Modal Analysis of Optimized Trapezoidal Stiffened Plates under Lateral Pressure and Uniaxial Compression

: This paper deals with the modal analysis of optimized trapezoidal stiffened plates with simple supported conditions on the four edges of the base plate. The main objective of the finite element analysis is to investigate the natural frequencies and mode shapes of some stiffened structures subjected to lateral pressure and uniaxial compression in order to identify any potentially dangerous frequencies and eliminate the failure possibilities. The natural frequencies and mode shapes are important parameters in the design of stiffened plates for dynamic loading conditions. In this study, the numerical analysis is performed for such a design of this kind of welded plates which have already been optimized for lateral pressure and uniaxial compression. The objective function of the optimization to be minimized performed with the Excel Solver program is the cost function which contains material and fabrication costs for Gas Metal Arc Welding (GMAW) welding technology. In this study, the eigenvalue extraction used to calculate the natural frequencies and mode shapes is based on the Lanczos iteration methods using the Abaqus software. The structure is made of two grades of steel, which are described with different yield stress while all other material properties of the steels in the isotropic elastic model remain the same. Drawing the conclusion from finite element analysis, this circumstance greatly affects the result.


Introduction
Welded stiffened plates are widely used in various load-carrying structures, e.g., ships, bridges, bunkers, tank roofs, offshore structures and vehicles. They are subject to various loadings, e.g., compression, bending, shear or combined load. The shape of plates can be square rectangular, circular, trapezoidal, etc. They can be stiffened in one or two directions with stiffeners of flat, L, box, trapezoidal or other shapes. From these structural versions, we select here rectangular plates that are uniaxially compressed, laterally pressed and stiffened in the direction of the compressive load.
Several researchers have studied the behaviour of stiffened plates, which remains a widely researched topic. Kim et al. [1] proposed a refined empirical formulation to predict the ultimate strength performance or ultimate limit state of flat-bar type steel stiffened panel under longitudinal compression. De Queiroz et al. [2] performed a structural numerical analysis to estimate the central deflection of thin stiffened plates and simple supported stiffened plates subjected to a transverse and uniformly distributed load, and the influence of parameters such as the number of longitudinal and transverse stiffeners and the ratio between their height and thickness, and searched for the optimal geometrical configuration of for naval and offshore applications by applying the exhaustive search technique and using Ansys software. Li et al. [3] investigated the ultimate strength of welded stiffened plates under the predominant action of axial compression with non-linear finite element analysis using the commercial finite element code Abaqus. Troina et al. [4] developed an approach associating the Constructal Design Method [5] and Exhaustive Search technique, merged in a computational model, validated and then applied with the scope to minimize the central deflections of these plates.
In addition, many studies have been published to reduce the total cost by minimizing the total weight and fabrication cost of the welded stiffened structure. To achieve the lowest costs, many optimization methods have been proposed and developed. Welding is a relatively expensive technology in the fabrication of stiffened plates. It is important to decrease the cost of the whole structure. Therefore, designers use a suitable structural version by comparison of structural solutions.
Structural optimization is a good basis not only for achieving weight and cost savings but also for helping designers select the most suitable structure version. Papers [6,7] elaborated a new optimization method for a totally FRP composite construction in which the single-objective weight optimization was solved by applying the Interior Point Algorithm of the Matlab software, the Generalized Reduced Gradient (GRG) Nonlinear Algorithm of the Excel Solver software, and the Laminator software. The Digimat-HC software solved the numerical models for the optimum structure. Their main contribution was developing a new method for optimizing a totally FRP composite sandwich structure. Zhou et al. [8] studied some distribution laws of the stress mode shapes (SMSs) from the layup and stress component perspectives for the analysis and optimization of dynamic composite laminates in aircraft structures. A study by Guan et al. [9] indicated that the four-node support can be used in a free vibration test to determine the elastic properties of full-sized wood composite panels. Kim et al. [1] studied the ultimate limit state (ULS) behaviour of stiffened panels under longitudinal compression by a non-linear finite element method (NLFEM) using the Ansys software and considering different types of stiffeners mainly being used in shipbuilding, i.e., T-bar, flat-bar and angle-bar.
This paper contains the minimum cost design of longitudinally stiffened plates using strength calculation methods and finite element analysis, which is a powerful technique that is used, among other things, for the dynamic response of structures. Numerous studies have discussed the inclusion and efficiency of the finite element method in the analysis of structures; therefore, the case of stiffened plates is also studied successfully in this way [10]. More detailed descriptions of finite element procedures can be found in [11,12]. Numerical simulation possibilities employ finite element methods for the modal and buckling analysis of plates to investigate the effect of initial geometric imperfection on the loaddisplacement response. It is worthwhile to mention the use of FEA if there are necessary cutouts in a thin-walled structure as a functional requirement that weaken the affected part of the structures, thus compromising the structure's integrity, or if the stiffeners are reinforced the structure. Paper [13] is devoted to the natural frequencies of plates with square holes when subjected to in-plane uniaxial, biaxial or shear loading. Jafari et al. [14] introduced the optimal values of effective parameters on the stress distribution around a circular, elliptical and quasi-square cutouts in perforated orthotropic plates under inplane loadings. In case the plates also are reinforced with stiffeners, paper [15] provides an overview for the buckling and free vibration analysis. Wu et al. [16] investigated the case of rectangular plates subjected to preloads. The vibrational properties of the structures can also be investigated by numerical and experimental modal analysis with dynamic loads. For instance, in paper [17], mechanical vibrations of the IPM motor components were detected and analyzed via a numerical, analytical and experimental investigation. Modal analysis methods can be useful for vibration control to identify the transient natural frequencies and transient modal shapes online and in real-time [18].
In the authors' previous paper [19], various Young's moduli were used, and it was assumed that the structural parts (the base plate and ribs) were made of different steel materials. The natural frequencies were investigated, and a linear perturbation analysis for the stiffened plate with flat stiffeners was performed in the commercial software Abaqus [12], where a flat stiffened plate was only uniaxially compressed. Simple, practical design procedures and calculation methods are not available due to the anisotropy of the stiffened plates having longitudinally single-sided, closed stiffeners. Therefore, it is especially worthwhile to use finite element analysis to determine mode shapes. In this study, the numerical investigations on the vibration analysis of trapezoidal stiffened plates under uniaxial compression and lateral pressure are carried out to identify any potentially dangerous frequencies.

