Plastic Response and Failure Prediction of Stiffened ...

Proceedings of the ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

OMAE2013-11043

PLASTIC RESPONSE AND FAILURE PREDICTION OF STIFFENED PLATES PUNCHED BY A WEDGE Bin Liu Centre for Marine Technology and Engineering (CENTEC) Instituto Superior Técnico, Technical University of Lisbon, Portugal

Richard Villavicencio Centre for Marine Technology and Engineering (CENTEC) Instituto Superior Técnico, Technical University of Lisbon, Portugal

C. Guedes Soares Centre for Marine Technology and Engineering (CENTEC) Instituto Superior Técnico, Technical University of Lisbon, Portugal

ABSTRACT Experimental tests have been performed to examine the response of small-scale stiffened plates laterally punched by a wedge. The specimens are supported at two opposite edges and the indenter is located at the mid-span. In the unsupported edges, the ends of the stiffeners are either connected to transverse stiffeners or remain free. The obtained forcedisplacement responses show a good agreement with the simulations performed by the LS-DYNA finite element solver. The finite element model includes defining the experimental boundary condition so as to simulate small axial displacements of the specimen at the supports. The strain hardening of the material is defined using experimental data of quasi-static tension tests and the critical failure strain is evaluated using tensile test simulations. The results show that the response of the specimens is highly sensitive to the amount of restraint provided at the supports. In addition, simplified calculations are proposed to evaluate the contribution of each structural component on the energy absorbed by the stiffened plate specimens. Keywords: Stiffened plate, punching test, numerical simulation, axial displacement, force-displacement response, failure strain. INTRODUCTION The common structural elements of ships and offshore structures are the stiffened plates. These elements can be exposed to dynamic loads, such as dropped objects on deck structures and collision of ships with floating structures.

Therefore, the design of marine structures requires an accurate prediction of the extent of damage within the stiffened plates subjected to large deformations and failure, incorporating the effect of permanent deformation and maximum force, as well as the stress and strain distributions at the impact point and at the supports, among others. Most of the experimental investigations on stiffened plates under impact loading propose analytical expressions to describe ship-to-ship collision events; see for example [1-3]. Other types of abnormal actions have been also studied, such as the extent of damage in stiffened plates due to dropped objects [4] or the residual damage under explosion loading [5]. Although the design of stiffened plates is mainly developed for marine structures, their use is extended to other areas of engineering. Thus, general theoretical analyses of their plastic behaviour have been proposed, such as to calculate the upper and lower bounds of loads causing plastic collapse for various boundary conditions [6] or to account for the stresses generated in the stiffeners due to their orthogonally interdependence of strain [7]. The finite element analysis is a useful tool to predict the extent of damage in the marine structural components. Though stiffened plates are common structural elements of ships and offshore structures, the available results for comparison between small-scale impact tests and finite element simulations are few. In most of the published works, the analyses are performed on structures that simulate the bottom or the side structures of the ship. In those cases, the experimentalnumerical impact response is examined by penetrating the panels using quasi-static lateral loads [8-11]. Nevertheless, few

