Nonlinear dynamic response of blast-loaded stiffened plates considering the strain rate sensitivity

Marine Structures 70 (2020) 102699

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Nonlinear dynamic response of blast-loaded stiffened plates considering the strain rate sensitivity Ying Peng b, Ping Yang a, b, *, Kang Hu b a b

Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China School of Transportation, Wuhan University of Technology, Wuhan 430063, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Nonlinear dynamic response Stiffened plates Singly symmetric beam model Strain rate sensitivity Instantaneous modes

The aim of the present paper is to develop a simple theoretical method which can quickly calculate the nonlinear dynamic response of stiffened plates under a blast loading. The large deformation behavior of the stiffened plate is analyzed by using a singly symmetric beam model as representative of the stiffened plate. The material is assumed to be rigid-perfectly plastic, and the strain rate sensitivity is considered by using the Cowper–Symonds constitutive model (CS model). By Lee’s extremum principle, the instantaneous modes of nonlinear structural response are determined. A series of calculations are performed to investigate the influence of pulse in­ tensity, pulse duration, plate thickness, stiffener spacing and material property on the displace­ ment response. The obtained results are in good agreement with those of numerical simulations performed by software package ABAQUS, and then a definition for the cases when the simplified method proposed here can be used is provided.

1. Introduction Stiffened plates have been widely used in civil engineering, aerospace and marine structures, because of light weight and high structural efficiency. Such structures are occasionally subjected to blast-type pulse such as gas explosion, air blast or underwater shock in the modern combat environments. The pressure loads generated by the explosion have some common characteristics involving high peak value, short duration and high propagation speed. Consequently, dynamic response of blast-loaded structures becomes more difficult to predict due to its dependence on both the loading history and boundary conditions. Over the past decades, the investigations on the nonlinear dynamic behavior of stiffened plates under blast loadings have received much attention of engineers and scholars. Houlston et al. [1–3] studied the dynamic plastic response of stiffened panels under air blast loading by a series of experiments, ADINA and the finite strip method, and the numerical results approximately agreed with the experimental data. Nurick et al. [4] and Rudrapatna et al. [5] conducted experimental and numerical works on stiffened square plates subjected to blast pressure loading, respectively. Louca and Pan [6,7] considered the effects of boundary conditions, initial imper­ fections and local stiffener buckling on the response of stiffened plates subjected to gas explosions. Yuen and Nurick [8] and Langdon et al. [9] compared the experimental and numerical results of built-in mild steel quadrangular plates with different stiffener config­ urations subjected to uniform and localised blast loading, respectively. The localised blast loading was found to result in higher displacements and lower tearing thresholds in contrast to uniform blast loading. Besides, for the localised blast loading, Bonorchis and * Corresponding author.Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China. E-mail addresses: [email protected] (Y. Peng), [email protected] (P. Yang). https://doi.org/10.1016/j.marstruc.2019.102699 Received 9 June 2019; Received in revised form 13 October 2019; Accepted 2 December 2019 Available online 7 January 2020 0951-8339/© 2019 Elsevier Ltd. All rights reserved.

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Nurick [10] investigated the effects of introducing welding into relatively well established laboratory scale stiffened plate experiments and predicted the effects numerically. The previous works are mainly focus on analytical or numerical deformation predictions and experimental studies. Although significant progress has been made in this field, theoretical analysis is still necessary to further understand the dynamic response mechanism of stiffened plates when subjected to blast loadings, and then is valuable for preliminary design, security studies and hazard assessments. Jones [11–13] was the first to develop a rigid, perfectly plastic method to predict the permanent displacements of ductile beams and plates. The method can be simplified with an approximate yield condition by circumscribing and inscribing the exact yield curve. Schubak et al. [14,15] and Olson [16] presented a simplified rigid-plastic modelling for blast-loaded stiffened plates. Subse­ quently, the method was extended to predict the nonlinear dynamic response of submerged stiffened plates subjected to underwater explosions by Jiang and Olson [17]. The obtained results agreed with the numerical results in general. More recently, the elastic-plastic dynamic response of stiffened plates under confined blast load was investigated by Zheng et al. [18,19] using experimental, theoretical and numerical approaches. The influences of elastic-plastic effect and stiffener location were taken into account in the theoretical prediction and numerical simulation respectively, and the results showed good agreement with the experimental data. However, the behavior of some materials, such as mild steel, is generally sensitive to the strain rate. Many theoretical results overestimate the permanent deflections due to the negligence of strain rate sensitivity. Therefore, in order to predict the dynamic response accurately, the effects of strain rate should be taken into account. Thus, the constitutive strain rate models have to be adopted. Jones [13] presented a theoretical method to predict the dynamic inelastic response for plates made from a strain rate sensitive material, and good agreement was obtained with the corresponding experimental data. Tanimura et al. [20] compared the rate-dependent constitutive models with experimental data. Liu et al. [21] carried out a series of high-speed tensile tests to obtain the stress-strain relationships at different strain rates, and inputted these data to analyze the dynamic response of stiffened plates numerically. Nevertheless, for simplicity, the influence of material strain rate sensitivity was mostly considered by means of numerical simulations. The nonlinear dynamic response of stiffened plates under a blast loading is analyzed on the basis of the simplified method introduced in the present study. The material strain rate sensitivity is taken into account in theoretical analysis through the commonly used CS model, and the comparison is made between the theoretical and numerical results. Thus, the accuracy and feasibility of the simplified beam model is confirmed. Furthermore, the effect of pulse intensity, pulse duration, plate thickness, stiffener spacing and material property on the displacement response is discussed in detail. The conclusions may be useful for clarifying the applied range of the present method, and then improving the accuracy in future rapid assessment of dynamic response. 2. Theoretical model and response mechanisms A stiffened plate spanning a distance 2L with uniform stiffener spacing d is considered (Fig. 1). The longitudinal edges of the stiffened plate are fully clamped. The plate is subjected to a lateral blast-type pulse p(t). The load is suddenly applied and uniformly distributed. Such a beam model with singly symmetric cross section is used to replace the stiffened plate in theoretical analysis that one-half of the distance to each neighboring stiffener is taken as the effective breadth, and the uniformly distributed pressure pulse applies through the beam’s axis of symmetry, as shown in Fig. 2 (a). 2.1. Assumptions The present theoretical analysis is based on the following assumptions: (1) The material of the plate and stiffeners is isotropic, rigid-perfectly plastic and strain rate sensitive, neglecting the effect of elasticity.

