# Enhanced Single-Degree-of-Freedom Analysis of Thin Elastic Plates Subjected to Blast Loading Using an Energy-Based Approach

Authors: M. D. Goel; T. Thimmesh; P. Shirbhate; C. Bedon
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Single-degree-of-freedom (SDOF) models are known to represent a valid tool in support of design. Key assumptions of these models, on the other hand, can strongly affect the expected predictions, hence resulting in possible overconservative or misleading estimates for the response of real structural systems under extreme actions. Among others, the description of the input loads can be responsible for major design issues, thus requiring the use of more refined approaches. In this paper, a SDOF model is developed for thin elastic plates under large displacements. Based on the energy approach, careful attention is given for the derivation of the governing linear and nonlinear parameters, under different boundary conditions of technical interest. In doing so, the efforts are dedicated to the description of the incoming blast waves. In place of simplified sinusoidal pressures, the input impulsive loads are described with the support of infinite trigonometric series that are more accurate. The so-developed SDOF model is therefore validated, based on selected literature results, by analyzing the large displacement response of thin elastic plates, under several boundary conditions and real blast pressures. Major advantage for the validation of the proposed SDOF model is obtained from experimental finite element (FE) and finite difference (FD) models of literature, giving evidence of a rather good correlation and confirming the validity of the presented formulation.


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## Body

## 1. Introduction

Based on continuous advances in material science and nanotechnology, modern lightweight design and engineering applications take often benefit of thin structures and load-bearing elements. Often, these structures can take the form of thin panels belonging to partitions and barriers (i.e., facades) that are subjected to severe dynamic operational conditions, such as blast loads, or extreme natural hazards. As far as these systems are characterized by a global size that is considerably large in comparison with their thickness, under blast they need to withstand high strain pressures and undergo large amplitudes of deformations. For these reasons, the response of thin plates is substantially different from classical bending and vibration formulations of linear theory [1], given that the out-of-plane deformations are no longer compatible with the small-deflection theory.Due to the intrinsic simplicity of application of single-degree-of-freedom (SDOF) models, several formulations have been proposed in the literature for the analysis of structures under blast (see for example [2–36]). Actually, SDOF models can represent a practical tool for engineers and structural consultants. In several cases, SDOF models have been specifically developed for thin elastic plates in the nonlinear regime. Solutions have been proposed to carry out simple SDOF analysis on various structural systems under blast pressure, such as windows [2–4], composite panels [5–12], doors or walls [13–18], and so on.With the advances in numerical analysis and computational technology, several numerical methods can be employed to obtain viable solutions (i.e., finite element (FE) or finite difference (FD) methods, etc.). However, these methods are notoriously computationally expensive and often demand high simulation expertise, both for their formulation and for the interpretation of results [19]. Consequently, the development of efficient and accurate SDOF models in support of first-hand investigations for thin plates (and structures in general) under large vibrations, without going into detailed numerical analysis, attracts continuous research studies. Moreover, it is worth of interest that for plates under blast loading, the Unified Facilities Criteria Design Manual [37] recommends the use of SDOF models for structural assessment purposes. The main reason for such a recommendation is that the SDOF model analysis can be developed on the basis of relatively few input parameters, in comparison with other refined but expensive calculation methods.In the past, a number of researchers investigated the structural response of plates using various SDOF formulations. Several studies have been presented for thin plates under large amplitude vibrations (see [1, 20–34]). Different solving techniques have been generally adopted, and the response of thin elastic plates has been mostly investigated under a specific set of boundary conditions. As a further common feature of the earlier studies, the input load has been generally assumed as a sinusoidal function, thus yielding (as also shown in this paper) approximate results for blast-loaded structures.Another key aspect is that real blast waves have a typical short duration and exponentially decaying trend, which includes both positive phase and negative (or suction) phase [10]. Blast pressures, in this regard, are commonly associated to random impulsive loads, whose description would require the use of infinite trigonometric series. Compared to simplified sinusoidal functions, the latter leads to more accurate solutions for blast-loaded structural systems in general and especially for thin structures. Accordingly, infinite trigonometric series are considered in the SDOF formulation presented herein, due the fact that it fully satisfies (differing from other approaches) a multitude of various boundary conditions. Furthermore, it is important to notice that the response of thin plates is generally more complex than other structural members. This depends on their relatively large deflections under blast and therefore the development of both membrane and bending stresses that should be properly investigated.Li and Jones investigated the dynamic response of clamped circular plates subjected to blast loading, using the Johansen yield criterion which includes the effect of bending moments along with transverse shear [35]. Based on literature review [2–18, 20–35], it is important to mention that most of the available SDOF models for thin elastic plates are based on bending theory only, wherein membrane effects were considered through a bending resistance function and therefore indirectly accounted in the form of an equivalent bending stiffness only. Earlier researchers, in some other cases, employed separately the membrane effect (as a nonlinear term of the equation of motion) and explored the SDOF response of thin elastic plates under blast loading [33].A final consideration must be spent, in conclusion, on the reliability of elastic SDOF analysis for systems under blast. Generally, structures under blast loads are primarily required to absorb part of the imposed energy in plastic regions so as to ensure acceptable and feasible design performances. Therefore, a detailed damage analysis is often required. On the other hand, the same issue also requires first-hand elastic analysis, which is the starting point of more complex elastic-plastic studies. In some cases (i.e., when brittle elastic materials are used, like for example glass), elastic analysis represents the reference calculation approach in support of design, to avoid fracture.In this paper, past literature efforts are further extended, and an enhanced SDOF model is developed based on energy principles, by employing infinite trigonometric series for the blast loading description. The proposed method is validated towards analytical, experimental, and numerical data of literature, for both static calculations (i.e., stiffness parameters) and dynamic loading configurations (i.e., field blast experiments of literature). The enhanced SDOF model is assessed for SS (simply supported) or CC (clamped) plates, with respect to available literature results. As shown, a rather good agreement is generally found for all of them, thus conforming the accuracy of the proposed method. Moreover, with suitable modification, the same SDOF procedure can be further adapted to predict the response of thin elastic plates under pulse loading, thus further enforcing the potential of the approach.


