Parametric study on dynamic responses of stiffened sandwich composite bridge deck panel

Recently, the use of sandwich composites in different fields of engineering such as aerospace, marine, automobile, pipelines, bridge structure, industrial work, has attracted significant attention. Sensitivity analysis of structures made of sandwich composites is necessary to design them properly and maintain their longevity. The present study analyzes stiffened sandwich composite bridge deck panels and focuses on its sensitivity analysis. The lack of control in the manufacturing of the sandwich composites may lead to non-uniform material properties, and thus variation in its behavior. The variation in the dynamic responses obtained through the free vibration analysis of the bridge deck panel models of stiffened composites due to is studied. The free vibration analysis is implemented using a finite element method. The analysis is carried out with the stiffeners located in different positions and alignments. The glass fiber-reinforced plastic (GFRP) and polyvinyl semi-rigid foam are considered in the face and core layer for modeling the deck panels, respectively. The sensitivity of the bridge deck panels is also observed with the presence of holes of different diameters in the core layer of the sandwich composite plate without stiffeners and with a transverse stiffener. It has been noticed that dynamic response, i.e. the eigenvalue, is sensitive concerning the in-plane parameters of the face layers compared to other parameters. Moreover, an increase in the size of the hole in the core layer results in a decrease in the dynamic response of the stiffened sandwich composite bridge deck panel. The knowledge of the sensitivity of the sandwich composites will be helpful to update the model and also to design the bridge deck for better performance and improved longevity.

transfer all types of loads to substructures. Moreover, the stiffened sandwich composite plate can carry more loads as compared to the conventional sandwich plate. The sandwich composite plate consists of a three-layers i.e. the bottom and top layer (face layers), and the core layer. The face layer materials are fiber-reinforced plastic (FRP) and the core layer material is polyvinyl semi-rigid foam. Sensitivity analysis is needed to evaluate the uncertainty performance in structure. It helps to predict the life span of the structure and also helps to improve the structure durability, strength, and load resistance capacity. Sensitivity analysis shows the effect of variation in individual independent variables which might arise during the manufacturing of the sandwich panels or due to poor quality of materials. Several pieces of research have been carried out on the dynamic analysis of composite structure in the last two decades. Meunier and Shenoi [1,2] conducted the free vibration analysis for the FRP sandwich composite plate by using first and higher-order shear deformation theories and investigated the impact of the material properties and the geometry of the sandwich plate on natural frequency. It was observed that with the decrease in the ratio of core to plate thickness, the natural frequency increased. Some studies illustrate the experimental and numerical investigations [3][4][5][6] examining the behavior of FRP sandwich composites. Mandal et al. [7] determined the orthotropic property of the FRP sandwich plate with various core configurations such as triangle, circular, and rib shape. Elastic constants were calculated using the analytical method and finite element model-based software ANSYS and the results from the theoretical analysis were compared with the results from finite element models. Mondal et al. [8] investigated a sandwich composite plate, with different thicknesses of the core layer and holes of different diameters in the core layer, experimentally using the impact hammer test and numerically by finite element modal analysis. Mukhopadhyay et al. [9] studied the effect of the irregularity of the honeycomb in a sandwich panel on the natural frequency of the panel. Azarafza [10] investigated composite sandwich structures along with cores made of the stiffened grid. Experimental modal testing and numerical analysis were conducted for the sandwich structures to determine the dynamic characteristics such as frequency, damping coefficient, and mode shapes. An acceptable matching between the numerical and experimental data was observed. The effect of material properties and geometry of components of sandwich composite structures on the behavior of the sandwich composites are also illustrated in several studies [11][12][13].
Some literature is available on the sensitivity analysis of the FRP composite structures considering the different independent variables such as material properties, geometric structures. Thompson et al. [14] developed a reliability-based optimization technique and implemented the same to optimize the weight of the stiffened bridge deck panels of FRP composite. Sensitivity analysis was also performed to identify the design parameters. Several studies on the sensitivity analysis of sandwich composites are published in previous literature [15][16][17]. Mukhopadhyay et al. [18] investigated the FRP sandwich deck with a web core based on the D-optimal design to minimize the weight of the sandwich bridge decks using the Nelder-Mead simplex algorithm. Sensitivity analysis was conducted with respect to the property of materials and various geometry of the bridge deck. Santhanakrishnan et al. [19] developed a stitching technique to stitch the sandwich panels and studied their performances under a low-velocity impact test. Akoussan et al.  [20] implemented the higher-order derivative for sensitivity analysis of the damping proprieties of the sandwich structure. The sensitivity analysis was carried out for the thickness of the sandwich structure with the viscoelastic core layer. The result showed that the variation in the stiffness of the structure depends on the core thickness. Liu et al. [21] performed an analytical sensitivity analysis with respect to the static response, eigenvalues, and eigenvectors of the laminated plate and shell. Mondal et al. [22] obtained the elastic material properties of the sandwich composite plate using the inverse eigen sensitivity method (IEM) from the natural frequency with or without the presence of random noise. The IEM determined the elastic parameters in the plane or out of the plane of the face sheet of the sandwich composite plate. Literature available on the composite structure indicates that very little research has been carried out related to stiffened sandwich deck slabs. In the present study, bridge deck panels of stiffened sandwich composites are analyzed to obtain the eigenvalues. Sensitivity analysis is performed for variation in various individual face layer material properties. Moreover, the effect of holes in the core layer of the panel is studied. The knowledge of the sensitive parameters will be helpful to control the manufacturing process as well as the quality of the stiffened sandwich composite panel.