Fig. 1. Stiffened plate subjected to uniformly distributed pressure pulse. 2

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Fig. 2. Beam model and mechanisms of deformation.

(2) The effects of shear deformation and rotary inertia are ignored. (3) The effect of axial force on yield condition must be taken into account for the finite deflection analysis of the beam with axial constraints at both ends, and the axial force is taken as constant along the beam. 2.2. General description of beam response The motion of a clamped beam subjected to uniformly distributed pressure pulse is assumed to proceed under the following three different mechanisms depending on the load intensity and the beam deflection: (1) If the load intensity is low, the deflected shape of the beam consists of stationary plastic hinges locating at the supports and the mid-span with two rigid connecting pieces. This type of response is referred to as the mid-span hinge mechanism shown in Fig. 2 (b). (2) If the load intensity is high, except for the stationary plastic hinges locating at the supports, the deflected shape of the beam consists of two travelling plastic hinges, which are symmetrically located at some distance X(t) to either side of the mid-span, as shown in Fig. 2 (c). The travelling hinges will move towards or outwards the mid-span as the motion proceeds with the variation of pressure pulse. This mode of response is referred to as the travelling hinge mechanism. (3) As the motion continues and the deflection increases, the axial force in the beam increases and the bending moment resistance decreases. When the entire beam section yields in tension, the beam can no longer resist moments. As a result, the beam re­ sponds as a plastic string with a constant tension. Therefore, this response mechanism is referred to as the string mode shown in Fig. 2 (d). For the blast-type pulse, the load intensity is usually very high (several times the static collapse load), and then the initial motion of the beam will proceed under the travelling hinge mechanism [22,23]. 3. The bending moment-axial force interaction relation and associated plastic flow rule In the dynamic plastic response analysis of axially constrained beams with large deformation, the axial force induced by the extension of centroidal axis cannot be ignored when the deflection reaches the magnitude of the beam height. The effect of axial force on yield condition must be taken into account, so the bending moment-axial force interaction relation should be used in the analysis. 3.1. The bending moment-axial force interaction relation For a singly symmetric cross section, the interaction relation between axial force N and bending moment M can be determined by the specific section, if the neutral axis does not parallel to the symmetric axis. Since the beam with a singly symmetric I-section is usually adopted as representative of the stiffened plate elements, the bending moment-axial force interaction curve is derived from the section shown in Fig. 3. It is assumed that the axial force is positive in tension and bending moment is positive if the outer fiber of the flange is in tension. 3

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If there’s no axial force in the cross section, the plastic neutral axis (PNA) is initially located in such a place that it divides the cross section into two equal areas. Assuming the plastic neutral axis is initially in the effective breadth, the ultimate bending moment Ms is: � � � � � �� �t 1 h Ms ¼ Ap tp α þ α2 þ Aw w þ g þ Af f þ hw þ g σs (1) 2 2 2 where σ s is the static yield stress, Ap ¼ bptp is the area of effective breadth, Aw ¼ hwtw is the area of web and Af ¼ bf tf is the area of flange. α ¼ (Ap-Aw-Af)/2Ap is a constant determined by the cross section. g ¼ α⋅tp is the distance from the plastic neutral axis to the inner side of

the effective breadth. As the beam deflection increases, both bending moment and axial force will exist in the cross section, and then the plastic neutral axis will offset by the distance a from the initial position. When the section yields, the axial force N is: (2)

N ¼ 2bp aσ s With A representing the total area of the cross section, the ultimate axial force Ns is:

(3)

Ns ¼ Aσs The non-dimensional axial force is defined as: � � n ¼ N Ns ¼ 2bp a A

(4)

When the section yields, the bending moment M is: � � � � A2 2 1 8 bp g2 þ bp t2p þ n nA hw þ tf hc þ g þ bp tp 2g hw þ tf hc > 2 4bp > > > > > > > � � � > � > > hw tf � 2bp g > > �n�1 h h t t t σs þ b h > w w c f f f c > > A 2 2 > > > > � � � � � > > � > A2 2 bp g hw > > þ t n nA þ h þ t h g h þ t h h t 2b h > w f c p w f c w w c f > tw 4tw 2 > > < M¼ # � > > � � > b2p g2 2 hw tw þ bp g tp � tf � 2bp g > > > þ b t þ t h þ t σ h h �n� þ b p p w f c f f c s > > A A 2 2 > tw > > > > >� � � > � > � A2 2 bp g þ hw tw tp � > > > n þ nA hc tf þ bp tp hw þ tf hc þ þ 2bp g hc tf > > 4b b 2 > f f > > > > > > � > � � � : b g þ h t �2 � 2 hw tw þ bp g hw tf � p w w þ hw tw hc tf þ σs 1 � n � þ bf tf hc bf 2 2 A

(5)

� � � �� � � At t where hc ¼ Ap 2p þhw þtf þAw h2w þtf þ 2f f A is the distance from the centroidal axis to the outer side of the flange. Similarly, the non-dimensional bending moment is defined as: (6)

m ¼ M=Ms

Substituting Eqs. (1) and (5) into Eq. (6), the relation between m and n, viz. the bending moment-axial force interaction relation is obtained, as shown in Fig. 4 (a). With the beam deflection increasing, the axial tension increases gradually, and the plastic neutral axis

Fig. 3. Cross section of a singly symmetric I-beam. 4

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offsets accordingly. When the neutral axis coincides with the centroidal axis, the ultimate bending moment M0 occurs at the supports, and the axial force of the beam is: (7)

N ¼ λNs where λ is the asymmetry parameter, which can be determined by the given cross section as: � � �� λ ¼ 2 hc tf tw þ Af Ap Aw A For a doubly symmetric section, λ will be zero. Simultaneously, the ultimate bending moment M0 can be obtained as: � � � � � � � �2 tp hw tf � M0 ¼ Ap þ hw þ tf hc þ Aw σs þ tf hc þ Af hc þ tw hc tf 2 2 2

(8)

(9)