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studies using relatively low energies to impact small stiffened plate specimens scaled from typical decks of marine structures have been performed, proposing numerical models to predict the deflection and the failure of the specimens; see for example [12, 13]. The present paper investigates the plastic response of small-scale longitudinally stiffened plates punched by a wedge. This is done through experiments and finite element simulations. The plates stiffened with three flat bar profiles are supported at two opposite edges. In the unsupported edges, the ends of the stiffeners are either connected to transverse stiffeners or remain free. Thus, the influence of the end connection of the stiffeners is studied. Since one of the most important finite element definitions are the boundary conditions, the emphasis in the numerical modelling is put on the representation of the experimental support condition. The numerical model includes part of the experimental support and uses loads and restraints to represent its remaining components. This representation of the supports is similar to the model proposed by Villavicencio and Guedes Soares [14] to simulate the boundary conditions of small-scale stiffened plates subjected to lateral impact along their span. Another important definition that must agree with the characteristics of each particular impact test and specimen is the material nonlinearity. It is known that plastic strain hardening, plastic strain rate and true fracture strain are required for prediction of the extent of damage in structures under impact loads. In the current calculations, the strain hardening is defined by the exact true stress-strain relationship until the maximum load [15] and beyond necking is approximated by a power law relation [16]. This combined material is proposed by Villavicencio and Guedes Soares [17] and used to predict the numerical plastic response and critical failure strain of quasi-static tensile tests and dynamic models of pre-notched beams struck transversely by a mass. In the present work, the strain rate is not of interest since the specimens are subjected to quasi-static loads. However, the definition of the critical failure strain is of special concern, having in mind that it influences the response of the specimens. Here, the failure strain is estimated by numerical simulations of tensile tests following the procedure summarised in Refs. [17, 18]. The experimental and numerical results are discussed in terms of their force-displacement response, which is sensitive to the amount of restraint provided at the supports. In addition, simplified calculations are proposed to evaluate the contribution of each structural component on the energy absorbed by the stiffened plate specimens. These calculations also include the effect of the axial displacement at the supports. EXPERIMENTAL DETAILS The experimental program evaluates the large plastic deformation of small-scale stiffened plates of length 250x250 mm with free and end connected stiffeners. The specimens are punched by a wedge at the mid-span. The

geometry of the specimens is shown in Fig. 1. The plate thickness is 4.0 mm and the stiffeners are 25x4 mm flat bar profiles. In both cases the stiffener’s weld joint is double fillet with throat 2.8 mm. The specimens are supported at two opposite edges, i.e. the edges in the longitudinal direction (specimen’s length) are supported, whereas the edges in the transverse direction (specimen’s width) remain free (unsupported). The supported edges could represent the restraints of main supporting members of a ship structure, such as transverse web frames and bulkhead. The unsupported edges could represent the behaviour of a stiffened plate with a much longer span in the longitudinal direction. The supported part of the specimens is represented by the phantom lines in the plan views of Fig. 1. The ends of the stiffeners are free (Fig. 1a) or end connected to 25x4 mm flat bars (Fig. 1b). The specimen with ends of the stiffeners free is designated as ‘free specimen’ while the specimen with ends of the stiffeners connected is named ‘end connected specimen’.

Figure 1: Stiffened plate specimens. The mechanical properties of the plate and stiffener material are obtained by in-house quasi-static tensile tests using standard tensile specimens and procedures [19]. The dimensions of the flat bar machined test pieces are shown in Fig. 2. Three tensile tests are conducted for each steel material at a rate of 1.0 mm/min until fracture occurs. The displacement-controlled tension tests are carried out with a tensile test machine INSTRON 3369. It records the forceelongation curve, which is used to determine the engineering stress-strain behaviour of the materials. The mechanical properties of the plate and the stiffener materials are summarized in Table 1. The specimens are supported between two thick steel plates (upper and lower support plates) and are compressed by two M16 bolts (diameter D = 16 mm) at each support, as shown in Fig. 3. The lower support plates are actually I-beams stiffened by brackets and fixed to a strong structural base to prevent any movement. The torque T = 60 Nm applied to screw the bolts and compress the supported area of the specimens provides the clamping force: Pz = 18.75 kN (Pz = T/KD, where K = 0.2 is the torque coefficient; [20]).

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Table 1. Mechanical properties of plate and stiffener materials (normal structural steel) PL. FB. Property Units 4.0 mm 25x4 mm Mass density kg/m3 7850 7850 Young’s modulus GPa 206 206 Poisson’s ratio 0.26 0.26 Yield stress MPa 286 367 Ultimate tensile strength MPa 426 488 Fracture stress MPa 322 384 Fracture strain 0.21 0.18

The specimens are punched by a knife edge indenter (see Fig. 3). The panel deformation is enforced by a hydraulic cylinder at a rate of 5 mm/min, approximately. The forcedisplacement response is measured, using a load cell (80 ton) and a displacement transducer (50 mm). The test of the end connected specimen stops once the transverse stiffeners fracture. At this point the instantaneous force drops to zero. The test of the free specimen stops when the indenter reaches a displacement few millimetres larger than that measured in the end connected specimen. The free specimen only suffers large plastic deformation. NUMERICAL MODEL The computations are carried out using the finite element package LS-DYNA (Version 971, [21]). The numerical model is designed with the following components (Fig. 4): specimen, indenter and supports.