Designed Constraints
Paper [20] developed the minimum cost design of longitudinally stiffened plates using strength calculation methods. Its results show that the trapezoidal stiffeners are more cost-effective than open section ribs, e.g., the cost savings can reach 40%, although the higher strength steel is 8-10% more expensive. These procedures can be also applied to many other problems in engineering practice; for example, for composite sandwich structures [6,21] and plastic composite sandwich construction in aircraft structures [7].

Calculation of the Deflection due to Compression and Lateral Pressure
Paper [22] used differential equations of large deflection orthotropic plate theory and the Galerkin method to derive the following cubic equation for elastic deflection of stiffened plate loaded by lateral pressure p and uniaxial compression. They have continued this line for initial deflection in [23].
where assuming that the orthotropic model is in a ′ − ′ plane and provided that the structural reference axes ( , , and ) are parallel to the principal material axes ( ′ , ′ , and ′ ), the following equations are written as: in which m is the number of half buckling length, B is the width of the base plate under compression, L is the length of the base plate, n = − 1 is the number of ribs, and are the thicknesses of the base plate and the stiffener, respectively, and the longitudinal Young's modulus and the transverse modulus are given by For given applied compression the uniaxial normal stress is obtained as and is according to Equation (16). The self-weight is taken into account; consequently, the lateral pressure is modified as where is the material density, is the volume of the structure, 0 is the uniformly distributed load, and the gravitational acceleration is denoted by .
The torsional and flexural stiffnesses of the orthotropic plate can be given in a relatively simple form if we introduce the following notations: in which is the Poisson's ratio, is the distance of the center of gravity which is derived from Equation (37), ′ , given by Equation (38), describes the moment of inertia of cross-section containing a stiffener and a base plate strip of width b, and , given by Equation (41), is the stiffener torsional moment of inertia.
The deflection due to lateral pressure is calculated as The solution of Equation (1) is where

Deflection due to the Shrinkage of Longitudinal Welds
The deflection of the stiffened plate due to the longitudinal welds is as follows where the curvature for steel is where is the heat input.
denotes the weld eccentricity. The heat input for stiffeners for the weld size is