However, it is very difficult to utilize the interaction curve shown in Fig. 4 (a) due to its nonlinearity. Therefore, the linearized curve by an inscribed polygon is used to simplify the calculation in this paper, as shown in Fig. 4 (b). Thus, the bending moment at the plastic hinges can be easily determined as: MX ¼

1 λ M0 ; ML ¼ 1þλ

(10)

M0

where the subscripts X and L denote variables at the travelling hinges (x ¼ �X) and at the support hinges (x ¼ �L), respectively. Combining Eq. (10), the static ultimate load q0 is obtained by the principle of virtual work as: q0 ¼

2ðMX ML Þ 4M0 ¼ L2 ð1 þ λÞL2

(11)

3.2. The associated plastic flow rule The plastic hinges formed by bending moment and axial force are defined as the generalized plastic hinges. Then both extensional strain rate and curvature rate are involved at the generalized plastic hinges. Some geometric relations for the case of beam mechanism 2 are given as follows: For the travelling hinge mechanism response (Fig. 2(c)), the total centroid extension of the beam with finite deflections occurs at the plastic hinges at the rate: e_ ¼ 2e_X þ 2e_L ¼

2W W_ L X

(12)

where W and W_ are the mid-span displacement and velocity, respectively. Meanwhile, the beam is rotated through the finite angles θX at the travelling plastic hinges and θL at the support plastic hinges. Assuming these angles positive in sagging, the rotations change at the rate:

Fig. 4. Interaction curve and associated plastic flow rule for a singly symmetric I-beam: (a) actual curve; (b) linearized curve. 5

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θ_ X ¼

θ_ L ¼

W_ L

(13)

X

Combining Eqs. (12) and (13), an expression for W in terms of the ratio of extensional strain rate to curvature strain rate at the plastic hinges can be written as: _ ðe= _ θÞ X

(14)

_ ¼W ðe= _ θÞ L

_ of a hinge must be outwardly normal to the _ Ms θ) According to the associated plastic flow rule, the plastic deformation vector (Ns e, yield curve at the considered stress state. Due to the kinematic constraints, stress states can exist only at points of slope discontinuity of the approximate yield curve, because the deformation ratios can lie within a range of values at these points [14], as shown in Fig. 4 (b). Once finite displacements of the beam occur, the stress states will correspond to the points B and B’ at the travelling and support hinges, respectively. The deformation ratios at these points can be given as: Ns e_X M0 =Ms ¼ ; 1þλ Ms θ_ X

M0 =Ms Ns e_L M0 =Ms � � 1 λ 1þλ Ms θ_ L

and hence _ ¼ ðe= _ θÞ X

M0 Ns ð1 þ λÞ

(15)

M0 M0 _ � � ðe= _ θÞ L Ns ð1 λÞ Ns ð1 þ λÞ

(16)

Substituting Eqs. (15) and (16) into Eq. (14), the hinge mechanism response occurs when the displacement is in the range of: (17)

0 � W � Ws

2M0 where Ws ¼ N ð1 is the critical displacement that a plastic string occurs. For displacements outside of this range, the stress states λ2 Þ s

jump instantaneously to point C. Consequently, the moment resistance is reduced to zero, and the beam behaves as a plastic string under constant tension Ns. When the load intensity is low or the travelling plastic hinges overlap into a single hinge at the mid-span, the beam will deform in accordance with beam mechanism 1, and the derivation is similar as above. 4. Analytical methods 4.1. Governing equations of motion In order to simplify the nonlinear dynamic problems, the instantaneous mode approach proposed by Lee and Martin [24] is used here. Thus the dynamic response process is based on a sequence of instantaneous modes. For the blast-type pulse response, due to the high load intensity, the instantaneous modes are assumed to approximatively conform to the travelling hinge mechanism, shown in Fig. 2(c). Then the response of the beam can be described by: wðx; _ tÞ ¼ VðtÞφðx; tÞ

(18)

w € ðx; tÞ ¼ AðtÞφðx; tÞ

(19)

where V(t) and A(t) are the mid-span velocity and acceleration, respectively, and φ(x,t) is the instantaneous mode shape function. Only one half of the beam is considered because of its symmetry. The mode shape function can be written as: 8 1 0 � x � XðtÞ < φðx; tÞ ¼ (20) L x : XðtÞ < x � L L XðtÞ Recalling Eqs. (12) and (13), respectively, the total centroid extensional strain rate and curvature strain rate at the plastic hinges can be rewritten as: e_ ¼ 2e_X þ 2e_L ¼ θ_ X ¼

θ_ L ¼

2WV L X

(21)

V L

(22)

X

During the instantaneous mode response, the kinetic energy K of the beam is:

6

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Z

L

K¼2 0

1 1 m½wðx; _ tÞ�2 dx ¼ mðL þ 2XÞV 2 2 3

(23)

and it changes at the rate: Z L 2 K_ ¼ 2 mwðx; _ tÞ€ wðx; tÞdx ¼ mðL þ 2XÞVA 3 0

(24)

where m is the mass per unit length of the beam. The rate of external work done by the load q(t) is: Z L E_ ¼ 2 qðtÞwðx; _ tÞdx ¼ qðtÞðL þ XÞV

(25)

0

The rate of energy dissipation due to plastic deformations is: D_ ¼ N

n X

n X

e_i þ i¼1

2ðMX Mi θ_ i ¼

i¼1

L

ML þ NWÞ V X

(26)

where ei and θi are the extension and rotation of the ith plastic hinge in the beam, respectively. By Lee’s extremum principle [25], the kinetic energy K is kept constant during the variation of X, and the instantaneous mode extremizes the functional: _ Jðφ; tÞ ¼ Dðφ; tÞ

_ Eðφ; tÞ ¼

(27)

_ KðtÞ

Substituting Eqs. (24)–(26) into Eq. (27), the functional J is rewritten as: � � MX ML þ NW 2 JðXÞ ¼ 2 qðtÞðL þ XÞ V ¼ mðL þ 2XÞVA L X 3 The governing equation of motion of the beam can then be given by Eq. (28) as: � � � qðtÞ L2 X 2 2ðMX ML Þ 3N 3 � WðtÞ ¼ AðtÞ þ L2 þ XL 2X 2 2m m L2 þ XL 2X 2

(28)

(29)