## 2. Problem Definition

It is well known that a SDOF system requires the definition of only one coordinate to describe its position at any instant of time,t. Moreover, if the basic components of the vibratory system behave linearly (i.e., spring stiffness, mass, and damping (if any)), the resulting behavior is known to be linear (Figure 1(a)). In this case, the differential equation that governs the mechanical response can be solved by means of the principle of superposition.Figure 1
Governing differential equations for (a) linear and (b) nonlinear SDOF models.
(a)(b)However, if any of the basic components behave nonlinearly, the overall behavior of the SDOF system is obviously nonlinear, and appropriate calculation tools are required (Figure1(b)). Further, while considering the bending stiffness only, the system is still known as a linear system, wherein the system along with membrane effect is an example of a nonlinear system in this study.The present paper focuses on thin rectangular, isotropic elastic plates with dimensionsa × b and thickness h (Figure 2(a)), under various boundary conditions (Figure 2(b)). The input blast wave is described in the form of a distributed, time-varying pressure q(x, y, t), with x, y being the Cartesian coordinates and t being the time parameter.Figure 2
(a) Reference thin elastic plate under uniform pressure (q), with evidence of (b) the selected edge boundary conditions.
(a)(b)The nonlinear elastic dynamic behavior of a representative plate (see Figure2(a)) can be efficiently expressed by von Karman’s equations:(1)ρh∂2w∂t2+Eh3121−ν2∇4w−h∂∂xβσαβ∂w∂xα−qx,y,t=0,(2)∂σαβ∂xβ=0,where E is Young’s modulus, ν is Poisson’s ratio, and ρ is the mass density for the material in use. Moreover, w is the out-of-plane deflection (middle plane of the plate) under the external normal force per unit area q, while σαβ is Cauchy’s stress tensor (with α, β being the subscripts that take the value of 1 or 2). The bending stiffness D of the plate is given by(3)D=Eh3121−ν2.Finally, the two-dimensional biharmonic operator in equation (1) is defined as(4)∇4w=∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4.Equation (4) can be derived from kinematic assumptions and constitutive relations for the plate and assumes that the out-of-plane stresses (σ33, σ13, and σ23) are zero. Together with equations (1) and (2), it is able to express the conservation of linear momentum in two dimensions. Further, the three Föppl–von Karman equations [1, 21, 26] can be reduced to two, by introducing Airy’s stress function F(x, y, t), as defined by(5)σxx=∂2F∂x2,σxy=∂2F∂x∂y,σyy=∂2φ∂y2,where σxx, σxy, and σyy are membrane stresses induced in the plate. Accordingly, the above equations can be rewritten as(6)Dw,F=Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y,−qx,y,t=0,∇2F+E∂2w∂x2∂2w∂y2−∂2w∂x∂y2=0.Further, the corresponding membrane forces in the plate can be expressed as(7)N11=h∂2F∂y2,N12=−h∂2F∂x∂y,N22=h∂2F∂x2,where Airy’s stress function represents the membrane action in the plate that is induced by large displacements only.As also schematized in Figure2(b), the boundary conditions of technical interest considered in the present investigation are defined in equations (8)–(20). For all the examined edge restraints, the coordinate system in Figure 2(a) is taken into account.For out-of-plane displacement of the plate, the following boundary configurations are defined (see also Figure2(b)):(i)
Simply supported (SS) edge condition: all the edges are linearly simply supported.(8)wx,y,t|x=0x=a=0,∂2wx,y,t∂x2+ν∂2wx,y,t∂y2|x=0x=a=0,(9)wx,y,t|y=0y=b=0,∂2wx,y,t∂y2+ν∂2wx,y,t∂x2|y=0y=b=0.(ii)
Clamped (CC) edge condition: all the edges are linearly clamped.(10)wx,y,t|x=0x=a=0,∂wx,y,t∂x|x=0x=a=0,(11)wx,y,t|y=0y=b=0,∂wx,y,t∂y|y=0y=b=0.As far as the membrane stresses are also taken into account, the reference boundary conditions are defined as follows (see also Figure2(b)):(iii)
Immovable edge condition: all the edges are assumed to be immovable, according to equations (12) and (13). It is important to notice that this edge condition incorporates membrane stresses.(12)ux,y,t|x=0x=a=0,σmxy=∂2F∂x∂y|x=0x=a=0,(13)vx,y,t|y=0y=b=0,σmxy=∂2F∂x∂y|y=0y=b=0.(iv)
Membrane stress-free edge condition: all the edges are assumed free from membrane stresses, and this condition is defined by equations (14) and (15).(14)σxxm=∂2F∂y2|x=0x=a=0,σxym=∂2F∂x∂y|x=0x=a=0,(15)σyym=∂2F∂x2|y=0y=b=0,σxym=∂2F∂x∂y|y=0y=b=0.(v)
Movable edge condition: all the edges are assumed movable, as defined by equations (16)–(19). Further, it is to be noted that the movable edges are the edges which are kept straight by a distribution of normal stresses, whose resultant is zero.(16)ux,y,t|x=0x=a≠0Constant,σmxy=∂2F∂x∂y|x=0x=a=0,(17)vx,y,t|y=0y=b≠0Constant,σmxy=∂2F∂x∂y|y=0y=b=0,(18)Pxx,t=h∫0b∂2F∂y2dy|x=0x=a=0,(19)Pyy,t=h∫0a∂2F∂x2dy|y=0y=b=0,wherePx and Py are the resultant forces in the x and y directions, respectively.Further, at the timet = 0 (i.e., the time instant at which the plate is at rest), the out-of-plane deflections, w of the middle plane, of the plate are defined as(20)wx,y,t|t=0=0,∂wx,y,t∂t|t=0=0.Therefore, the investigation of thin elastic plates under large deflections can be generally solved using two nonlinear differential equations (i.e., equations (1) and (2)) with various edge boundaries and membrane stress configurations of technical interest (i.e., equations (8)–(19)), along with their initial condition (as given by equation (20)).


## 3. Blast Load Description

Among the available SDOF formulations for thin elastic plates under impulsive loads, various formulations and assumptions can be found in the literature. Besides, it is generally recognized that the analytical solution of the complex governing equations, as well as their numerical solution, involves considerable computational difficulties.Within the multitude of existing SDOF formulations, a critical examination highlights that most of them are developed on the basis of a sinusoidal load function (see, for example, [25–27, 33]). Minimum variations can be found in the reference Cartesian system. As far as the reference x, y axes are located at the central point of the midplane of the plate in Figure 2(a), for example, the out-of-plane deflections, w, are commonly expressed as [25–27, 33](21)wx,y,t=w0t∫0a∫0bsinπxasinπybdxdy.Conversely, when the external load is described considering reference coordinate system from Bayles et al. [26], the resulting displacement of the plate is given by (i.e., [1, 33])(22)wx,y,t=w0t∫−a/2a/2∫−b/2b/2cosπxacosπybdxdy.As such, all the above SDOF formulations are equivalent. Moreover, they are able to provide solutions for plates under sinusoidal loads only.The latter assumption represents one of the major issues for most of the SDOF formulations of literature, given that the assumed functions, even for sinusoidal loads only, typically result in rather approximate predictions. The reason for this weakness (as also discussed in this paper) derives from the lack of convergence of finite trigonometric series that take the form of a sinusoidal function whenm = 1 and n = 1. Otherwise, the convergence is always ensured by infinite trigonometric series. Such a concept is also in contrast with blast-related issues, given that blast pressures are recognized to act as random impulsive loads, thus recommending the use of infinite trigonometric series for enhanced solutions.Accordingly, the present research study focuses on the derivation of linear and nonlinear governing parameters for SDOF systems subjected to blast loading, based on energy concepts and infinite trigonometric series.To this aim, the adopted functions for SS and CC plates are expressed by equations (23) and (24), respectively.(23)φx,y=∑m=1∞∑n=1∞sinmπxasinnπyb,(24)φx,y=∑m=1∞∑n=1∞sin2mπxasin2nπyb.Using this function, the blast response of thin elastic plates is expected to vary in time, but the governing parameters will remain constant. Moreover, the advantage of equations (23) and (24) is that the chosen function is able to satisfy all the boundary conditions in Figure 2(b), for both SS and CC plates (for any value of m and n). Further, the components of the three-dimensional Green–Lagrangian strain tensor are defined similar to [33].


## 4. Nonlinear SDOF Model

In the present investigation, it is assumed that the out-of-plane, transverse deflection,w, for the plate in Figure 2(a) can be described as(25)wx,y,t=Ytφx,y,where Y(t) is a function of time only, which needs to be determined, and φx,y is the deflection shape function, as defined by equations (23) and (24) for the SS and CC conditions, respectively. The deflection shape functions are assumed in terms of sine function (whereas, Feldgun et al. assumed in [33] a cosine function). As such, the following ordinary nonlinear differential equation is obtained to describe the behavior of the reference SDOF:(26)mY¨t+K1Yt+K3Y3t=Pt.