Figure 4: Details of finite element model (End connected specimen). Figure 2: Standard dimensions of the flat bar machined test pieces.

Specimen The plate and the stiffeners are modelled by four-node shell elements with five-integration points through thickness, defining the Belytschko-Lin-Tsay shell element formulation [21]. Shell elements are adopted since they decrease the computation time and find practical application for the analysis of marine structures subjected to large deformations. The mesh size is 2.0 mm, which is evaluated according to its ability to predict the experimental force-displacement response and shape of the deformation. The supported part is modelled with a coarser mesh of 2.0x5.0 mm. The weld joints of the stiffeners are accounted for in the simulations by increasing the plate and stiffener thicknesses at their intersection. This is necessary when simulating small-scale structural elements, because the weld increases the resistance of the stiffener and smoothes the transverse transition between the stiffener and the plate. The fillet weld cross-section takes a triangle shape and the leg length is chosen as 4.0 mm. Thus, equivalent thicknesses of 5.0 and 6.0 mm are considered for the plate and the stiffeners, respectively (see Fig. 5 for illustration). A similar procedure for modelling welds is reported in Refs. [11, 14].

Figure 3: Experimental supports. Figure 5: Weld elements of the plate-stiffener intersection.

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Material The mechanical properties of the material (Table 1) used in the finite element model are obtained from in-house quasistatic tensile tests. The engineering and true stress-strain curves are shown in Fig. 6. The true material relation is divided in two parts. In the true curve until the onset of necking, the true stress σt and the true strain εt are expressed in terms of the engineering stress σe and the engineering strain εe as [15]:

σ t = σ e (ε e + 1)

(1)

ε t = ln(ε e + 1)

(2)

taking into consideration that Eqs. (1) and (2) are valid only until the maximum load, the true stress-strain relationship beyond necking is better represented by [16]:

σ t = Cε t n

specimens between the clamping edges is modelled, see Fig. 7. For initiating necking, the width of the specimen at the centre is gradually reduced by 0.5 %. The mesh is diagonally orientated to avoid zero deformation modes of energy. The translational degrees of freedom are restricted at one end and a constant displacement of 100 times the experimental speed is prescribed at the other. The force of the displaced nodes at the free end is obtained and then plotted versus the applied displacement, and these values used to give the engineering stress-strain behaviour. The resulting engineering stress–strain curve is presented in Fig. 8. The critical failure strains assigned in the true material definition of the tensile model results from successive numerical simulations run until they give the engineering fracture strain [18]. For the stiffener material the engineering fracture strain is 0.18 (Table 1); the failure strain defined in the tension piece model is 0.8 (numerical curve in Fig. 8).

(3)

where variables C and n are determined by the maximum uniform strain. The material relation is named ‘combined material’ since it combines two approximations for the flow curve. This true material was proposed by Villavicencio and Guedes Soares [17] to predict the numerical plastic response and the critical failure strain of quasi-static tensile test specimens and dynamic models of transversely impacted prenotched beams. Moreover, the combined material has been selected to predict the impact behaviour of laterally impacted small-scale rectangular plates [22], stiffened plates [14] and tanker side panels [18]. Complete details of this material model can be found in Refs. [14, 18].

Figure 7: Tensile test model. Mesh size 2.0 mm.

Figure 8: Engineering stress-strain curves obtained from tensile test simulation (stiffener material). Figure 6: Engineering and true stress-strain curves of plate (4.0 mm) and stiffener (flat bar 25×4) steel material. Since the transverse stiffeners of the end connected specimens fracture, the failure strain should be included in the material model of the stiffeners. Therefore, the critical failure strain is determined by tensile test simulations that follow the procedure presented by Villavicencio [18]. Failure strain denotes the strain value when fracture occurs, and the finite element calculations remove elements when their average strain reaches the defined critical failure strain. In the tensile test simulations [18], only the length of the tensile test