In addition, making use of the relationship between K and V in Eq. (23), the functional J in Eq. (28) can also be expressed as: 3 2 rffiffiffiffiffiffi � 2 2 X 7 3K 6 7 62ðMX ML þ NWÞ qðtÞ1 L JðXÞ ¼ (30) 5 m4 ðL XÞðL þ 2XÞ2 It’s worth noting that K is independent of the variation of X and held constant at the instant considered. Moreover, by Lee’s principle, X will correspond to the extrema of J in Eq. (30). Thus, the travelling hinge positions can be obtained by setting dJ/dX ¼ 0 as: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 � > 6ðMX ML þ NWÞ : 0 otherwise 4.2. Response solution Since the instantaneous modes conform to a travelling hinge mechanism for the beam subjected to blast loadings, the coefficients of Eq. (29) are changing accordingly with the variation of travelling hinge position X. Thus the analytical solution can hardly be obtained. Therefore, it is necessary to find a semi-analytical solution for the coupled equations. Through dividing the response period into short enough time steps, the travelling hinge position can be considered as a constant _ € within the step. Then the mid-span velocity and acceleration become VðtÞ ¼ WðtÞ and AðtÞ ¼ WðtÞ, respectively. Consequently, the governing equation of motion for the step is decoupled and simplified to a linear, second-order differential equation: � � � 2ðMX ML Þ qðtÞ L2 X 2n 3N 3 € þ � WðtÞ WðtÞ ¼ (32) 2m L2 þ Xn L 2X 2n m L2 þ Xn L 2X 2n where Xn is the travelling hinge position of the nth time step, which is determined at the beginning of the step and held fixed throughout the step. Note that the mid-span displacement W and the beam’s kinetic energy K at the instant considered are held constant, whereas the beam’s velocity field will have a finite change through mode changes. Meanwhile, by Lee’s principle, the instantaneous modes are chosen to maximize the rate of change of the kinetic energy. Consequently, the displacement response of the beam may be 7

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overestimated due to the conservation of the kinetic energy. _ _ þ Here, denoting the mid-span velocity at an instant before t ¼ tn as Wðt n Þ and at an instant after as Wðt n Þ, a function δ(t) is introduced to describe the difference between two velocity fields. When the mode of the beam changes between the nth and (nþ1)th time steps, the function is presented as: Z � ��2 1 L � δðtn Þ ¼ w_ tþ m w_ tn dx (33) n 2 L _ Substituting Eq. (18) into Eq. (33), and noting VðtÞ ¼ WðtÞ in each time step, δ(tn) is rewritten as: Z L � � � �2 1 δðtn Þ ¼ m W_ tn φn ðxÞ W_ tþ n φnþ1 ðxÞ dx 2 L

(34)

_ þ Þ ¼ 0, δ(tn) is minimized, and then the two velocity fields are most similar. Thus, recalling the mode shape By setting dδðtn Þ=dWðt n function, Eq. (20), and integrating it, the transformation relation between the two velocity fields is obtained as: � 2 � � 3 L2 X 2nþ1 ðL Xn Þ _ ¼ W t W_ tþ (35) n n 2ðL þ 2Xnþ1 ÞðL Xnþ1 Þ 4.3. Strain rate sensitivity It is well known that mild steel will become stronger due to the effect of strain rate. As an important influential factor, the strain rate sensitivity should be taken into account to accurately predict the dynamic response for structures. Various constitutive models have considered the effect of strain rate in the literatures. Among these models, the Cowper-Symonds constitutive model (CS model) is most widely used, which is uniaxial, perfectly plastic, and can be empirically described as follows [13]: � ε_ �P σd ¼1þ σs C 1

(36)

where σd is the dynamic yield stress associated with the plastic strain rate ε_ . σs is the static yield stress. C and P are Cowper-Symonds coefficients. For mild steel, the values C ¼ 40.4/s and P ¼ 5 are usually adopted in calculations [26]. In order to evaluate the strain rates, it is necessary to approximate the geometry of plastic hinges. Otherwise, the strain rates cannot be calculated because of the zero length of point hinges in rigid-plastic bending. For simplicity, as adopted in Ref. [15], the following analysis is also based on the assumption proposed by Nonaka. Then, the plastic hinges act like uniform slip-line fields according to quasi-static slip-line theory in plane strain. Thus, these slip-line fields result in magnitudes of strain rate and stress that are constant through the beam depth, as indicated in Fig. 5. Consequently, M and N in above equations need only be modified by multiplying the functions of strain rate. When the plastic hinge rotates at the rate θ_ (assuming it is positive), the fiber at a distance of z from the neutral axis is extended at _ Note that the length of the hinge at this location is 2z, as shown in Fig. 5. The strain rate is considered as a constant through the rate zθ. the beam depth, and can be obtained as:

ε_ ¼

zθ_ 1 _ ¼ θ 2z 2

(37)

As mentioned above, the mid-span velocity becomes V ¼ W_ in the instantaneous mode response. Substituting Eq. (22) into Eq. (37), and then putting the resulting strain rate in Eq. (36), the CS constitutive relation is transformed into the following form:

Fig. 5. Model of a plastic hinge. 8

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�

�P1

σd W_ ¼1þ σs 2CðL XÞ

(38)

Obviously, the dynamic yield stress in the plastic hinges corresponds to the mid-span velocity and the travelling hinge position at _ n 1 Þ can be the instant considered. Within the nth time step, both the travelling hinge position Xn ¼ X(tn-1) and mid-span velocity Wðt determined at the beginning of the step and kept constant throughout the step. Thus, the fully plastic, rate sensitive section capacities Mdn and Ndn can be obtained as: � �1 _ n 1Þ P Wðt Mdn Ndn ¼ ¼1þ 2CðL Xn Þ M0 Ns

(39)