### 4.1. Simply Supported (SS) Plate

Assume a deflection shape function for a thin elastic rectangular plate under blast loading as per equation (23). The mass m is therefore defined as(27)m=ρabhab∫0a∫0bφ2x,ydxdy,or(28)m=ρabh4ab∫0a∫0b∑m=1∞∑n=1∞1−cos2mπxa−cos2nπyb+cos2mπxacos2nπybdxdy.Integrating equation (28), it is found that(29)m=ρabh4ab∑m=1∞∑n=1∞xy−ysin2mπx/a2mπ/a−xsin2mπy/b2mπ/b+sin2mπx/asin2mπy/b2mπ/a2mπ/b0a0b.By applying the above limits, equation (29) reduces to(30)m=ρabh4.Therefore, equation (30) expresses the mass of a thin, rectangular, and elastic isotropic plate with dimensions a × b, thickness h, and density ρ under a given SS edge condition. Further, a given external normal force per unit area of the plate, P, can be expressed as(31)Pt=∫0a∫0bqx,y,tφx,ydxdy.In this study, it is assumed that the distributed pressureq(x, y, t) may be described as(32)qx,y,t=Qx,yPt,where in case of uniform lateral pressure, it is Q (x, y) = 1, and therefore:(33)Pt=∫0a∫0b∑m=1∞∑n=1∞sinmπxasinnπybdxdyP1tdt,where P1(t) represents an exponentially decaying real blast function agreeing with [10]. Integrating the above equation and applying the reference limits, it is therefore found that(34)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1tdt,where equation (34) represents the external normal force per unit area of the plate in trigonometric series.Accordingly, the plate bending stiffnessK1 can be expressed as(35)K1=D∫0a∫0b∂2φx,y∂x2+∂2φx,y∂y22−21−ν∂2φx,y∂x2∂2φx,y∂y2−∂2φx,y∂x∂y2dxdy.Applying the assumed deflection function (equation (23)) and solving equation (34), K1 for SS plates is finally expressed as(36)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22.In order to obtain the membrane stiffness function, a suitable stress function for large deflections of SS plates needs first to be considered. The substitution of deflection (i.e., equations (23) and (25) for the SS condition) into equation (2) yields(37)∂4F∂x4+2∂4F∂x2∂y2+∂4F∂y4=E∂2w∂x∂y2−∂2w∂x2∂2w∂y2.Herein:(38)∂2wx,y∂x2=−∑m=1∞∑n=1∞mπa2sinmπxasinnπyb,(39)∂2wx,y∂y2=−∑m=1∞∑n=1∞nπb2sinmπxasinnπyb,(40)∂2wx,y∂x∂y=∑m=1∞∑n=1∞mπanπbcosmπxacosnπyb.Combining equation (38) through equation (40), finally, the general equation for SS plates without any edge condition can be written as(41)∇4F=−Eh2π4m2n22a2b2∑m=1∞cos2mπxa+∑n=1∞cos2nπyb.Equation (41) is a general solution which is the sum of complimentary function (CF) and particular integral (PI). Now, the PI in equation (41) is expressed as(42)F1=Eh2η1∑m=1∞cos2mπxa+η2∑n=1∞cos2nπyb,wherein η1 and η2 are unknown constants and are determined by comparing equations (41) and (42), thus leading to(43)η1=−a232b2,η2=−b232a2.The PI of equation (41) is therefore further expressed as(44)F1=−Eh232a2b2∑m=1∞cos2mπxa+b2a2∑n=1∞cos2nπyb.Moreover, the CF in equation (41) is given by(45)F0=12Pxy2+12Pyx2+Eh2∑r=1∞Arr2sinh/rπβcosh/rπβ+/rπβsinhrπβ+rπβcoshrπβcosh2rπay−2rπaysinhrπβsinh2rπaycosh2rπax+Brβ2r2sinhrπβcoshrπβ+rπβsinhrπβ+rπβcoshrπβcosh2rπbx−2rπbysinhrπβsinh2rπbycosh2rπby.Therefore, in conclusion, the solution to equation (41) is expressed as(46)F=F0+F1.It is important to notice thatPx and Py denote tensile loads on the sides of a given thin rectangular plate of general size, varying from 0 to a (in the x direction) and from 0 to b (in the y direction), respectively. To derive the final expression for Px and Py, it is assumed that all the edges of the plate remain straight after deformation.This condition implies that the elongation of the plate along thex direction is independent of y, and hence(47)δx=∫0a∂u∂xdx=∫0a1E∂2F∂y2−ν∂2F∂x2−12∂w∂x2dx,(48)δx=∫0a1E∂2F∂y2−ν∫aa1E∂2F∂x2−∫0a12∂w∂x2.Herein,(49)∫0a1E∂2F∂y2=1Eh2∫0aPx−nπb2∑m=1∞sinmπxa∑n=1∞sinnπybdx,∫0a1E∂2F∂y2=1EPxx−nπb2amπ∑n=1∞cosmπxa∑n=1∞sinnπyb0a.After applying the above limits and solving equation (48), it is found that(50)δx=PxaEh2−νPyaEh2−∑m=1∞m2π28a.Similarly, it is found that(51)δy=PybEh2−νPxbEh2−∑n=1∞n2π28b.Accordingly, to obtain the final solution, a final set of in-plane force boundary conditions is used (i.e., all the edges are immovably constrained). Such an assumption implies that the elongation alongx and y directions is zero, and thus(52)Px=π2Eh28a2∑m=1∞m2+ν2β2∑n=1∞n21−ν2,Py=π2Eh28a2∑n=1∞n2β2+ν∑m=1∞m21−ν2.In conclusion, the stress function for SS plates with immovable edge conditions is defined as(53)F=−Eh2322π21−ν21a2∑m=1∞m2+νβ2∑n=1∞n2y2+ν∑m=1∞m2+β2∑n=1∞n2x2+β2∑m=1∞cos2mπxa+1β2∑n=1∞cos2nπyb.Now, assume that the functionF0 for stress-free edge condition is expressed as given in [21]:(54)φ¯pq=4βπp2+β2q22×p−1qεqsinh2pπ/βsinhpπ/βcoshpπ/β+pπ/βAp+q−1pεpsinh2qπβsinhqπβcoshqπβ+qπβBq,εp=0.5,p=0,1.0,p>0,εq=0.5,q=0,1.0,q>0.The coefficientsApp=0,1 and Bqq=0,1 are the solutions of the linear systems.It is to be noted that the functionF0 for SS plates and movable edges is equal to zero.For easy understanding, equation (1) is now expressed as I(w, F). The solutions for the membrane stiffness of the plate are detected as(55)∫0a∫0bIw,Fsinmπxasinnπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,tsinmπxasinnπybdxdy=0.Employing first the Ritz–Galerkin method for the solution of equation (58), in particular, the bending stiffness K1 is determined as(56)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22.Moreover, the membrane stiffnessK3 (having a fundamental role for the nonlinear behavior of the examined plates) is defined as(57)K3=−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂ysinmπxasinnπyb.Solving equation (57), K3 can be therefore expressed for a SS plate with immovably constrained edge as(58)K3=ab4π4Eh38a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m4+121+a4b4∑m=1∞∑n=1∞m2n2.Similarly, for the membrane stress-free edge condition,K3 is defined as(59)K3=ab42π4Eh3a2b2φ¯01+φ¯10−132a2b2+b2a2f3.For the movable edge condition, finally, the membrane stiffnessK3 is defined as(60)K3=ab4π4Eh316a41+a4b4∑m=1∞∑n=1∞m2n2.Now, for the forcePt, it is assumed that(61)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1tdt.Further, the bending strains in thex and y directions are defined as(62)εxx=1E∂2F∂y2−ν∂2F∂x2,εyy=1E∂2F∂x2−ν∂2F∂y2.The shear strains in thex-y direction are zero, due to assumed function, and such an assumption is in close correlation with various literature researchers (see, for example, [21, 26]). This means that(63)γxy=−21+νE∂2F∂x∂y=0.Finally, the strains inx direction are therefore defined as(64)εxx=h2324π2a2∑m=1∞m2+∑n=1∞n2cos2nπyb−νβ2∑m=1∞m2cos2mπxa.Assuming thatm = 1 and n = 1, the latter results in(65)εxx=h2π28a21+cos2πyb−νβ2cos2πxa.For a square plate (a/b = 1), it is therefore found that(66)εxx=h2π28a21+cos2πya−νcos2πxa.It is noteworthy that the above equation as herein derived by using infinite trigonometric series equals the strain equation given by Bauer [25] for square plates.Similarly, the strain along they direction is defined as(67)εyy=1EEh2322π21−ν21a2ν∑m=1∞m2+β2∑n=1∞n2×2+β22mπa2∑m=1∞cos2mπxa−ν2π21−ν21a2∑m=1∞m2+νβ2∑n=1∞n2×2+1β22nπb2∑n=1∞cos2nπyb.Again, assumingm = 1 and n = 1, it is found that(68)εyy=h2π28b21+cos2πxa−νβ2cos2πyb.Therefore, for a square plate,(69)εyy=h2π28a21+cos2πxa−νcos2πya,or(70)εyy=h2π28b21+cos2πxb−νcos2πyb,and equation (70) coincides with the strain equation given by Bauer [25] for square plates.Now, merging all the so-obtained expressions in equation (55), it is found that(71)∫0a∫0bIw,Fsinmπxasinnπybdxdy=ρh2f¨+∑m=1∞∑n=1∞mπa4+nπb4+2mπa2nπb2Dhf−4π2∑m=1∞∑n=1∞1−cosmπ−cosnπ+cosmπcosnπmn+π4Eh48a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m41−ν2+121+a4b4∑m=1∞∑n=1∞m2n2f3.Substitutingm = 1 and n = 1 in equation (71), the solution is therefore obtained for various boundary conditions.The solution for a SS plate with immovably constrained edges, in particular, is given by(72)ρh2f¨+Dhπa4+πb4+2πa2πb2f+π4Eh48a41+2νa2/b2+a4/b41−ν2+121+a4b4f3=16π2.For a SS plate with movable edges, it is(73)ρh2f¨+∑m=1∞∑n=1∞mπa4+nπb4+2mπa2nπb2Dhf−4π2∑m=1∞∑n=1∞1−cosmπ−cosnπ+cosmπcosnπmn+π4Eh48a4121+a4b4∑m=1∞∑n=1∞m2n2f3=0.The solution for a SS plate with movable constrained edges is(74)ρh2f¨+Dhπa4+πb4+2πa2πb2f−π4Eh48a4121+a4b4f3=16π2.Finally, the solution for the SS plate with stress-free edges is given by(75)ρh2f¨+Dhπa4+πb4+2πa2πb2f−2π4Eh4a2b2φ01′+φ10′−132a2b2+b2a2f3=16π2.