Indenter The indenter is modelled with solid elements to avoid initial interpenetrations of the plate. A rigid material is defined to ensure no deformation, assigning steel mechanical properties. As no information is obtained from the indenter elements, zero integration points are defined. It should be mentioned that the rigid elements are bypassed in the processing and there is no storage of history variables. The material ‘Mat.020-Rigid’ is selected from the library of LSDYNA. The contact between the indenter and the specimen is defined as automatic surface to surface [21]. In this LSDYNA’s automatic contact, the nodes on the slave side are first

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checked for penetration through the master surface and then the master nodes are checked for penetration through the slave surface. In the contact card, the definition of the static coefficient of friction is unnecessary since the indenter allows only vertical translation. The wedge is subjected to a constant displacement of 1000 times the experimental speed. This constant speed does not provoke inertial effects during the early response and oscillations of the instantaneous forces. Supports In order to provide an accurate representation of the experimental supports, as well as to simplify the numerical model, part of the structural support is modelled while the remaining components are represented simply as nodal boundary conditions and prescribed loads. The numerical representation of the experimental supports follows the model proposed by Villavicencio and Guedes Soares [14] to simulate the experimental restraints of stiffened plates struck laterally by a mass.

Figure 9: Numerical modelling of the experimental supports. The model of the supports includes the upper and lower support plates as illustrated in Figs. 4 and 9. Both support plates are modelled by four-node shell elements with twointegration points throughout the thickness. The material is defined as elastic with steel mass density, Young’s modulus and Poisson’s ratio. The reduced number of integration points and the elastic properties of the material are adopted since the support plates are stiff enough, so they do not suffer any important deformation during the experiments. The lower support plates are modelled between the webs of the I-beams. The webs and central brackets of the I-beams are represented by fully constrained nodes (see Fig. 9). No gap between the support plates and the supported part of the specimen is considered. The upper support plates compress the specimen as in the experiments. Since this compression

depends on the torque applied on the bolts, the experimental clamping force Pz = 18.75 kN is applied at the nodes where the bolts pass through the plate. The pressure transmitted by the upper support plate to the specimen is not uniformly distributed, because the force is applied only over few nodes. However, the pressure is strong enough to maintain the supported part of the specimen without any vertical displacement during the impact. The clamping force is constant during the entire simulation and represents the vertical constraints. The axial restraint is provided by the strong compression over the small area between the nut and the head of the bolts and by the static friction between the support plates and the supported part of the specimen. However, during the experiments the specimens may have slipped a little, as the visual inspection of the bolts demonstrated. In both specimens the bolts themselves restrain the axial displacement of the specimens when the supported part (holes in the plate) gets into contact with the bolts. The magnitude of the axial displacement at the supports is measured after the experiments (~ 3.0 - 4.0 mm). In the numerical model, the axial restraint at the supports is represented by two conditions. First, prescribed nodal axial forces are applied at the bolt locations during the entire simulation (Fig. 9). The magnitude of this force is Px = µPz, where µ = 0.74 is a typical value of the Coulomb friction for static contact of mild steel on mild steel (Table 26.1 in Ref. [21]). Here, the bolt clamping force Pz represents the normal force over the small area between the nut and the head of the bolt. Second, the LS-DYNA’s automatic surface to surface contact between the surfaces of the support plates and the supported part of the specimen is defined. The contact includes a static coefficient of friction equal to 0.2, as proposed by Villavicencio and Guedes Soares [23]. They estimate this value for a similar support-specimen contact in sliding mild steel on mild steel surfaces when the specimen is struck transversely by a mass. RESULTS The experimental and numerical force-displacement responses are plotted in Fig. 10. In general, the plastic behaviour is well predicted by the simulations, particularly the tendencies of the curves and the magnitudes of the maximum force. The experimental curves show oscillations during the deformation process due to irregular vertical and axial movements at the support within the elastic limit. This oscillatory response disappears in the numerical curves since the axial and vertical restraints remain stable during the entire simulation. The main characteristics of the recorded forcedisplacement responses can be summarised as follows. The initial slope for the free specimen reaches 13 kN and for the end connected specimen reaches 27 kN. This slope is much larger for the end connected specimen since the transverse stiffeners contribute to the bending strength of the panel.