By substituting Eq. (39) into Eq. (32), the governing equation of motion considering the effect of strain rate can be obtained, and it may be solved as an initial value problem. 5. Examples and numerical verification In the theoretical results presented in Refs. [14,22], the strain rate effect of material was not taken into account for simplicity. However, the plastic flow of mild steel is very sensitive to the strain rate. Thus the strain rate effect definition is important for esti­ mating the dynamic response accurately. The stiffened plate with four fully clamped edges in Refs. [14,22] is still adopted to confirm the method developed in this paper. The geometric configuration of the stiffened plate is indicated in Fig. 6, and the dimensional parameters are listed in Table 1. The stiffened plate here is made of mild steel. The behavior of the material is assumed to be isotropic, rigid-perfectly plastic and strain rate sensitive. In the present theoretical analysis, the CS model is adopted to consider the effect of strain rate sensitivity. The detailed material properties are listed in Table 2. Based on Youngdahl’s impulse equivalent principle [27], the dependence of the pulse shape on the permanent deformation can be eliminated by introducing an effective load and the time at the centroid of the pulse, as discussed in Ref. [22]. Therefore, the blast load in this paper may be assumed to be a rectangular pulse for simplicity, and applied uniformly on the stiffened plate. For comparison, a rectangular pulse of constant intensity pm ¼ 1.779 MPa and of duration td ¼ 2 ms is also considered, as shown in Fig. 7. The pulse duration is significantly less than the fundamental period T0 of elastic vibration, which can be calculated to be approximately 6 ms by numerical simulations, as presented in Table 3. The response of the stiffened plate to this pulse is calculated with time step △t ¼ 0.05 ms. Numerical simulations are performed by the finite element software ABAQUS (Version 6.14), which is considered as an effective method to calculate the dynamic problems involving both geometric and material nonlinearities. The dynamic response is calculated by explicit analysis with the plate model and by implicit analysis with a single beam model, respectively. Among the library of available elements included in ABAQUS/Explicit, the reduced integration rectangular 4-node shell element (S4R) is selected to dis­ cretize the whole stiffened plate model. For the beam model, a 2-node linear beam element (B31) is adopted. The calculated dis­ placements and energies versus time curves can be easily extracted from the history output of ABAQUS. In order to determine the best size of elements based on a compromise between computational cost and accuracy, three types of mesh generations are considered. The detailed mesh sizes and number of elements are described in Table 4. The FE model of stiffened plate with moderate meshes is shown schematically in Fig. 8. The edges of the plate and stiffeners are fully clamped, including re­ straints against lateral and in-plane translations and rotations. The effect of elasticity is considered in FE calculations, which has been neglected in the theoretical analysis. Fig. 9 illustrates the comparison of the mid-span displacement responses. It can be seen that the

Fig. 6. Geometric configuration of stiffened plate. 9

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Table 1 Dimensional parameters of the stiffened plate (Unit: mm). Stiffener spacing

Span length

Plate thickness

Web height

Web thickness

Flange width

Flange thickness

d 914.4

2L 2438.4

tp 6.4

hw 135

tw 7.1

bf 74.9

tf 13.5

Table 2 Material properties of mild steel. Property

Values

Units 3

Mass density ρ Elastic modulus E Poisson’s ratio ν Static yield stress σs Dynamic yield stress σd Cowper-Symonds coefficient C Cowper-Symonds coefficient P

kg/m3 Pa – MPa MPa s 1 –

7.845 � 10 2.07 � 1011 0.3 310 372 40.4 5

Fig. 7. Rectangular pulse. Table 3 Summary of Ws, T0 and p0 with varied geometrical parameters. d (mm)

2L (mm)

tp (mm)

Ws (mm)

T0 (ms)

p0 (MPa)

914.4 914.4 914.4 762 1066.8 1219.2

2438.4 2438.4 2438.4 2438.4 2438.4 2438.4

4.8 6.4 7.9 6.4 6.4 6.4

119 117 117 118 116 116

5.8 6.0 6.5 5.9 6.4 6.7

0.227 0.231 0.234 0.275 0.199 0.174

Table 4 Mesh sizes and number of elements. Case

Coarse mesh Moderate mesh Fine mesh

Plate model

Beam model

Element size for the plate (mm)

Number of elements along the web direction

Number of elements along the flange direction

number of elements

150 100

2 4

2 2

20 40

45

6

4

60

10

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Fig. 8. Mesh for FE model.

Fig. 9. Mid-span displacement responses for different mesh sizes.

displacement responses from plate model and beam model are in good agreement. The mid-span displacements are affected by mesh sizes in numerical simulations, and they will increase with the decrease of mesh sizes. For the plate model, the response curve changes slightly if moderate meshes are further refined. Therefore, the moderate meshes (i.e., dimensions of 101.6 mm � 101.6 mm for the plate, 4 elements along the web direction and 2 elements along the flange direction) are adopted in the following numerical simu­ lations. However, for the beam model, if the number of beam elements along the length direction is specified to be 20, the displacement responses appear to converge. Then theoretical and FE results are compared to verify the accuracy of the theoretical approach pre­ sented in this paper.

Fig. 10. Mid-span displacement response of the stiffened plate under a rectangular pulse. 11