Calculation of Stress Constraint
The calculation of the stress constraint can include several effects of loads. These can be the following: average compression stress and bending stress caused by deflections due to lateral pressure, compression and the shrinkage of longitudinal welds.
where is according to Equation (7) and is the bending moment, in which the magnitude of compression force is denoted by N. According to [24], the calculation of local buckling stress of a faceplate strip of width is calculated taking into account the effect of residual welding stresses and initial imperfections:

Optimizing Method and Objective Function
The optimization algorithm methods have been inspired by natural or humanmade phenomena to introduce mathematical formulations that can solve problems in different fields of sciences. Specifically, optimization algorithms are used to find the maximum or minimum of a function. They have a wide range of applications in industry [25] and engineering problems such as structures [20]. Developers are more interested in phenomena that could inspire them to develop a new method that can solve new problems or find the best solutions for the existing ones. More descriptions of the optimum design of steel structures are found in [26,27].
In this paper, the Excel Solver program is used to minimize the cost function. This is a multi-objective optimization method in our case. The iterative algorithm is based on Weighted Normalized Method. The optimal design variables are the thicknesses of the base plate ( ) and the stiffener ( ) and the number of ribs (φ − 1) which are limited in size.
The objective function to be minimized is the cost function. It is defined as the sum of material ( ) and fabrication ( ) costs where and are cost factors, ρ is the density, is the volume of the structure and are the fabrication times: 1 is the time for preparation, tacking and assembly, 2 is the time of welding and 3 is the time of additional work such as deslagging, changing electrodes and chipping. The total time for welding is the sum of 1 , 2 and 3 , which is elaborated in [15]. In this paper, we calculated with GMAW welding technology.
The uniaxial compression at two opposite ends of the structure and the lateral pressure on the base plate act on the stiffened plate as shown in Figure 1. The magnitude of compression force is 1.974 × 10 7 N as a result of a uniformly distributed static load which can be described as loading over the length B and at the ends of the stiffeners. As shown in Table 1, three different values are employed for lateral pressure p on the base plate, namely 0.02, 0.01 and 0.005 MPa. The material is assumed to be isotropic elastic. In our investigation, the stiffened plate is made of two grades of steel, provided that the difference in steel quality is simply meant by yield stress = 235 MPa and = 355 MPa while Young's modulus of = 2.1 × 10 5 MPa, Poisson's ratio of = 0.3 and density of = 7.85 × 10 −9 t/mm 3 remain the same.
The welded plates studied were designed to have been optimized for loading while the following dimensions of stiffeners are fixed: and the other dimensions of stiffeners in Figure 2 can be obtained from the optimized results of Table 1 applying the formulas below: The distance of the centre of gravity of the rib from the y-axis of the mid-plane of the base plate is calculated as follows: is given from the dimensions in Figure 2, as are the stiffener moment of inertia and torsional moment of inertia The geometrical data of the base plate are width = 6000 mm and length = 4000 mm. The stiffeners are welded to the base plate with fillet welds to reinforce the plate. The numerical results for uniaxial compression and lateral pressure are summarized in Table 1, in which the columns contain the optimized main geometrical data and the optimised cost results. These results were used for the further modal analysis study. The result shows that the number of stiffeners φ -1 decreases if the lateral pressure is increased, but φ -1 increases if the yield stress of the material is increased.

Modal Analysis
The eigenvalue extraction to calculate the natural frequencies and the corresponding mode shapes is based on the subspace or Lanczos iteration methods described in [11], which are used to perform modal frequency response analysis or to investigate the eigenvalue for buckling prediction, and are applied to extract eigenvalues i.e., natural frequencies. Two analysis steps are used for natural frequencies belonging to the trapezoidal stiffened plate subjected to uniaxial compression and lateral pressure. At the first step, the applied loads and geometric nonlinearity were considered so that at the second step, the load stiffness is determined at the end of the first general analysis step and can be included in the eigenvalue extraction. To investigate the natural frequencies, a linear perturbation analysis for the stiffened plate with trapezoidal stiffeners is performed with these steps, using the commercial software Abaqus according to the model shown in Figure 3.  Table 1, also taking advantage of the symmetry.