### 4.2. Clamped (CC) Plate

Assuming the deflection shape function for a thin elastic plate under blast loading as per equation (24), the reference function satisfies all the boundary conditions. Here, the mass m is defined by equation (27) and can be further expressed as(76)m=ρabh16ab∫0a∫0b1−2cos2mπxa+121+cos4mπxa×1−2cos2nπyb+121+cos4nπybdxdy.Integrating the above equation, it reduces to(77)m=ρabh16abx−2sin2mπx/a2mπ/a+12x+sin4mπx/a4mπ/ay−2sin2nπy/b2nπ/b+12y+sin4nπy/b4nπ/b0a0b.Applying the limits, it is found that(78)m=9ρabh64,where the latter is suitable for a rectangular, elastic isotropic plate with dimensions a × b, thickness h, and density ρ, under a CC condition. Further, the external normal force per unit area of the plate, P, is expressed by equation (30). As mentioned earlier for the SS plate, in the case of uniform lateral pressure, it is Q (x, y) = 1; therefore,(79)Pt=∫0a∫0b∑m=1∞∑n=1∞sin2mπxasin2nπybdxdyP1tdt,or(80)Pt=14∫0a∫0b∑m=1∞∑n=1∞1−cos2mπxa1−cos2nπybdxdyP1tdt.Integrating the above equation, it reduces to(81)Pt=14∑m=1∞∑n=1∞x−sin2mπx/a2mπ/ay−sin2nπy/b2nπ/b0a0bP1tdt.Applying the limits, we get(82)Pt=ab4P1t,where the latter represents the external normal force per unit area of the plate, P, in trigonometric series (and is substantially different from the proposal in [33]).Now, the bending stiffnessK1 of the plate is expressed as(83)K1=D∫0a∫0b∂2φx,y∂x2+∂2φx,y∂y22−21−ν∂2φx,y∂x2∂2φx,y∂y2−∂2φx,y∂x∂y2dxdy.Applying the assumed deflection function and solving equation (87), K1 for a CC plate is therefore found in(84)K1=Dπ4ab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4.In order to obtain the membrane stiffness function for large deflections of rectangular CC plates, the substitution of deflection (i.e., equations (23) and (25) for the CC case) into equation (2) is first taken into account, so as to achieve a suitable stress function.This yields to(85)∇4F1=π4m2n22a2b2cos2mπxa+cos2nπyb+cos2nπybcos4mπxa−cos4mπxa+cos4nπybcos2mπxa−cos4mπxa−2cos2nπybcos2mπxa.The solution of equation (85) takes the form of CF and PI parts, wherein PI is given by(86)F1=132n2β2m2cos2mπxa+m2n2β2cos2nπyb+m2n2β24m2+n2β22cos2nπybcos4mπxa+m2n2β2m2+4n2β22cos4nπybcos2mπxa−n2β216m2cos4mπxa−m216n2β2cos4mπxa−m2n2β2m2+n2β22cos2nπybcos2mπxa.Therefore, the general solution isF=F0+F1, where(87)F0=Pxy22+Pyx22,and F0 for the stress-free edge condition is expressed as [21](88)φpq=4βπp2+β2q22×p−1qεqsinh2pπ/βsinhpπ/βcoshpπ/β+pπ/βAp+q−1pεpsinh2qπβsinhqπβcoshqπβ+qπβBq.Moreover:(89)εp=0.5,p=0,1.0,p>0,εq=0.5,q=0,1.0,q>0,while the coefficients Ap and Bq in equation (88) are the solutions of the linear systems.To derive the expression forPx and Py, all the edges of the plate are assumed to remain straight after the deformation. Following the same procedure earlier adopted for the SS plate, but with the reference CC function and boundary condition, it is found that(90)δx=PxaEh2−νPyaEh2−3Eh232∑m=1∞m2π2a2=0,δy=PybEh2−νPxbEh2−3Eh232∑n=1∞n2π2b2=0.The final solution is found for the immovably constrained edges, given that the elongation along thex and y directions is zero:(91)Px=3π2Eh232a2∑n=1∞m2+ν∑m=1∞n2β21−ν2,Py=3π2Eh232a2∑n=1∞n2β2+ν∑m=1∞m21−ν2.Finally, the stress function for a CC plate with immovable edge conditions is defined as(92)F=3π2Eh232a2∑n=1∞m2+ν∑m=1∞n2β21−ν2y22+3π2Eh232a2∑n=1∞n2β2+ν∑m=1∞m21−ν2x22+132n2β2m2cos2mπxa+m2n2β2cos2nπyb+m2n2β24m2+n2β22cos2nπybcos4mπxa+m2n2β2m2+4n2β22cos4nπybcos2mπxa−n2β216m2cos4mπxa−m216n2β2cos4mπxa−m2n2β2m2+n2β22cos2nπybcos2mπxa.For easy understanding, as in the case of the SS plate, equation (1) is expressed as I(w, F). The solution for the membrane stiffness of the plate is therefore derived as(93)∫0a∫0bIw,Fsin2mπxasin2nπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,t×sin2mπxasin2nπybdxdy=0.By employing the Ritz–Galerkin method to solve equation (53), the bending stiffness K1 is first calculated as(94)K1=π4Dab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4,where the mass is(95)m=9ρabh64.The membrane stiffnessK3 for a CC plate is therefore obtained by applying the same methodology used for the SS plate.For the immovably constrained edge condition, it is found that(96)K3=abπ4Eh3649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22.For the stress-free edge condition:(97)K3=abπ4Eh32a2b2φ¯01+φ¯10+φ¯11+12φ¯12+12φ¯21+φ¯02+φ¯20−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.For the movable edge condition, finally:(98)K3=abπ4Eh364a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22,where the force Pt is given by(99)Pt=ab4P1t.Therefore,(100)∫0a∫0b∂2F∂x2∂2w∂y2=−∑m=1∞∑n=1∞py3abπ2m216a2+m2n2β2×2nπb2mπa2ab8+132n2m2β24m2+n2β22×2nπb2mπa2ab16+n2m2β2m2+4n2β22×4nπb2mπa2ab32+m216n2β2×4nπb2mπa2ab32+n2m2β2m2+n2β22×2nπb2mπa2ab8,or(101)∫0a∫0b∂2F∂y2∂2w∂x2=−132∑m=1∞∑n=1∞px6abπ2n2b2+n2β2m2×nπb22mπa2ab8+n2m2β24m2+n2β22×nπb24mπa2ab32+n2m2β2m2+4n2β22×nπb22mπa2ab16n2β216m2×nπb24mπa2ab32+n2m2β2m2+n2β22×nπb22mπa2ab8.Here,∫0b∫0a2∂2F/∂x∂y∂2w/∂x∂y=0.Finally, summing up the above expressions, it is found that(102)9ρabh264f¨+Dhπ4ab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4f+π4Eh4649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22f3=ab4pt.By substitution ofm = 1 and n = 1, the expression for different CC edge conditions can be therefore finally obtained.The solution for a CC plate with immovable edges is given by(103)ρh2f¨+16Dhπ49a43+2ab2+3ab4f+π4Eh4a41+2νa2/b2+a4/b481−ν2+19a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2.The solution for the CC plate with movable edges is(104)ρh2f¨+Dhπ49a43+2ab2+3ab4f+π4Eh49a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2,while the solution for the CC plate with stress-free edges is(105)ρh2f¨+Dhπ49a43+2ab2+3ab4f+32π4Eh49a2b2φ01′+φ10′+φ11′+12φ12′+12φ21′+φ02′+φ20′−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2.