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In order to measure the effect of strain rate, it is necessary to compare the corresponding results neglecting the strain rate sensi­ tivity. The mid-span displacement responses calculated by the present semi-analytical method are compared with those of ABAQUS and Ref. [22], which is shown in Fig. 10. It can be seen that the predicted final displacements are all in excess of the displacement required for string response (Ws ¼ 117 mm). Without strain rate effect, the mid-span displacement calculated by the present method is greater than that of Ref. [22], which was obtained by an analytic method. This is due to the difference in the predicted travelling hinge positions. Another reason is that the travelling hinge mechanism is used to approximate the string response mode shape in the present analysis, while the cosine mode was used in Ref. [22]. Even so, the permanent mid-span displacement is within 10% of the prediction in Ref. [22]. In addition, compared to the beam model in theoretical analysis, both whole stiffened plate model and single beam model are adopted to calculate the dynamic response with ABAQUS. As can be seen from Fig. 10, the numerical solutions obtained from two kinds of FE models are in good agreement, and they are in between two theoretical solutions. Moreover, the stiffened plate does not become still but instead oscillates about some average value, when the mid-span displacement reaches the maximum value. This is due to the presence of elasticity in FE calculations. So the theoretical approach presented in this paper is verified to be feasible and acceptable. It is known that the closed-form solution can hardly be obtained if the strain rate effect is included. In order to further verify the accuracy of the present method, the mid-span displacement response is compared in Fig. 10 with that from a rate-sensitive FE analysis using ABAQUS. In the FE calculation, the CS model is also adopted, and then the yield stress is updated at every time increment. It is evident from Fig. 10 that the theoretical result is in good agreement with those of ABAQUS. Moreover, compared to the final mid-span displacements without strain rate effect, the present prediction is reduced by as much as 30% due to strain rate effect. In order to reveal the mechanism of dynamic response of stiffened plates exposed to blast loading, the movement of plastic hinge should be illustrated. Fig. 11 shows the comparison of travelling hinge positions. It can be seen that, if the strain rate sensitivity is considered, the initial hinge position almost coincides with that of Ref. [22], and the plastic hinges jump rapidly to the mid-span before the blast load is removed. That is to say, the travelling plastic hinges on both sides overlap into a single hinge at the mid-span, and the beam will deform in accordance with beam mechanism 1. However, ignoring the strain rate effect, the travelling plastic hinges continue to move towards the mid-span when the blast load is removed, and the plastic hinges have not met when the deflection becomes so large to initiate the string mode. Fig. 12 shows the energy curves with and without strain rate effect, which are obtained by the FE analysis. The total internal energy, plastic dissipation energy and elastic strain energy of the whole stiffened plate model are denoted by ALLIE, ALLPD and ALLSE, respectively. It is obvious that the elastic strain energy is much smaller than plastic dissipation energy in the total internal energy. So the effect of elasticity can be neglected for simplicity in the theoretical analysis. It also can be seen that the plastic dissipation with strain rate effect is less than that without strain rate effect. That is to say, the response becomes stiffer if the strain rate effect of material is taken into account. Therefore, the results will be overestimated when the strain rate sensitivity is neglected. It is confirmed that the strain rate effect should be considered to obtain more accurate predictions for mild steel. 6. Results and discussion In this section, a series of calculations are performed to investigate the nonlinear dynamic responses of stiffened plates subjected to blast loadings by using the present method and FE code ABAQUS, in which the strain rate sensitivity of material is taken into account. The stiffened plate shown in Fig. 6 is adopted here, and a rectangular pulse of high intensity is still considered for simplicity. For convenience, the beam nearest to the center of the stiffened plate is denoted as B1, and the beam nearest to the boundary as B2, as demonstrated in Fig. 6. Similar to the previous section, both whole stiffened plate model and single beam model are calculated in FE analysis. In order to further measure the accuracy of the simplified beam model adopted in theoretical analysis, a parametric study is carried

Fig. 11. Positions of the travelling plastic hinges. 12

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Fig. 12. Energy of the whole stiffened plate model.

out to analyze the effect of pulse intensity, pulse duration, plate thickness, stiffener spacing and material property on the displacement response. The pulse intensity pm is normalized to the static ultimate load intensity p0, which can be obtained from q0 divided by the stiffener spacing d. The calculated displacement responses are normalized with respect to the displacement Ws required for string response, and the response time is normalized to the fundamental period T0. A summary of Ws, T0 and p0 with varied geometrical parameters is listed in Table 3. Being different from the theoretical solution, the mid-span displacement predicted by ABAQUS is followed by an elastic oscillation after the maximum mid-span displacement is reached, and the mean of oscillation is the permanent displacement of the stiffened plate. The comparisons of the theoretical results with those of FEM are presented in Figs. 13–22. Thus, a definition for the cases when the blast-loaded stiffened plate can be represented by a singly symmetric beam model is provided in this paper. 6.1. Influence of pulse intensity In order to investigate the effect of pulse intensity on the displacement response, three dimensionless load intensities (pm/p0 ¼ 4, 6, 8) are selected for calculations, while the load duration is fixed at td ¼ 2 ms. The dimensionless mid-span displacements W/Ws versus dimensionless time t/T0 curves are plotted in Fig. 13. It is obvious that numerical results obtained by plate model and beam model agree well generally. Moreover, as the load intensity increases, the dis­ placements induced in the stiffened plate increase significantly. For the lower load intensity, such as pm/p0 ¼ 4 and 6, the dimensionless mid-span displacements are less than 1. That is to say, the permanent displacements induced by these pulses are less than that required for string response. However, for the higher load intensity with pm/p0 ¼ 8, a significant amount of string response occurs (W/Ws > 1). Simultaneously, the comparison between the displacement responses of B1 and B2 shows that slight difference begins to appear for this high pulse, and the displacement response obtained by the single beam model is more close to that of B1. Fig. 14 presents the dimensionless permanent mid-span displacement Wf/Ws versus dimensionless pulse intensity pm/p0 curves. It also can be seen from Fig. 14 that, the permanent mid-span displacements predicted by the present method are slightly greater than those of ABAQUS. As the load intensity increases, the permanent displacement almost increases linearly, and the difference between theoretical result and FEM’s becomes larger gradually. Therefore, it can be concluded that the permanent displacement predicted by the simplified beam model proposed in this paper will be further overestimated for the relatively high pulse, especially when the plastic string mode occurs. This is due to the approximation of the string response mode with the travelling hinge mechanism. Even so, the difference between theoretical and FE results is almost within 10%. As thus, to make the discussion more convincing, the dimensionless load intensity will be fixed at 8 in the following subsections. 6.2. Influence of pulse duration In order to check the influence of pulse duration on the displacement response, the load duration (td/T0) is varied in the range of 1/ 4-1/2, while the load intensity is kept constant with pm/p0 ¼ 8. Fig. 15 presents the dimensionless mid-span displacements W/Ws versus dimensionless time t/T0 curves. It is observed that the displacements induced in the stiffened plate also increase significantly with the increase of load duration. For the case when td/T0 � 1/ 3, a significant amount of string response occurs (W/Ws > 1). Simultaneously, the difference between the displacement responses of B1 and B2 increases obviously, and the displacement response obtained by the beam model is more close to that of B1. Also as seen from Fig. 15, the mid-span displacements calculated by the present method are in good agreement with FEM. For the pulses with load duration td/T0 � 1/3, the theoretical results are slightly higher than those of ABAQUS, which is safe for the preliminary assessment. However, for the longer load duration with td/T0 ¼ 1/2, the theoretical result is located between the final displacements of B1 and B2. 13

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Fig. 13. Dimensionless mid-span displacement response curves with varying pulse intensity.