FE Model Description
The uniform thickness base plate is reinforced by some trapezoidal-shaped stiffeners and simple support conditions are subjected on all the edges (SSSS). As described in Section 3, the stiffener number and geometry were optimized before the FE analysis. Figure  1 is depicted, the side lengths of the plate parallel to the -axis and -axis are denoted by and , respectively. Using the advantage of symmetry in geometry and loading, only one-quarter of the plate with boundary conditions is modelled in the FEA. The displacement boundary conditions on the FE model are as follows: Symmetry along the -axis, for which = = = 0 is required; symmetry along the -axis, which requires = = = 0; simple is prescribed support on the edge at = ± /2 according to = = 0 and at the edge of = ± /2 according to = = 0. Let , , and denote the displacements of a point in the mid-plane of the trapezoidal stiffened plate along the , , and directions, respectively and , , and are the rotations of the normal to the mid-plane at the same point of the structure (see in Figure 3).
The finite element method, which is a popular numerical technique, is used to numerically solve differential equations arising in engineering and modelling problems such as modal analysis problems [10,19]. The main concept of this technique is that the geometry of structure subdivides into non-overlapping small parts, so-called finite elements, which are implemented by the construction of a mesh. The conventional element types possess simple shaped geometry with well-defined stress displacement relationships. Due to the design, stress accumulation occurs at the supports along the side with length B, where the base plate meets the stiffeners, and therefore the mesh destiny in the FE model locally significantly affects the maximum stress values there. At the first step of modal analysis for the preload case, the applied loads and geometric nonlinearity were considered so that the best choice is the first-order element. Thus, the sufficiently refined mesh needs to ensure that the results from simulations are adequate. Accordingly, the trapezoidal stiffened plate is meshed into finite elements, which are four-node reduced integration shell elements (S4R in Abaqus), see details in [11,12]. The approximate global size is specified as 40 mm, as shown in Figure 3.

Modal Analysis Results
The numerical results of the FE analysis for natural frequencies are summarized in Table 2, which contains the first six natural frequencies from No. 1 to No. 6. in order to identify any potentially dangerous frequencies. In the optimized case 6, the first six mode shapes of the stiffened plate are depicted in Figures 4-9. The mode shapes for the other five cases of Table 2 are not shown separately here.   Table 1. According to Table  2, the reduction in the first natural frequencies no longer occurs at lateral pressure = 0.02 MPa if the grade of steel changes.
Comparing the first natural frequencies in Table 2 with the simulation results without loading, the rate of reduction is between 15% and 40%, and the first mode shapes are also similar to Figure 4a,b as expected. The same conclusions can be derived from [19]. Practice shows that the natural frequencies as numerical results of FEA are in good agreement with the experimental results. We can conclude that the plates become stiffer with more stiffeners, and the maximum displacement decreases so that the stiffeners can have a significant effect on global displacement.

Conclusions
The result of optimization for different manufacturing costs leads to structures with different geometry. The increased yield stress leads to a thickening of the base plate and a decrease in the number of ribs. From the optimization process and the finite element analyses carried out to examine the behaviour of trapezoidal stiffened plates under uniaxial compression and lateral pressure, the following conclusions can be drawn: that (i) the number of stiffeners φ -1 decreases if the lateral pressure is increased, but φ − 1 increases if the yield stress of the material is increased, (ii) the eigenfrequencies show a significant decrease based on our finite element analysis if the trapezoidal stiffened welded plates are subjected to given loadings, (iii) the structure can be made of two grades of steel, which are described with different yield stress while all other material properties in the isotropic elastic model remain the same, then this condition is greatly influencing the results, and (iv) the influence of initial geometric imperfections allowed by designed constraints on frequencies is less significant, which is similar to the results in [19]. Therefore, it is important to know that the structures are properly designed, made from proper materials, and constructed considering loading during their lifespan. The FE analysis has provided the natural frequencies and shape modes so that the dynamic behaviour of the stiffened plates can be further investigated by mode superposition analysis.
Author Contributions: Conceptualization, Z.V. and S.S.; methodology, Z.V. and S.S.; software, S.S.; formal analysis, Z.V. and S.S.; writing-original draft preparation, Z.V. and S.S.; writing-review and editing, Z.V. and S.S.; visualization, Z.V. and S.S. All authors have read and agreed to the published version of the manuscript.