## 4.1. Simply Supported (SS) Plate

Assume a deflection shape function for a thin elastic rectangular plate under blast loading as per equation (23). The mass m is therefore defined as(27)m=ρabhab∫0a∫0bφ2x,ydxdy,or(28)m=ρabh4ab∫0a∫0b∑m=1∞∑n=1∞1−cos2mπxa−cos2nπyb+cos2mπxacos2nπybdxdy.Integrating equation (28), it is found that(29)m=ρabh4ab∑m=1∞∑n=1∞xy−ysin2mπx/a2mπ/a−xsin2mπy/b2mπ/b+sin2mπx/asin2mπy/b2mπ/a2mπ/b0a0b.By applying the above limits, equation (29) reduces to(30)m=ρabh4.Therefore, equation (30) expresses the mass of a thin, rectangular, and elastic isotropic plate with dimensions a × b, thickness h, and density ρ under a given SS edge condition. Further, a given external normal force per unit area of the plate, P, can be expressed as(31)Pt=∫0a∫0bqx,y,tφx,ydxdy.In this study, it is assumed that the distributed pressureq(x, y, t) may be described as(32)qx,y,t=Qx,yPt,where in case of uniform lateral pressure, it is Q (x, y) = 1, and therefore:(33)Pt=∫0a∫0b∑m=1∞∑n=1∞sinmπxasinnπybdxdyP1tdt,where P1(t) represents an exponentially decaying real blast function agreeing with [10]. Integrating the above equation and applying the reference limits, it is therefore found that(34)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1tdt,where equation (34) represents the external normal force per unit area of the plate in trigonometric series.Accordingly, the plate bending stiffnessK1 can be expressed as(35)K1=D∫0a∫0b∂2φx,y∂x2+∂2φx,y∂y22−21−ν∂2φx,y∂x2∂2φx,y∂y2−∂2φx,y∂x∂y2dxdy.Applying the assumed deflection function (equation (23)) and solving equation (34), K1 for SS plates is finally expressed as(36)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22.In order to obtain the membrane stiffness function, a suitable stress function for large deflections of SS plates needs first to be considered. The substitution of deflection (i.e., equations (23) and (25) for the SS condition) into equation (2) yields(37)∂4F∂x4+2∂4F∂x2∂y2+∂4F∂y4=E∂2w∂x∂y2−∂2w∂x2∂2w∂y2.Herein:(38)∂2wx,y∂x2=−∑m=1∞∑n=1∞mπa2sinmπxasinnπyb,(39)∂2wx,y∂y2=−∑m=1∞∑n=1∞nπb2sinmπxasinnπyb,(40)∂2wx,y∂x∂y=∑m=1∞∑n=1∞mπanπbcosmπxacosnπyb.Combining equation (38) through equation (40), finally, the general equation for SS plates without any edge condition can be written as(41)∇4F=−Eh2π4m2n22a2b2∑m=1∞cos2mπxa+∑n=1∞cos2nπyb.Equation (41) is a general solution which is the sum of complimentary function (CF) and particular integral (PI). Now, the PI in equation (41) is expressed as(42)F1=Eh2η1∑m=1∞cos2mπxa+η2∑n=1∞cos2nπyb,wherein η1 and η2 are unknown constants and are determined by comparing equations (41) and (42), thus leading to(43)η1=−a232b2,η2=−b232a2.The PI of equation (41) is therefore further expressed as(44)F1=−Eh232a2b2∑m=1∞cos2mπxa+b2a2∑n=1∞cos2nπyb.Moreover, the CF in equation (41) is given by(45)F0=12Pxy2+12Pyx2+Eh2∑r=1∞Arr2sinh/rπβcosh/rπβ+/rπβsinhrπβ+rπβcoshrπβcosh2rπay−2rπaysinhrπβsinh2rπaycosh2rπax+Brβ2r2sinhrπβcoshrπβ+rπβsinhrπβ+rπβcoshrπβcosh2rπbx−2rπbysinhrπβsinh2rπbycosh2rπby.Therefore, in conclusion, the solution to equation (41) is expressed as(46)F=F0+F1.It is important to notice thatPx and Py denote tensile loads on the sides of a given thin rectangular plate of general size, varying from 0 to a (in the x direction) and from 0 to b (in the y direction), respectively. To derive the final expression for Px and Py, it is assumed that all the edges of the plate remain straight after deformation.This condition implies that the elongation of the plate along thex direction is independent of y, and hence(47)δx=∫0a∂u∂xdx=∫0a1E∂2F∂y2−ν∂2F∂x2−12∂w∂x2dx,(48)δx=∫0a1E∂2F∂y2−ν∫aa1E∂2F∂x2−∫0a12∂w∂x2.Herein,(49)∫0a1E∂2F∂y2=1Eh2∫0aPx−nπb2∑m=1∞sinmπxa∑n=1∞sinnπybdx,∫0a1E∂2F∂y2=1EPxx−nπb2amπ∑n=1∞cosmπxa∑n=1∞sinnπyb0a.After applying the above limits and solving equation (48), it is found that(50)δx=PxaEh2−νPyaEh2−∑m=1∞m2π28a.Similarly, it is found that(51)δy=PybEh2−νPxbEh2−∑n=1∞n2π28b.Accordingly, to obtain the final solution, a final set of in-plane force boundary conditions is used (i.e., all the edges are immovably constrained). Such an assumption implies that the elongation alongx and y directions is zero, and thus(52)Px=π2Eh28a2∑m=1∞m2+ν2β2∑n=1∞n21−ν2,Py=π2Eh28a2∑n=1∞n2β2+ν∑m=1∞m21−ν2.In conclusion, the stress function for SS plates with immovable edge conditions is defined as(53)F=−Eh2322π21−ν21a2∑m=1∞m2+νβ2∑n=1∞n2y2+ν∑m=1∞m2+β2∑n=1∞n2x2+β2∑m=1∞cos2mπxa+1β2∑n=1∞cos2nπyb.Now, assume that the functionF0 for stress-free edge condition is expressed as given in [21]:(54)φ¯pq=4βπp2+β2q22×p−1qεqsinh2pπ/βsinhpπ/βcoshpπ/β+pπ/βAp+q−1pεpsinh2qπβsinhqπβcoshqπβ+qπβBq,εp=0.5,p=0,1.0,p>0,εq=0.5,q=0,1.0,q>0.The coefficientsApp=0,1 and Bqq=0,1 are the solutions of the linear systems.It is to be noted that the functionF0 for SS plates and movable edges is equal to zero.For easy understanding, equation (1) is now expressed as I(w, F). The solutions for the membrane stiffness of the plate are detected as(55)∫0a∫0bIw,Fsinmπxasinnπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,tsinmπxasinnπybdxdy=0.Employing first the Ritz–Galerkin method for the solution of equation (58), in particular, the bending stiffness K1 is determined as(56)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22.Moreover, the membrane stiffnessK3 (having a fundamental role for the nonlinear behavior of the examined plates) is defined as(57)K3=−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂ysinmπxasinnπyb.Solving equation (57), K3 can be therefore expressed for a SS plate with immovably constrained edge as(58)K3=ab4π4Eh38a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m4+121+a4b4∑m=1∞∑n=1∞m2n2.Similarly, for the membrane stress-free edge condition,K3 is defined as(59)K3=ab42π4Eh3a2b2φ¯01+φ¯10−132a2b2+b2a2f3.For the movable edge condition, finally, the membrane stiffnessK3 is defined as(60)K3=ab4π4Eh316a41+a4b4∑m=1∞∑n=1∞m2n2.Now, for the forcePt, it is assumed that(61)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1tdt.Further, the bending strains in thex and y directions are defined as(62)εxx=1E∂2F∂y2−ν∂2F∂x2,εyy=1E∂2F∂x2−ν∂2F∂y2.The shear strains in thex-y direction are zero, due to assumed function, and such an assumption is in close correlation with various literature researchers (see, for example, [21, 26]). This means that(63)γxy=−21+νE∂2F∂x∂y=0.Finally, the strains inx direction are therefore defined as(64)εxx=h2324π2a2∑m=1∞m2+∑n=1∞n2cos2nπyb−νβ2∑m=1∞m2cos2mπxa.Assuming thatm = 1 and n = 1, the latter results in(65)εxx=h2π28a21+cos2πyb−νβ2cos2πxa.For a square plate (a/b = 1), it is therefore found that(66)εxx=h2π28a21+cos2πya−νcos2πxa.It is noteworthy that the above equation as herein derived by using infinite trigonometric series equals the strain equation given by Bauer [25] for square plates.Similarly, the strain along they direction is defined as(67)εyy=1EEh2322π21−ν21a2ν∑m=1∞m2+β2∑n=1∞n2×2+β22mπa2∑m=1∞cos2mπxa−ν2π21−ν21a2∑m=1∞m2+νβ2∑n=1∞n2×2+1β22nπb2∑n=1∞cos2nπyb.Again, assumingm = 1 and n = 1, it is found that(68)εyy=h2π28b21+cos2πxa−νβ2cos2πyb.Therefore, for a square plate,(69)εyy=h2π28a21+cos2πxa−νcos2πya,or(70)εyy=h2π28b21+cos2πxb−νcos2πyb,and equation (70) coincides with the strain equation given by Bauer [25] for square plates.Now, merging all the so-obtained expressions in equation (55), it is found that(71)∫0a∫0bIw,Fsinmπxasinnπybdxdy=ρh2f¨+∑m=1∞∑n=1∞mπa4+nπb4+2mπa2nπb2Dhf−4π2∑m=1∞∑n=1∞1−cosmπ−cosnπ+cosmπcosnπmn+π4Eh48a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m41−ν2+121+a4b4∑m=1∞∑n=1∞m2n2f3.Substitutingm = 1 and n = 1 in equation (71), the solution is therefore obtained for various boundary conditions.The solution for a SS plate with immovably constrained edges, in particular, is given by(72)ρh2f¨+Dhπa4+πb4+2πa2πb2f+π4Eh48a41+2νa2/b2+a4/b41−ν2+121+a4b4f3=16π2.For a SS plate with movable edges, it is(73)ρh2f¨+∑m=1∞∑n=1∞mπa4+nπb4+2mπa2nπb2Dhf−4π2∑m=1∞∑n=1∞1−cosmπ−cosnπ+cosmπcosnπmn+π4Eh48a4121+a4b4∑m=1∞∑n=1∞m2n2f3=0.The solution for a SS plate with movable constrained edges is(74)ρh2f¨+Dhπa4+πb4+2πa2πb2f−π4Eh48a4121+a4b4f3=16π2.Finally, the solution for the SS plate with stress-free edges is given by(75)ρh2f¨+Dhπa4+πb4+2πa2πb2f−2π4Eh4a2b2φ01′+φ10′−132a2b2+b2a2f3=16π2.