Methodology
This study uses stiffeners at the bottom of the composite plate with different numbers and alignments, i.e., transverse, longitudinal, and inclined alignment of stiffeners. The sandwich composite deck as well as the stiffeners is made of a core layer made of polyvinyl semi-rigid foam sandwiched between two FRP face layers. Dynamic analysis is carried out to get the mode shape and natural frequency of the stiffened deck slab. A comparison study is made for the natural frequency of the sandwich composite deck panels. Sensitivity analysis is then conducted for the natural frequencies of sandwich composite plates with different positions of stiffeners with respect to various elastic parameters of face layers. In addition to this, sensitivity analysis of the stiffened sandwich deck slab is conducted concerning the holes of different diameters in the core layer of the sandwich composite plate with a transverse stiffener.

Finite element formulation of the stiffened sandwich composite plate
The sandwich plate of size 350 mm × 250 mm with the thickness of the core and face layers as 30 mm and 2.5 mm respectively is considered for the investigation as shown in Fig. 1. Glass fiber-reinforced plastic (GFRP) is considered for the face layers. Polyvinyl semi-rigid foam is considered for the core. Stiffeners are located on the bottom of the bridge deck panel with different alignments, specifically in the transverse, longitudinal, and inclined direction of the sandwich plate. This figure presents the x, y, and z-axes in the transverse, vertical, and longitudinal directions. Figure 1 shows the dimension of the sandwich composite plate without stiffener and the cross-sectional dimension of the stiffener, respectively. The typical sandwich composite plates with stiffeners (three numbers) in longitudinal, transverse, and inclined directions of the sandwich composite plate are shown in Fig. 2a- The material constants elastic modulus chosen (as per Mondal et. al. 2015) for the face layers are E x = 14.46 GPa, E y = 13.5 GPa, E z = 60 MPa, and Poisson's ratio are υ xy = 0.32, υ xz = 0.1, υ yz = 0.1 for the face sheets. The shear moduli are selected as G xy = 3.27 GPa, G xz = 5.47 MPa, and G yz = 4.831 MPa. The modulus of elasticity of the core layer is taken as E = 0.119 GPa, and the Poisson's ratio of the core layer is taken as υ = 0.3. The density of the core is considered to be 100 kg/m 3, and that of the face layers is assumed to be 1400 kg/m 3 . Stiffeners considered are also sandwich composites with the same material properties as a sandwich composite. The depth of stiffener is taken as 50 mm meshing size of the sandwich composite plate is 5 mm for each layer. In the present study, 3D eight-node linear brick element (C3D8R) is considered for the sandwich composite plate. the sandwich plates without stiffeners and the plates with a single stiffener in the transverse direction are modeled with holes of different diameters in the core layer of the sandwich composite plate as well as in the stiffeners. The diameters of the holes considered are 10 mm, 15 mm, 20 mm, and 30 mm, and the spacing between in two holes is fixed to 10 mm. The typical core layers, with 15 mm diameter holes, in the panel and stiffener are presented in Fig. 3.