Fig. 14. Dimensionless permanent mid-span displacement versus pulse intensity curves with td ¼ 2 ms.

Fig. 15. Dimensionless mid-span displacement response curves with varying pulse duration.

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Fig. 16. Dimensionless permanent mid-span displacement versus pulse duration curves with pm/p0 ¼ 8.

Fig. 17. Dimensionless mid-span displacement response curves with varying plate thickness.

Fig. 18. Dimensionless permanent mid-span displacement versus plate thickness curves with pm/p0 ¼ 8 and td ¼ 2 ms.

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Fig. 19. Dimensionless mid-span displacement response curves with varying stiffener spacing.

Fig. 20. Dimensionless permanent mid-span displacement versus stiffener spacing curves with pm/p0 ¼ 8 and td ¼ 2 ms.

Fig. 21. Dimensionless mid-span displacement response curves with varying Cowper-Symonds coefficients.

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Fig. 22. Dimensionless mid-span displacement response curves considering the strain hardening.

The dimensionless permanent mid-span displacement Wf/Ws versus dimensionless pulse duration td/T0 curves are compared in Fig. 16. It can be seen that, when the dimensionless load duration is longer than 1/3, the calculated permanent mid-span displacements increase more rapidly, and a significant discrepancy between the final mid-span displacements of B1 and B2 occurs. When the dimensionless load duration is near to 1/2, the prediction of B1 begins to be slightly higher than theoretical result. However, the numerical result calculated by the beam model approaches to the theoretical result gradually. Thus it can be concluded that, if the load duration becomes more longer, the difference between the plate model and beam model will become larger. Compared to the nu­ merical result calculated by the plate model, the permanent mid-span displacement calculated by the theoretical method may be underestimated, and the prediction by the simplified beam model may be dangerous. Therefore, it is confirmed that the rigid-plastic theory in theoretical analysis is suitable for the pulse with load duration significantly less than the natural period, such as in the range of 0.25T0 and 0.5T0. 6.3. Influence of plate thickness Different plate thicknesses as listed in Table 3 are considered to investigate their influence on the displacement response of stiffened plate. The load intensity is kept constant with pm/p0 ¼ 8, and the load duration is fixed at td ¼ 2 ms. The dimensionless mid-span displacements W/Ws versus dimensionless time t/T0 curves are plotted in Fig. 17. As expected, the displacements induced in the stiffened plate decrease significantly with the increase of plate thickness. It is visible from Fig. 17 that the plastic string mode almost occurs (W/Ws > 1) for all the considered cases, and the comparison of theoretical results with those of ABAQUS is good generally. However, as the plate thickness becomes thinner, such as tp ¼ 4.8 mm, the difference between the plate model and beam model becomes larger in the numerical simulation. Fig. 18 presents the dimensionless permanent mid-span displacement Wf/Ws versus plate thickness tp curves. It is found that the calculated permanent mid-span displacement almost decreases linearly as the plate thickness increases, and the difference between theoretical and numerical results calculated by the beam model nearly remains unchanged. However, if the plate model is adopted in the numerical simulation, the difference between theoretical and FE results increases gradually. Besides, a slight difference of final mid-span displacements between B1 and B2 begins to appear when the plate thickness is larger than 6.4 mm. For the case when tp ¼ 7.9 mm, the dimensionless final mid-span displacements calculated by ABAQUS are slightly smaller than 1, while the theoretical method predicts a significant amount of string response. This indicates that, compared to the FE result, the solution by the simplified beam model may be conservative for the stiffened plate with thick plate. 6.4. Influence of stiffener spacing In order to investigate the effect of stiffener spacing on the displacement response, the stiffener spacing (d/L) is altered in the range of 5/8-1 (detailed dimensions listed in Table 3), with the pulse and plate thickness kept unchanged (pm/p0 ¼ 8, td ¼ 2 ms, tp ¼ 6.4 mm). Since the theoretical results agree well with those of ABAQUS for the case when d/L ¼ 6/8 (d ¼ 914.4 mm), as mentioned above, the dimensionless mid-span displacement responses with other three stiffener spacings (d/L ¼ 5/8, 7/8, 1) are compared here in Fig. 19. It is evident that the displacements induced in the stiffened plate decrease significantly with the increase of stiffener spacing. Moreover, the numerical results calculated by the beam model are more close to those of B1. If the stiffener spacing is equal to the half-span length of stiffened plate (d/L ¼ 1), a string response just occurs by the theoretical method. However, the maximum dimensionless dis­ placements calculated by ABAQUS are larger than 1, and then the mid-span displacements oscillate about some average values, which are less than 1. That is to say, the permanent displacements from ABAQUS show no string response occurring. Fig. 20 demonstrates the variation of permanent mid-span displacement Wf/Ws with the change of stiffener spacing d/L. As can be seen, the permanent mid-span displacements almost decrease linearly with the increase of stiffener spacing, and the final 17