## 4.2. Clamped (CC) Plate

Assuming the deflection shape function for a thin elastic plate under blast loading as per equation (24), the reference function satisfies all the boundary conditions. Here, the mass m is defined by equation (27) and can be further expressed as(76)m=ρabh16ab∫0a∫0b1−2cos2mπxa+121+cos4mπxa×1−2cos2nπyb+121+cos4nπybdxdy.Integrating the above equation, it reduces to(77)m=ρabh16abx−2sin2mπx/a2mπ/a+12x+sin4mπx/a4mπ/ay−2sin2nπy/b2nπ/b+12y+sin4nπy/b4nπ/b0a0b.Applying the limits, it is found that(78)m=9ρabh64,where the latter is suitable for a rectangular, elastic isotropic plate with dimensions a × b, thickness h, and density ρ, under a CC condition. Further, the external normal force per unit area of the plate, P, is expressed by equation (30). As mentioned earlier for the SS plate, in the case of uniform lateral pressure, it is Q (x, y) = 1; therefore,(79)Pt=∫0a∫0b∑m=1∞∑n=1∞sin2mπxasin2nπybdxdyP1tdt,or(80)Pt=14∫0a∫0b∑m=1∞∑n=1∞1−cos2mπxa1−cos2nπybdxdyP1tdt.Integrating the above equation, it reduces to(81)Pt=14∑m=1∞∑n=1∞x−sin2mπx/a2mπ/ay−sin2nπy/b2nπ/b0a0bP1tdt.Applying the limits, we get(82)Pt=ab4P1t,where the latter represents the external normal force per unit area of the plate, P, in trigonometric series (and is substantially different from the proposal in [33]).Now, the bending stiffnessK1 of the plate is expressed as(83)K1=D∫0a∫0b∂2φx,y∂x2+∂2φx,y∂y22−21−ν∂2φx,y∂x2∂2φx,y∂y2−∂2φx,y∂x∂y2dxdy.Applying the assumed deflection function and solving equation (87), K1 for a CC plate is therefore found in(84)K1=Dπ4ab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4.In order to obtain the membrane stiffness function for large deflections of rectangular CC plates, the substitution of deflection (i.e., equations (23) and (25) for the CC case) into equation (2) is first taken into account, so as to achieve a suitable stress function.This yields to(85)∇4F1=π4m2n22a2b2cos2mπxa+cos2nπyb+cos2nπybcos4mπxa−cos4mπxa+cos4nπybcos2mπxa−cos4mπxa−2cos2nπybcos2mπxa.The solution of equation (85) takes the form of CF and PI parts, wherein PI is given by(86)F1=132n2β2m2cos2mπxa+m2n2β2cos2nπyb+m2n2β24m2+n2β22cos2nπybcos4mπxa+m2n2β2m2+4n2β22cos4nπybcos2mπxa−n2β216m2cos4mπxa−m216n2β2cos4mπxa−m2n2β2m2+n2β22cos2nπybcos2mπxa.Therefore, the general solution isF=F0+F1, where(87)F0=Pxy22+Pyx22,and F0 for the stress-free edge condition is expressed as [21](88)φpq=4βπp2+β2q22×p−1qεqsinh2pπ/βsinhpπ/βcoshpπ/β+pπ/βAp+q−1pεpsinh2qπβsinhqπβcoshqπβ+qπβBq.Moreover:(89)εp=0.5,p=0,1.0,p>0,εq=0.5,q=0,1.0,q>0,while the coefficients Ap and Bq in equation (88) are the solutions of the linear systems.To derive the expression forPx and Py, all the edges of the plate are assumed to remain straight after the deformation. Following the same procedure earlier adopted for the SS plate, but with the reference CC function and boundary condition, it is found that(90)δx=PxaEh2−νPyaEh2−3Eh232∑m=1∞m2π2a2=0,δy=PybEh2−νPxbEh2−3Eh232∑n=1∞n2π2b2=0.The final solution is found for the immovably constrained edges, given that the elongation along thex and y directions is zero:(91)Px=3π2Eh232a2∑n=1∞m2+ν∑m=1∞n2β21−ν2,Py=3π2Eh232a2∑n=1∞n2β2+ν∑m=1∞m21−ν2.Finally, the stress function for a CC plate with immovable edge conditions is defined as(92)F=3π2Eh232a2∑n=1∞m2+ν∑m=1∞n2β21−ν2y22+3π2Eh232a2∑n=1∞n2β2+ν∑m=1∞m21−ν2x22+132n2β2m2cos2mπxa+m2n2β2cos2nπyb+m2n2β24m2+n2β22cos2nπybcos4mπxa+m2n2β2m2+4n2β22cos4nπybcos2mπxa−n2β216m2cos4mπxa−m216n2β2cos4mπxa−m2n2β2m2+n2β22cos2nπybcos2mπxa.For easy understanding, as in the case of the SS plate, equation (1) is expressed as I(w, F). The solution for the membrane stiffness of the plate is therefore derived as(93)∫0a∫0bIw,Fsin2mπxasin2nπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,t×sin2mπxasin2nπybdxdy=0.By employing the Ritz–Galerkin method to solve equation (53), the bending stiffness K1 is first calculated as(94)K1=π4Dab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4,where the mass is(95)m=9ρabh64.The membrane stiffnessK3 for a CC plate is therefore obtained by applying the same methodology used for the SS plate.For the immovably constrained edge condition, it is found that(96)K3=abπ4Eh3649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22.For the stress-free edge condition:(97)K3=abπ4Eh32a2b2φ¯01+φ¯10+φ¯11+12φ¯12+12φ¯21+φ¯02+φ¯20−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.For the movable edge condition, finally:(98)K3=abπ4Eh364a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22,where the force Pt is given by(99)Pt=ab4P1t.Therefore,(100)∫0a∫0b∂2F∂x2∂2w∂y2=−∑m=1∞∑n=1∞py3abπ2m216a2+m2n2β2×2nπb2mπa2ab8+132n2m2β24m2+n2β22×2nπb2mπa2ab16+n2m2β2m2+4n2β22×4nπb2mπa2ab32+m216n2β2×4nπb2mπa2ab32+n2m2β2m2+n2β22×2nπb2mπa2ab8,or(101)∫0a∫0b∂2F∂y2∂2w∂x2=−132∑m=1∞∑n=1∞px6abπ2n2b2+n2β2m2×nπb22mπa2ab8+n2m2β24m2+n2β22×nπb24mπa2ab32+n2m2β2m2+4n2β22×nπb22mπa2ab16n2β216m2×nπb24mπa2ab32+n2m2β2m2+n2β22×nπb22mπa2ab8.Here,∫0b∫0a2∂2F/∂x∂y∂2w/∂x∂y=0.Finally, summing up the above expressions, it is found that(102)9ρabh264f¨+Dhπ4ab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4f+π4Eh4649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22f3=ab4pt.By substitution ofm = 1 and n = 1, the expression for different CC edge conditions can be therefore finally obtained.The solution for a CC plate with immovable edges is given by(103)ρh2f¨+16Dhπ49a43+2ab2+3ab4f+π4Eh4a41+2νa2/b2+a4/b481−ν2+19a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2.The solution for the CC plate with movable edges is(104)ρh2f¨+Dhπ49a43+2ab2+3ab4f+π4Eh49a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2,while the solution for the CC plate with stress-free edges is(105)ρh2f¨+Dhπ49a43+2ab2+3ab4f+32π4Eh49a2b2φ01′+φ10′+φ11′+12φ12′+12φ21′+φ02′+φ20′−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22f3=16π2.