Numerical modeling of the plates with holes in core layers
The free vibration analysis is carried out for both sandwich composite plates with stiffener and without a stiffener in the transverse direction with the holes in core layers. The natural frequencies are mentioned in Tables 4 and 5 and from both tables, it is observed that the frequency value decreases with increasing the diameter of the hole in the core layer of the sandwich composite deck panel and stiffeners.

Free vibration analysis
The mathematical formulation for the free vibration analysis of sandwich composite bridge deck panels is briefly discussed in this section. Equilibrium equation for the stiffened sandwich composite plate can be written as The undamped equation of motion is written in the eigenvalue problem format, as shown in Eq. 2, and is solved using an appropriate Eigen solver to obtain the frequencies and mode shapes.
Here, λ i =ω i 2 is the eigenvalue, ω is the frequency of the structure, and φ i is the corresponding eigenvector (mode shape). (1)

Result and discussion
Free vibration analysis is carried out for the stiffened sandwich composite bridge deck panels. The first ten-mode natural frequencies of the sandwich composite plate without stiffener and with stiffeners are obtained, mentioned in Table 1, by free vibration analysis. The nomenclatures along with the masses of the models are mentioned in Table 2. From the free vibration analysis, corresponding mode shapes are also drawn. The first five-mode shapes for the sandwich plates with multiple stiffeners are demonstrated in Figs. 4, 5, and 6.

Comparison study of dynamic responses of stiffened sandwich composite plates
A comparison study of natural frequencies is carried out for the sandwich composite plate models. Firstly, the comparison is carried out for the plates with the different alignment of stiffeners keeping the number of stiffeners constant. The comparison of frequencies between the sandwich composite plate without stiffener and with a single stiffener in transverse, longitudinal, and inclined direction is shown in Fig. 7. It is observed that for the first three modes, the inclined stiffened sandwich composite plate gives higher frequency values. The sandwich composite deck without stiffener gives higher frequency values for the next three (4-6) modes. For the 7th to 10th modes, the longitudinal stiffened sandwich composite deck gives higher frequency values as compared to all other models. However, the difference between the frequency values for a particular mode is low. The comparison of frequencies for the cases CSWS, CS2TS, CS2LS, and CS2IS is presented in Fig. 8. Higher frequency is observed in the first three and 8th to 10th modes      of the sandwich composite plate with two stiffeners in the transverse direction. Figure 9 shows the comparison of frequencies between the plate models CSWS, CS3TS, CS3LS, and CS3IS. The sandwich plate CS3TS is observed to have higher frequencies in the 2nd and 3rd modes. A little difference between the frequencies of plates CSWS and CS3TS is observed. It has been noticed the sandwich composite plate with transverse stiffeners performed better with relatively higher frequencies. Moreover, it is also observed that the plates with multiple stiffeners give better results. The increase in the performance of the sandwich composite panels can be observed because of the added stiffness by the stiffeners. From the overall comparison study, it is observed that the plate models with stiffeners in the transverse direction performed better as compared to other cases and the plate models with one stiffener (CS1TS) in different directions give better-performing results with higher frequency values. The nomenclature for the sandwich composite plates (with holes in the core layer) without stiffener and with a transverse stiffener is presented in Table 3. The natural frequencies for the sandwich plate CSWS and CS1TS with holes in the core are obtained and presented in Tables 4 and 5, respectively.