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Figure 10: Force-displacement response. (a) Free specimen. (b) End connected specimen. The free specimen suffers large plastic deformation while the end connected specimen fractures at the transverse stiffeners. The end connected specimen absorbs larger energy than that absorbed by the free specimen (see Table 2). This difference implies larger reaction forces during the entire deformation process which are attributed to the strength provided by the transverse stiffeners. Table 2. Summary of experimental results Specimen Values at Maximum Deflection Force Defln Energy (kN) (mm) (J) Free 30.4 38.7 788.6 End-conn 53.8 33.4 1403.6

Remark

Deformation Fracture

During the rebound process of the free specimen (Fig. 10a), the numerical force decreases faster although the permanent deflection is well predicted. The end connected specimen (Fig. 10b) predicts accurately the onset of fracture where the difference with the experiments is about 3.0 %. Reasonable prediction of the impact response is not the only criterion. It is also beneficial to predict well the shape of deformation including the failure of the stiffeners. The deformed shape of the specimens is captured accurately by the numerical simulations (Fig. 11). The free specimen shows large stresses at the position of contact. Its deformation is mainly due to global bending, suggesting that the influence of

the stiffeners is very small. The end connected specimen fails at the transverse stiffeners close to the central-longitudinal stiffener. This is due to the increased strength at the cross longitudinal-transverse stiffener provided by the weld. The crack starts in the lower edge of the stiffener and extends upward resulting in a strength failure (Fig. 12). The transverse stiffener provides most of the bending stiffness to the end connected specimen. The first failing element deforms severely and elongates along the longitudinal direction of the stiffener. This stretching rises into an axial membrane strain and an associated membrane force. Consequently, the bending moment and transverse shear force do not play an important role and the stiffener deforms plastically due to the membrane forces. Therefore, it is reasonable to use the critical failure strain calibrated by tensile test simulations.

Figure 12: Details of the fracture position. End connected specimen.

Figure 11: Shape of the deformation. (a) Free specimen. (b) End connected specimen.

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The good correspondence between the numerical simulations and the experimental results indicates that the material describes the stress and strain state of the impact specimens with sufficient accuracy. This is due to the fact that the true stress and strain relationships used as input in the material definition are based on the tensile test experiments. Moreover, the critical failure strain predicted by numerical simulations of the tensile tests describes sufficiently the failure mode of the end connected specimen. Although the material relation has strong influence on the numerical results, it does not determine the response of laboratory specimens under large plastic deformation. Certainly the most important aspect in this analysis is the modelling of the boundary conditions. The proposed restraints are capable of representing the small axial displacement experienced at the supports, which adds a permanent transverse deflection in the specimens at the impact point. It should be mentioned that the axial displacements at the supports predicted by the numerical models are similar to the one observed in the experiments, as illustrated in Fig. 13. This figure plots the axial displacements at the supports versus the vertical displacement of the indenter. It is observed that the longitudinal displacements for the free and end connected specimens are similar during the entire deformation process. This condition is used as the basis to describe the following simplified analysis.

at the supports. These two parameters also influence the energy absorbed by the end connected specimen. However, the plastic deformation of the transverse stiffener determines the difference between the energy absorbed by the end connected and the free specimen. By a simple geometric relation, the change in the length of the plate lg can be obtained from the transverse span l0 (250 mm) and the deflection δ (33.4 mm):

lg = 2 (l0 / 2) 2 + δ 2 − l0

The change in the length of the plate (8.8 mm) is not only due to the elongation of the plate (le) but also due to the axial displacement at the supports (la), i.e. lg = le + la. The elongation of the plate and the axial displacement at the supports can be estimated by the plastic absorbed energy and the external work done by the axial forces, respectively. As the plate experiences plastic deformation due to axial extension, the average stress σe (356 MPa) in the cross-sectional area of the plate A0 (1000 mm2) is estimated from the mean value of the yield stress and the ultimate tensile stress (Table 1). The axial force Ff (17.625 kN) during the deformation process considers the axial restraint of the bolts (µ = 0.74) and the friction between the surfaces of the supports and the supported part of the specimen (µ = 0.20), i.e. Ff = 0.74Pz + 0.2Pz. For the free specimen, the stiffeners do not experience deformation, and thus the absorbed energy (632.8 J) is only dissipated by the deformation of the plate component and the axial displacement at the supports. The absorbed energy in terms of the axial displacements at the supports can be expressed by: E = σ e A0 (l g − la ) + Ff la