## 5. Static Analysis

As a further extension of the proposed SDOF approach, static loads were also taken into account. Besides, such a loading condition has no correlation with blast events, and the same equations ((25) and (26)) with slight modifications can be in fact efficiently used to predict the static response of a given elastic plate.The governing equation is obtained from equation (25), by assuming Y¨t=0 and replacing Y(t) with the static maximum deflection Ym. Therefore, original equations (25) and (26) take the following form:(106)wx,y,t=Ymφx,y,t,K1stYm+K3stYm3=P0.


### 5.1. Simply Supported (SS) Edges

For general plates with all simply supported edges, the bending stiffness is defined as(107)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22,while the membrane stiffness K3 modifies with the edge condition.For the immovably constrained edge condition, in particular, it is found that(108)K3=ab4π4Eh38a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m41−ν2+121+a4b4∑m=1∞∑n=1∞m2n2.For the stress-free edge condition:(109)K3=ab42π4Eh3a2b2φ¯01+φ¯10−132a2b2+b2a2f3.For the movable edge condition, it is(110)K3=ab4π4Eh316a41+a4b4∑m=1∞∑n=1∞m2n2,where the force P(t) is expressed as(111)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1t.


### 5.2. Clamped (CC) Edges

The bending stiffnessK1 is defined as(112)K1=π4Dab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4.Regarding the membrane stiffnessK3, for the immovably constrained edge condition, it is given by(113)K3=abπ4Eh3649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22.For the stress-free edge condition:(114)K3=abπ4Eh32a2b2φ¯01+φ¯10+φ¯11+12φ¯12+12φ¯21+φ¯02+φ¯20−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.For the movable edge conditions:(115)K3=abπ4Eh364a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.The forcePt is given by(116)Pt=ab4.Again, considering equation (26) in a different form:(117)Y¨t+K1mYt+K3mY3t=Ptm.The latter can be further expressed in a normalized form:(118)Y¨t+ω2Yt+ω2∈Y3t=q0t.The governing parameters calculated from the proposed SDOF model (equation (118)), for both SS and CC plates with various edge conditions, are finally reported in Table 1.Table 1
Derivation of the governing parameters for SS or CC square plates, based on the SDOF model herein developed.
Boundary conditionEdge conditionq0tω2∈Simply supported (SS)Immovably constrained16/π2ρh2ptπ4Eh2/3ρ1−ν2a43/83−ν1+νStress-free16/π2ρh2ptπ4Eh2/3ρ1−ν2a40.194761−ν2Movable edges16/π2ρh2ptπ4Eh2/3ρ1−ν2a43/81−ν2Clamped (CC)Immovably constrained16/π2ρh2pt32π4Eh2/27ρ1−ν2a427/1281+ν+8.49/321−ν2Stress-free16/9ρh2pt32π4Eh2/27ρ1−ν2a40.223551−ν2Movable edges16/9ρh2pt32π4Eh2/27ρ1−ν2a40.26531−ν2


## 5.1. Simply Supported (SS) Edges

For general plates with all simply supported edges, the bending stiffness is defined as(107)K1=π4Dab4∑m=1∞∑n=1∞ma2+nb22,while the membrane stiffness K3 modifies with the edge condition.For the immovably constrained edge condition, in particular, it is found that(108)K3=ab4π4Eh38a4∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m41−ν2+121+a4b4∑m=1∞∑n=1∞m2n2.For the stress-free edge condition:(109)K3=ab42π4Eh3a2b2φ¯01+φ¯10−132a2b2+b2a2f3.For the movable edge condition, it is(110)K3=ab4π4Eh316a41+a4b4∑m=1∞∑n=1∞m2n2,where the force P(t) is expressed as(111)Pt=4abπ2∑m=1,3,5,…∑n=1,3,5,…1mnP1t.


## 5.2. Clamped (CC) Edges

The bending stiffnessK1 is defined as(112)K1=π4Dab4∑m=1∞∑n=1∞3m4a4+2m2n2a2b2+3n4b4.Regarding the membrane stiffnessK3, for the immovably constrained edge condition, it is given by(113)K3=abπ4Eh3649∑n=1∞n4β4+2νβ2∑m=1∞∑n=1∞m2n2+∑m=1∞m48a41−ν2+m4a4+m48a4+n4b4+n48b4+2m4n4b2m2+a2n22+m4n44b2m2+a2n22+m4n4b2m2+4a2n22.For the stress-free edge condition:(114)K3=abπ4Eh32a2b2φ¯01+φ¯10+φ¯11+12φ¯12+12φ¯21+φ¯02+φ¯20−132a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.For the movable edge conditions:(115)K3=abπ4Eh364a2b2a4b4+98+2a/b41+a2/b22+a48b4+a/b41+4a2/b22+a/b44+a2/b22.The forcePt is given by(116)Pt=ab4.Again, considering equation (26) in a different form:(117)Y¨t+K1mYt+K3mY3t=Ptm.The latter can be further expressed in a normalized form:(118)Y¨t+ω2Yt+ω2∈Y3t=q0t.The governing parameters calculated from the proposed SDOF model (equation (118)), for both SS and CC plates with various edge conditions, are finally reported in Table 1.Table 1
Derivation of the governing parameters for SS or CC square plates, based on the SDOF model herein developed.
Boundary conditionEdge conditionq0tω2∈Simply supported (SS)Immovably constrained16/π2ρh2ptπ4Eh2/3ρ1−ν2a43/83−ν1+νStress-free16/π2ρh2ptπ4Eh2/3ρ1−ν2a40.194761−ν2Movable edges16/π2ρh2ptπ4Eh2/3ρ1−ν2a43/81−ν2Clamped (CC)Immovably constrained16/π2ρh2pt32π4Eh2/27ρ1−ν2a427/1281+ν+8.49/321−ν2Stress-free16/9ρh2pt32π4Eh2/27ρ1−ν2a40.223551−ν2Movable edges16/9ρh2pt32π4Eh2/27ρ1−ν2a40.26531−ν2


## 6. Validation of the Enhanced SDOF Model

For the validation of the presently developed SDOF model, the analytical results reported in [33] by Feldgun et al. for elastic thin plates were first taken into account. Figure 3, in this regard, shows the calculated stiffness coefficient K1 for SS and CC plates, along with the past SDOF results from [33]. For both SS and CC plates and variable a/b values, it can be observed that the results of the present SDOF approach are in close agreement with [33], and a mostly perfect match can be observed for various geometrical configurations.Figure 3
Comparison of bending stiffness predictions (K1), as obtained from the current SDOF model and literature [33].Similarly, further validation was carried out for additional edge restraint conditions, namely, represented by immovable, movable, and stress-free edges (for both the SS and CC plates).The corresponding results for theK3 membrane stiffness coefficient are proposed in Figure 4. Also, in this case, a close match can be perceived from the present SDOF model and the past calculations presented in [33].Figure 4
Comparison of membrane stiffness predictions (K3), as obtained from the current SDOF model and literature [33]. (a) Immovable edges. (b) Movable edges. (c) Stress-free edges.
(a)(b)(c)Finally, the current SDOF model estimates were also validated for additional combinations of edge conditions, as reported in Figure5.Figure 5
Variation of the membrane stiffness coefficient (K3) as a function of a/b, as obtained with m = 1 and n = 1 for (a) SS and (b) CC plates with different edge conditions.
(a)(b)


