**1. Introduction**

Fiber-reinforced polymer (FRP) has the advantages of being lightweight and having high strength, corrosion resistance, and tailorability. It is a new type of material that exhibits excellent performances, and has been extensively used in civil engineering, shipbuilding, aerospace and so on. Under the same load capacity, the weight of an FRP bridge is only 30% of that of a steel bridge and 5% of that of a reinforced concrete bridge. Due to the excellent properties of FRP, it has gradually become a substitute for traditional building materials (e.g., steel and concrete). Compared with traditional steel and concrete bridges, FRP bridges are not only lightweight but are also more convenient to manufacture, transport, and install. Furthermore, they have long service lives of more than 100 years [1].

The orthogrid-stiffened FRP panel (OSFP) is characterized by a lattice of rigid, interconnected stiffeners, which is a generic structural element in weight-sensitive structure applications [2]. Its stiffness, buckling, and vibration characteristics are not only related to the stiffening forms but are also closely related to the structural and material parameters, which increases the difficulty of analysis. At present, most analysis methods for stiffened panels can be summarized as numerical methods, analytical methods, and a combination of the two methods. The calculation models for the effective analysis of stiffened panels can be divided into finite element models (FEMs) [3–8], discrete stiffener models (DSMs) [9], and smeared stiffener models (SSMs) [10]. The stiffeners are modeled as members with axial bending/torsion stiffnesses on the attached skin in DSM, and they can only be effectively

**Citation:** Wang, P.; Zhong, Y.; Shi, Z.; Luo, D.; Yi, Q. Estimating of the Static and Dynamic Behaviors of Orthogrid-Stiffened FRP Panel Using Reduced-Order Plate Model. *Materials* **2021**, *14*, 4908. https:// doi.org/10.3390/ma14174908

Academic Editor: Krzysztof Schabowicz

Received: 26 July 2021 Accepted: 25 August 2021 Published: 28 August 2021

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applied for the analysis of stiffened panels with thin skins and rigid stiffeners. The geometric features of the stiffeners and skin are contained in FEM, which is more flexible and accurate than DSM. However, it not only requires a long simulation time due to the large computational workload, but also the inability to set different configurations and materials during preliminary design and optimization [11]. The basic idea of SSM is to smear the stiffnesses of the stiffeners into the panel and calculate the effective properties [12–14]. This may be applicable to the overall analysis of stiffened panels with intensive interconnected stiffeners but not to the stress and strain analysis of stiffeners [15].

The global buckling is considered as the main failure mode of stiffened panels under axial compression or/and external pressure according to the failure mode map [16,17]. Among the methods used to study the global buckling of stiffened panels, extensive studies have concentrated on the FEM and SSM [18,19]. The extension–bending coupling interaction caused by the eccentricity of the stiffeners was neglected in the work by Chen et al. [20], which resulted in the imprecise buckling prediction results [21]. Abhijit et al. [22] used the FEM to study the free vibration characteristics of stiffened plates with symmetric stiffeners. Later, the isoparametric stiffened-plate element was introduced to analyze the free vibration of eccentrically stiffened plates [23,24]. Lam et al. [25] explored the vibrations of stiffened plates by dividing the plate domain into appropriate rectangular segments.

On the basis of literature review, there were some issues in the existing methods for predicting the effective performance of stiffened panels. First, the shear stiffness matrix cannot be predicted using the homogenized elastic constants and plate thickness together with classic plate theory. Second, due to the assumptions for defining the kinematics, the models were usually applicable to some specific form of stiffeners. Third, few plate models can accurately predict the local stress and strain distributions of stiffeners, which were of great significance to the failure analysis of stiffened panels.

Recently, Yu and Zhong [26–29] put forward a new multiscale modeling technique to deal with dimensionality-reducing structures based on the variational asymptotic method (VAM) [30]. With this method, the small structural parameters (such as the thickness– width ratio) were used to asymptotically expand the energy functional, and the higherorder terms were removed to obtain the approximate energy of different orders and the corresponding dimension reduction model, which achieved a good tradeoff between accuracy and effectiveness. In this work, a VAM-based reduced-order plate model of stiffened FRP panel was established to solve the three issues mentioned above. The influences of geometric parameters (such as height, thickness, length-width ratio, and period length) on the effective performance of the OSFP were investigated. Finally, the displacements and free-vibration modes of stiffened FRP panels with different stiffening forms, such as the orthogrid-, T- and blade-stiffened panels, were compared. To the author's knowledge, this method has never been used for this purpose.

#### **2. Theoretical Formulation for Reduced-Order Plate Model Using VAM**