Figure 13: Axial displacement at the supports of one node located at the centreline of the supported edges. SIMPLIFIED ANALYSIS Simplified calculations are proposed to analyse the relationships between the elongation, axial displacement and energy absorption of the stiffened plates. The calculations accounts for the contribution of the plate and the stiffeners to the total strength of the stiffened plate specimens. First, it is considered the energy absorbed by both specimens at a deflection of 33.4 mm which coincides with the deflection at failure of the end connected specimen (Table 2). The respective absorbed energy (E) for the free and the end connected specimen is 632.8 and 1403.6 J. For the free specimen, the energy is mainly absorbed by the plastic deformation although has some influence from the axial work

(4)

(5)

Eq. (5) should include the effect of axial displacements since it is impossible to satisfy the zero displacement condition at the supports when structural components are subjected to large deformation. In practice, the axial displacements depend on the boundary conditions addressed in a particular experimental setup. An axial displacement of 3.7 mm can be calculated by Eq. (5) at each support (la/2) when the absorbed energy is 632.8 J. Also, the elongation of the plate is estimated at 1.4 mm (le = lg – la). The results of the elongation, axial displacement and the absorbed energy for the free specimen are shown in Table 3. It is observed that the predicted displacement at each support is similar to that observed in the experiments and simulations.

Table 3. Results of elongation, axial displacement and absorbed energy for the free specimen. Results Elongation Axial dis. Absorbed (mm) (mm) Energy (J) Experimental 3.0 – 4.0 632.8 Numerical 3.7 661.0 Analytical 1.4 3.7 632.8

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For the end connected specimen, the energy absorbed by the plate element should be similar to that absorbed by the plate of the free specimen. Thus, the absorbed energy by the transverse stiffeners represents the difference in the absorbed energy (770.8 J) between the end connected and free specimens.

be derived. In the calculations, Eq. (5) indicated that the energy absorbed by the specimens is very sensitive to the axial restraint provided at the supports.

Figure 14: Elongation of the end connected specimen.

REFERENCES [1] Hagiwara K, Takanabe H, Kawano H. A proposed method of predicting ship collision damage. International Journal of Impact Engineering 1983; 1 (3): 257-279. [2] Manolakos DE, Mamalis AG. On ship collisions: The plastic collapse of longitudinally framed shell plating subjected to oblique loading. International Journal of Impact Engineering 1985; 3 (1): 41-55. [3] Wang G, Arita K, Liu D. Behavior of a double hull in a variety of stranding or collision scenarios. Marine Structures 2000; 13: 147-187. [4] Cho S-R, Lee H-S. Experimental and analytical investigations on the response of stiffened plates subjected lo lateral collisions. Marine Structures 2009; 22: 84-95. [5] Park B-W, Cho S-R. Simple design formulae for predicting the residual damage of unstiffened and stiffened plates under explosion loadings. International Journal of Impact Engineering 2006; 32: 1721-1736. [6] Manolakos DE, Mamalis AG. Limit analysis for laterally loaded stiffened plates. International Journal of Mechanical Sciences 1988; 30 (6): 441-447. [7] Boot JC, Moore DB. Stiffened plates subjected to transverse loading. International Journal of Solid and Structures 1988; 24 (1): 89-104. [8] Wu F, Spong R, Wang G. Using numerical simulation to analyze ship collision. In. Proc. 3rd International Conference on Collision and Grounding of Ships: Izu, Japan 2004; p. 27-33. [9] Paik JK, Seo JK. A method for progressive structural crashworthiness analysis under collision and groundings. Thin-walled Structures 2007; 45: 15-23. [10] Alsos HS, Amdahl J, Hopperstad O. On the resistance of stiffened plates, Part I: Experiments. International Journal of Impact Engineering 2009; 36: 799-807. [11] Alsos HS, Amdahl J, Hopperstad O. On the resistance of stiffened plates, Part II: Numerical analysis. International Journal of Impact Engineering 2009; 36: 875-887. [12] Choung J, Cho S-R, Kim KS. Impact test simulations of stiffened plates using the micromechanical porous plasticity model. Ocean Engineering 2010; 37: 749-756.