Sensitivity analysis of sandwich plates without holes w.r.t elastic parameters
Sensitivity analysis is carried out for all the cases by a 5% increment of the elastic properties of the individual face layer. The free vibration analysis is performed for the plates with increasing elastic parameters of the face layers to determine the first ten-mode natural frequencies for all the cases. Table 6 represents the natural frequencies of the plate with a single stiffener in the transverse direction with a 5% increment in the individual elastic parameters. The sensitivity analysis for the above case and normalized value is shown in Fig. 10. It can be observed that the in-plane Young's modulus in x and z direction (E x and E z ) of the bridge deck panel are more sensitive as compared to all other elastic parameters. Similarly, sensitivity analysis for the case CS1LS is carried out, and it is observed that E x and E z are more sensitive. Moreover, the shear modulus G xz is also observed to be a sensitive parameter as shown in Fig. 11. The sensitivity for the face layer elastic parameters for the case CS1IS is shown in Fig. 12. The in-plane elastic modulus (E x , E z ) and shear modulus G xy are found to be more sensitive as compared to other elastic parameters.
Accordingly, the sensitivity analysis is performed for the sandwich composite panels with multiple stiffeners at their bottom. Figure 13 shows the sensitivity analysis for the case CS2TS and it is observed that E x and E z are significantly sensitive. The shear modulus G xz and G yz are also noticed to be sensitive. The sensitivity of the elastic parameters for the case CS2LS and CS2IS are shown in Figs. 14 and 15, respectively. It is noticed that E z is highly sensitive in the first eight modes compared to E x for the plate CS2LS. The shear modulus G xz and G yz are also observed to be less sensitive in comparison to the in-plane elastic moduli. For the plate CS2IS, the elastic modulus E x and E z are highly sensitive. The shear moduli G xy , G xz , and G yz are found to be moderately sensitive as compared to the in-plane Young's moduli. The sensitivity analysis for the sandwich composites plates (with multiple stiffeners) CS3TS, CS3LS, and CS3IS are presented in Figs. 16, 17, and 18, respectively. It has been observed that the in-plane elastic moduli E x and E z are sensitive. However, the shear moduli are comparably less sensitive than in-plane elastic properties. Sensitivity analysis is carried out for the sandwich composite plates with holes of diameter 10 mm and 20 mm in the core layer of sandwich composite plates without stiffener and with a transverse stiffener w.r.t a 5% increment of the elastic parameters. The natural frequencies for the sandwich plate WS-H10 with a 5% increment of individual elastic properties of face sheets are shown in Table 7.
The sensitivity analysis of the sandwich composite plates WS-H10 and WS-H20 are shown in Figs. 19 and 20, respectively, which describes that in-plane parameters E z , E x , and G xz are sensitive compared to other elastic parameters. Similarly, Figs. 21 and 22 represent the sensitivity in the sandwich composite plates TS-H10 and TS-H20, respectively. It is observed that the E x , E z , and G xz are sensitive parameters to the natural frequency of the sandwich composite plate. However, the other shear moduli G xy and G yz are also observed to be moderately sensitive.   Tables 3 and 4. A sensitivity analysis is carried out for all the cases by the normalization of the natural frequencies.
It is observed that the plates with 30 mm diameter holes in the core layers give decreased natural frequency values showing that the size of the holes is also a sensitive parameter. The sensitivity analysis is conducted for the plates with and without transverse stiffeners and with holes in the core layer. The sensitivity for the cases WS-H10, WS-H15, WS-H20, and WS-H30 is shown in Fig. 23. The sensitivity for the  cases TS-H10, TS-H15, TS-H20, and TS-H30 is shown in Fig. 24. It can be observed that for larger dia holes (e.g., 30), all modes are equally sensitive for a plate with transverse stiffeners and without stiffeners. However, for smaller dia holes, sensitivity can be observed for a plate with transverse stiffeners.

Conclusion
The present study performed the dynamic analysis of a sandwich plate with and without stiffeners. The stiffeners are provided along a transverse, longitudinal, and inclined direction at the bottom of the panel surface of the sandwich composite bridge deck. The sensitive analysis has been performed for various scenarios, e.g., sensitivity analysis with respect to the alignment of stiffeners, individual materials properties, and hole diameter in the core layers. A comparison study of eigenvalues concludes that the sandwich composite deck with transverse stiffeners shows a good result with higher eigenvalues as compared to other stiffened sandwich composite bridge deck panels. It is noticed that the in-plane elastic parameters E x and E z are more sensitive as compared to all other elastic parameters in all stiffened sandwich composite bridge decks. Besides, the shear moduli are less sensitive as compared to Young's moduli, but G xz is sensitive than the other two shear moduli. Moreover, sensitivity analysis with different hole sizes in the core layer of the sandwich composite plate without stiffener and with a single transverse stiffener. It is observed that natural frequencies are sensitive for smaller dia holes present in the core. The repeated real-life experiment is very timeconsuming and costly. On the other hand, accurate finite element modeling gives satisfactory results and is accepted worldwide. The work emphasizes that the sensitivity analysis performed numerically with respect to different parameters is a better alternative in comparison to full-scale modeling and updating all parameters. The result will also provide insights to design the sandwich composite structures for economic and improved performance.