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displacements of B1 and B2 calculated by the plate model diverge obviously. Except for the case when d/L ¼ 6/8, the numerical results calculated by the beam model are very close to those of B1. For the case when d/L ¼ 5/8, the difference between theoretical result and B1 is within 10%. However, the final displacement of B2 is too low compared to the theoretical result, with the difference in excess of 10%. As the stiffener spacing continues to increase until it is near to the half-span length of stiffened plate, such as d/L ¼ 7/8 and 1, the final displacement of B2 exceeds that of B1. Thus, although the difference between theoretical result and B2 is reduced, the difference between theoretical result and B1 is increased. This indicates that the stiffener location has an effect on the mid-span displacement when the stiffener spacing becomes larger. Therefore, the simplified beam model proposed in this paper is limited to cases when the stiffeners are suitably spaced closely, i.e. 1/2 < d/L < 1. Otherwise, the difference between FE and theoretical predictions is in excess of 10%, and then the accuracy of the present method may be reduced. 6.5. Influence of material property The Cowper-Symonds constitutive model is adopted to consider the effect of strain rate sensitivity theoretically and numerically in this paper, and the values of C ¼ 40.4/s and P ¼ 5 are obtained from the published literature. For mild steel, the value of P is approximately 5, but a wide range of values for C are found to describe the strain rate sensitive effect in experimental tests [28]. In order to assess the effect of strain rate sensitivity, a comparison is made to estimate the sensitivity of C by using the values of C ¼ 3200/s and P ¼ 5, as reported in Ref. [29]. Besides, the strain hardening of the material is neglected in previous sections, which may also contribute to the plastic dissipation during the blast loading events. Therefore, for comparison, the material behavior is taken to be elastic-plastic-strain hardening in the numerical simulation, with the strain hardening modulus Et ¼ 1240 MPa [14]. Other structural and pulse parameters are kept unchanged (pm/p0 ¼ 8, td ¼ 2 ms, tp ¼ 6.4 mm, d/L ¼ 6/8) in this subsection. Fig. 21 shows the dimensionless mid-span displacement responses with different Cowper-Symonds coefficients. It can be seen that good agreement is achieved generally between the theoretical and numerical results, with the theoretical results slightly higher than those of ABAQUS. As the value of C increases, the calculated displacement responses increase significantly. That is to say, if the value of P ¼ 5 remains unchanged, the effect of strain rate sensitivity becomes smaller for a larger value of C. The dimensionless mid-span displacement responses considering the strain hardening are compared in Fig. 22 with those neglecting the strain hardening. It is observed that the calculated displacement responses decrease if the strain hardening is taken into account. This is due to the contribution of strain hardening to the plastic dissipation. Therefore, the predicted permanent displace­ ments will also be overestimated when the strain hardening of the material is neglected. 7. Conclusions The nonlinear dynamic response of stiffened plates under a blast loading is analyzed based on a simplified beam model in this paper. The material strain rate sensitivity is taken into account in theoretical analysis. The linearized yield curve by an inscribed polygon is used to simplify the calculation, and a semi-analytical solution can be obtained by Lee’s extremum principle. In order to confirm the accuracy of the theoretical method, a series of FE simulations are performed by using ABAQUS. The influence of pulse intensity, pulse duration, plate thickness, stiffener spacing and material property on the displacement response is discussed in detail. A definition for the case when the present simplified method can be used is then provided. According to the obtained results, the major conclusions can be summarized as follows: (1) The mid-span displacements calculated by the theoretical method and ABAQUS agree well in general, with the theoretical results higher than those of ABAQUS. However, most of the difference between them is within 10%. Therefore, although the theoretical predictions may overestimate the permanent displacement, the obtained results are still acceptable for the pre­ liminary estimation. (2) The effect of strain rate should be considered for the mild steel. Compared to the results without strain rate effect, the present prediction is reduced by as much as 30% due to strain rate effect. Keeping the coefficient P ¼ 5 unchanged, a smaller strain rate sensitive effect is observed for a larger value of C. Besides, the predicted permanent displacements will also be overestimated when the strain hardening of the material is neglected. (3) The pulse has a significant influence on the dynamic response of stiffened plate. With the increase of load intensity, the per­ manent mid-span displacement almost increases linearly. Moreover, as the load duration becomes longer, the permanent midspan displacement increases more rapidly, and the calculated displacements of beam near to the center (B1) and boundary (B2) of stiffened plate begin to diverge. Therefore, to reduce the difference between the plate model and beam model, the load duration must be significantly less than the fundamental period of stiffened plate, such as in the range of 0.25T0 and 0.5T0. (4) The plate thickness also has an obvious effect on the dynamic response of stiffened plate. With the increase of plate thickness, the calculated permanent mid-span displacement almost decreases linearly. Although the difference between theoretical and numerical results calculated by the beam model nearly remains unchanged, the difference between theoretical results and FE results from plate model increases gradually. Consequently, the solution by the simplified beam model may be conservative for the stiffened plate with thick plate. (5) The stiffener spacing affects the dynamic response of stiffened plate significantly. As the stiffener spacing increases, the per­ manent mid-span displacement almost decreases linearly. Moreover, when the stiffener spacing is near to the half-span length of stiffened plate, the final displacement of B2 begins to exceed that of B1. Thus the effect of stiffener location occurs, and the 18

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accuracy of solution by the beam model may be reduced. Therefore, the simplified beam model proposed here is suitable for the stiffened plate with stiffeners properly close spaced, i.e. 1/2 < d/L < 1. Due to complexity and nonlinearity involved in the dynamic analysis, the obtained results are limited to the case studied in this paper, which may not be directly extended to other cases with different material properties, load types or boundary conditions. Declaration of competing interest The authors declare that they have no commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgements The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant No. 51779198). References [1] Houlston R, Slater JE. A summary of experimental results on square plates and stiffened panels subjected to air-blast loading. 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Experimental and numerical studies on the dynamic response of steel plates subjected to confined blast loading. Int J Impact Eng 2018;113:144–60. [20] Tanimura S, Tsuda T, Abe A, Hayashi H, Jones N. Comparison of rate-dependent constitutive models with experimental data. Int J Impact Eng 2014;69:104–13. [21] Liu K, Wang ZL, Tang WY, Zhang YC, Wang G. Experimental and numerical analysis of laterally impacted stiffened plates considering the effect of strain rate. Ocean Eng 2015;99:44–54. [22] Peng Y, Yang P. Dynamic plastic response of clamped stiffened plates with large deflection. J Mar Sci Appl 2008;7(2):82–90. [23] Peng Y. Research on nonlinear dynamic response and buckling of stiffened plates under impact loading. Ph.D. Thesis. Wuhan University of Technology; 2007. [24] Lee LSS, Martin JB. Approximate solutions of impulsively loaded structures of a rate-sensitive material. J Appl Math Phys 1970;21(6):1011–32. [25] Lee LSS. Mode responses of dynamically loaded structures. J Appl Mech 1972;39:904–10. [26] Jones N. The credibility of predictions for structural designs subjected to large dynamic loadings causing inelastic behaviour. Int J Impact Eng 2013;53:106–14. [27] Yu TX, Stronge WJ. Dynamic models for structural plasticity. Beijing: Peking University Press; 2002. p. 122–3. [28] Zhu L, Shi SY, Jones N. Dynamic response of stiffened plates under repeated impacts. Int J Impact Eng 2018;117:113–22. [29] Villavicencio R, Guedes Soares C. Impact response of rectangular and square stiffened plates supported on two opposite edges. Thin-Walled Struct 2013;68: 164–82.

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