## 7. Accuracy of SDOF Solutions for SS and CC Plates

In order to check the accuracy of SDOF results, another derivation was carried out for SS and CC plates. In particular, it is shown that (for any value ofm and n) the results from the proposed SDOF approach converge and are exact. This is the intrinsic advantage of infinite trigonometric series function and therefore the major issue of sinusoidal load-based SDOF formulations of literature.To this aim, the deflection for SS or CC plates is first defined by equations (119) and (120), respectively.(119)∫0a∫0bDw,Fsinmπxasinnπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,tsinmπxasinnπybdxdy=0,(120)∫0a∫0bDw,Fsin2mπxasin2nπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−h∂2F∂y2∂2w∂x2+∂2F∂x2∂2w∂y2−2∂2F∂x∂y∂2w∂x∂y−qx,y,tsin2mπxasin2nπybdxdy=0.The above equation can be rewritten in the following form:(121)∫0a∫0bDw,Fsinmπxasinnπybdxdy=∫0a∫0bρh∂2w∂t2+Eh3121−υ2∇2w−Nxx∂2w∂y2+∂2w∂x2Nyy−2Nxy∂2w∂x∂y−qx,y,tsinmπxasinnπybdxdy=0,where(122)Nxx=h∂2F∂y2,Nxy=−h∂2F∂x∂y,Nyy=h∂2F∂x2,are the membrane forces. Using von Karman equations and focusing on the membrane and bending actions, for the SS plate, it is found that(123)mπa2Nxxcr+nπb2Nyycr=π4Dab4∑m=1∞∑n=1∞ma2+nb22,(124)m2Nxxcrπ2a2D+n2Nyycrπ2b2D=m2a2+n2b22.Therefore, in order to predict the out-of-plane response of the plate,m and n are chosen in such a way that the values Nxx and Nyy are minimized for any given value of Nxx/Nyy.In the limit case of a square plate subjected to a uniform pressureP(t) on its middle plane, it is a = b and therefore Nxx = Nyy. Accordingly, equation (124) becomes representative of a critical value (Nxx)cr that is defined as(125)Nxxcr=π2Da2m2+n2min.For this limit condition only, the minimum (Nxx)cr value of equation (125) is achieved for m = 1 and n = 1. On the other hand, in the general case of rectangular plates, neither Nxx nor Nyy is able to converge at m = 1 and n = 1.Hence,(126)F=−Eh232a2b2∑m=1∞cos2mπxa+b2a2∑n=1∞cos2nπyb,Nyy=h∂2F∂x2=Eh3322mπa2a2b2,Nxx=h∂2F∂y2=Eh3322nπb2b2a2.The comparison of the above equations leads to(127)Nyy=2mπ/b22nπ/a2,Nxx=m2n2a2b2Nxx,m2Nxxcrπ2a2D+n2Nxxcrπ2b2Dm2n2a2b2=m2a2+n2b22,Nxxcr=π2Dm2/a2+n2/b22m2/a2+m2a2/b4.Finally, given that the (Nxx)cr is obtained when n = 1, it follows that(128)Nxxcr=π2Dm2/a2+1/b22m2/a2+m2a2/b4,or(129)Nxxcr=π2Dm2+a2/b22m2+m2a4/b4,Accordingly, the smallest value ofNxx at which the denominator of one of the terms in equations (128) and (129) becomes zero represents the critical value (Nxx)cr.Figure6(a) shows the variation of Nxx with the a/b ratio for thin elastic plates. Comparative results are proposed for m = 1, 2, …, 6. From the collected plots, it is possible to observe that all Nxx are converging, but with the exception of the m = 1 case. It must also be noted that the deflection expression for SS plates is an odd function. Consequently, it is necessarily required to choose the minimum converged odd m value, to achieve the exact solution. For the calculation examples in Figure 6(a), the minimum critical value for SS plates is therefore m = 3.Figure 6
Variation ofNxx as a function of a/b: (a) SS plates and (b) CC plates.
(a)(b)By following a similar procedure, the variation ofNxx and Nyy with a/b can be found for CC plates. In the latter case, the maximum values of infinite trigonometric series function are represented by equations (130) and (131), respectively:(130)Nxx=h∂2F∂y2=−132m2n2β22nπb2+m2n2β24m2+n2β222nπb2+m2n2β2m2+4n2β224nπb2−m216n2β24nπb2−m2n2β2m2+n2β222nπb2,(131)Nyy=h∂2F∂x2=−132n2β2m22mπa2+m2n2β24m2+n2β224mπa2+m2n2β2m2+4n2β222mπa2−n2β216m24mπa2−m2n2β2m2+n2β222mπa2.Therefore, by comparing each term in the above equations, it can be seen that they are defined by(132)Nyy=m2n2a2b2Nxx,m2Nxxcrπ2a2D+n2Nxxcrπ2b2Dm2n2a2b2=123m4a4+2m2n2a2b2+3n4b4,Nxxcr=1/23m4/a4+2m2n2/a2b2+3n4/b4m2/π2a2D+n2/π2b2Dm2/n2a2/b2.The critical value (Nxx)cr is obtained by assuming n = 1. Accordingly, the above equation becomes(133)Nxxcr=C13m4/a4+2m2/a2b2+3/b4m2/a2+m2a2/b2,or(134)Nxxcr=C13m4+2a2m2/b2+3a4/b4m2+m2a4/b4,The so-calculated values are proposed in Figure6(b).As in the case of the SS plates, it can be still observed that the collectedNxx values are not converging for m = 1, but they are indeed converging for all other cases (m = 2, 3, 4, 5, 6). The exact solution for CC plates is given by the minimum converged value, that is, m = 2 from Figure 6(b).


## 8. Experimental Validation for Square Plates under Blast

In order to further assess the reliability of SDOF estimates, additional comparative calculations were carried out by taking into account the experimental study reported in [38] for ship panels subjected to air blast waves.In [38], in particular, Houlston et al. carried out a series of field experiments (12 repetitions in total) on square steel plates with an effective span of 508 mm and a thickness of 3.4 mm. Young’s modulus of elasticity was E = 207 GPa, with ν = 0.3 being Poisson’s ratio and ρ = 7770 kg/m3 being the mass density. No dynamic increase factor (DIF) has been taken into account in [38], for the reference material properties. In this paper, for comparative and validation purposes, the same input parameters were taken into account, and any DIF was still disregarded.The dynamic response of the reference steel plate has been explored in [38] under a typical blast pressure-time history agreeing with Figure 7. The measured pressure from the experimental trial (three transducers) is reported, together with an average fitting pressure history. The typical test setup has been thus arranged so as to include the steel target plate, with fully clamped (CC) edges and subjected to uniform pressure. The restraint details have been realized by means of four bolted steel box beams, clamped to a concrete base seated on the ground.Figure 7
Reference field blast pressure time-history (adapted from [38] with permission from Elsevier®, Copyright license number: 4853550044917, June 2020).As far as the fitting pressure history of Figure7 is taken into account and uniformly distributed on the plate surface for the SDOF model herein proposed (with both the positive and negative phases included), major calculation results are collected in Figure 8, where the out-of-plane displacement at the center of the plate is shown. More in detail, both the linear and nonlinear approaches are compared for the CC boundary configuration. Together with the past experimental estimates from [38], additional comparative data are proposed for the same steel plate, as obtained from FE nonlinear dynamic analysis reported in [38], as well as from the SDOF formulation presented in [33].Figure 8
Out-of-plane displacement for a square steel plate under field blast loads, as obtained from the presently developed SDOF model and from literature [33, 38]. In evidence, the results are given by the (a) linear and (b) nonlinear solutions for the CC boundary condition.
(a)(b)In Figure8, it can be observed that the presently developed SDOF model is in close agreement with the displacement predictions from earlier researchers. Further, for both the linear and nonlinear CC plates, the current SDOF model estimates are in rather good agreement with the SDOF results given in [33]. As such, it can be concluded that the present SDOF model predicts the measured time history very well and provides good estimates both for displacement amplitudes and periods of vibration.As far as the SS condition is also assessed for the same steel plates, as also expected, Figure9 shows an increased scatter of the SDOF calculations from the corresponding experimental estimates. As such, this further emphasizes the need of analytical solutions that could be adapted to multiple restraint conditions of technical interest. On the other hand, the agreement of the displacement plots with the past SDOF predictions from [33] is still good.Figure 9
Out-of-plane displacement for a square steel plate under field blast loads, as obtained from the presently developed SDOF model and from literature [33, 38]. In evidence, the results are given by the linear and nonlinear solutions for the SS boundary condition.


## 9. Conclusions

Single-degree-of-freedom (SDOF) methods can offer a robust support in the analysis of structural systems. On the other hand, the basic assumptions can often involve approximations and misleading estimates. This can be the case of thin elastic plates under blast loads, where special care among other aspects should be given for the load description.In this paper, an enhanced SDOF formulation was developed for the linear and nonlinear analysis of thin elastic plates under blast waves. The key governing parameters have been proposed, more in detail, for different boundary conditions of technical interest. Based on an energy approach, both the linear and nonlinear solutions were explored, by accounting, respectively, for the bending stiffness only or for both the bending and membrane stiffness contributions.Various parameters (such as deflections, membrane stresses, and bending stresses) have been assessed for blast-loaded plates with simply supported or clamped restraints, under different edge conditions (including movable, immovable, or membrane stress-free cases), under the effects of a uniform blast pressure.Compared to existing SDOF formulation, as shown, the major advantage was represented by the use of infinite trigonometric series for the description of incoming blast pressures. The developed SDOF model has been therefore compared, for validation purposes, in terms of predicted stiffness coefficients as well as out-of-plane deflections in time, for a given blast wave. In doing so, major advantage was taken from available experimental and numerical solutions of literature. For both static and dynamic nonlinear solutions reported in the paper, the validity and accuracy of the proposed SDOF model have therefore been discussed.



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