm. This value is an interesting parameter in this study, which is proposed to use mortar layer thickness in construction projects, especially in lightweight composite wall materials. Although SNI 2847-2019 [1] requires an optimum thickness of 15 mm, this does not apply to lightweight concrete combined with conventional mortar.

Material Homogeneity
The material mixing process has a higher level of difficulty compared to conventional concrete. Adding styrofoam by 70 % of the volume of the mixture results in a loss of mixing mass (especially styrofoam). The stirring strategy is carried out by processing between water and styrofoam first, then adding to the dry mix using a special tool to achieve a good level of homogeneity.
Furthermore, the normality test of the data with ANOVA analysis was carried out to determine the level of the normal distribution with a significance level of 5%, as proposed by [21]. It is necessary because the assumption of the material forming the specimen consists of various characteristics (fine aggregate, water, polystyrene), and it is necessary to accomplish a homogeneity test [21]. The results in Table 4 show a good level of test variation with average scores below 10 % based on the significance level. This shows that the method of carrying out the work of mixing sandwich panel walls is acceptable (homogeneous). Furthermore, the analysis was carried out using the Kolmogorov-Smirnov method to determine the data distribution with a normal distribution.  continued for analysis because if the significance of the data to be tested has a significant difference from standard normal data, it means that the data is normal.

Statistic Test
The results of the Kolmogorov-Smirnov test showed that the data were normally distributed. In contrast, the results of the ANOVA analysis showed a value of 0.011 < 0.05, which means that the mixing material is homogeneous. Therefore, the test results are continued by analyzing the effect of core thickness on wave propagation speed. Two-tail t-test analysis was carried out with 0.026 < 0.05, which means that there is a significant effect due to differences in the thickness of the core and wall layers.

Lagrangian Approach
Lagrange analysis can be used to predict paired data in various applications, such as particulate two-phase flow [25], discrete particle behavior prediction [26], and coupled program finite element method [27]. In principle, if the data pair (xi, yi), 0 ≤ i ≤ n is available, then the Based on the experiment for a thickness of 70 mm, data is obtained as shown in Table 6.   For the thickness of 90 mm, the maximum speed is 1.912km/s as shown in Figure 5.
Source: Author's analysis (2022)   The Lagrangian approach emphasizes optimization problems that cannot be solved due to constraints that limit the objective function. Lagrange's characteristic approach is being able to transform the optimization problem due to the influence of two variables into an optimization problem without constraints. Therefore, the optimization problem can be solved as indicated by the effective core and layer thickness of 80 mm and 20 mm, respectively.

Conclusion
The analysis results show that the homogeneity of the mixture can be determined based on the ultrasonic pulse velocity, which is proven by ANOVA analysis with 0.011 < 0.05.
Meanwhile, the results of the t-test report that there is a significant effect due to differences in the thickness of the core and wall layers. The proposed Lagrange analysis can reveal the behavior of the propagation speed. Based on the results of the Lagrange approach, the highest velocity is obtained at a thickness of 80 with a maximum speed of 2.395 km/s. At the same time, cores with dimensions of 70 mm and 90 mm are not recommended for use because they have a low testing accuracy. New methods need to be developed to predict wave refraction due to the influence of EPS. The non-destructive tests using the Ultrasonic Pulse Velocity method and the Lagrangian numerical test successfully predict the sandwich panel's thickness. The results of this study contribute to the non-destructive test procedure, especially in determining homogeneity and the dimensions of the effective thickness of the structural walls (cores and layers) that have been installed in the field quickly, cheaply, accurately, and briefly.

Acknowledgement
The Financial Support from the Faculty of Engineering, Universitas Negeri Yogyakarta and Beihang University are sincerely acknowledged by the authors.