The transverse stiffener elongates 0.7 mm at each side, as occurs in the attached plate. Since the width of the stiffener is 25 mm, the elongation of the lower edge of the stiffener can be obtained geometrically (14.8 mm), as shown in Fig. 14. The energy dissipated by one transverse stiffener (346.3 J) is obtained from its elongation at the neutral axis (8.1 mm), the average stress (427.5 MPa) and the cross-sectional area (100 mm2). The energy dissipated by both transverse stiffeners (692.6 J) results in a difference of 10.1 % with the one obtained in the experiment (770.8 J). This difference could be attributed to the bending experienced by the stiffeners during the experiments. CONCLUSIONS The numerical results are in good agreement with the experimental tests when proper boundary conditions are represented in the finite element models. The proposed boundary conditions are capable of representing the small axial displacement experienced at the supports, which adds a permanent transverse deflection in the specimens at the indentation point The stress and strain relationship until fracture used as input in the material definition is based on tensile test experiments. The good numerical results until fracture indicates that the material describes the stress and strain state of the tensile specimen with sufficient accuracy. The simulations of the tensile tests to determine the critical failure strain are valid for this particular experiment, because some elements in the original cross-section have deformed severely and the most important effect is the development of membrane forces. Simplified calculations are proposed to evaluate the contribution of each structural component on the energy absorbed by the stiffened plate specimens. Moreover, the effect of the axial displacement at the supports can be accounted for. The contribution of the elements and the influence of the restraint are easily determined since the specimens are subjected to axial extension during the indentation and, consequently, geometrical dependencies can

ACKNOWLEDGMENTS The work of the first author has been financed by a PhD scholarship from ABS, the American Bureau of Shipping. The authors are grateful to Dr. George Wang for his initiative to promote this scholarship. The second author has been financed by the Portuguese Foundation for Science and Technology, under contract SFRH/BD/46369/2008.

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[13] Villavicencio R, Guedes Soares C. Numerical prediction of impact loads in rectangular panels. In: Guedes Soares, Fricke, editors. Advances in Marine Structures. London: Taylor & Francis Group 2011; p. 399-409. [14] Villavicencio R, Guedes Soares C. Numerical modelling of laterally impacted plates reinforced by free and end connected stiffeners. Engineering Structures 2012; 44:46-62. [15] Dieter GE. Mechanical behavior under tensile and compressive loads. ASM Handbook 1986; 8: 99-110. [16] Zhang L, Egge ED, Bruhns H. Approval procedure concept for alternative arrangements. In. Proc. 3rd International Conference on Collision and Grounding of Ships: Izu, Japan 2004; p. 87-96. [17] Villavicencio R, Guedes Soares C. Numerical plastic response and failure of a pre-notched transversely impacted beam. Ships and Offshore Structures 2012; 7(4): 417-429. [18] Villavicencio R. Response of ship structural components to impact loading. Ph.D. thesis on Naval Architecture

and Marine Engineering, Technical University of Lisbon 2012. [19] ASTM (American Society for Testing and Materials). Standard methods of tension testing of metallic materials, E 8. [20] Portland Bolt & Manufacturing Company, Inc. http://www.portlandbolt.com/faqs/tension-vs-torque. [21] Hallquist JO. LS-DYNA Theory Manual 2010. Livermore Software Technology Corporation. [22] Liu B, Villavicencio R, Guedes Soares C. Experimental and numerical plastic response and failure of laterally impacted rectangular plates. In. Proc. 31st International Conference on Ocean, Offshore and Artic Engineering (OMAE 2012): Rio de Janeiro, Brazil 2012; Paper: OMAE2012-84015. [23] Villavicencio R, Guedes Soares C. Numerical modelling of the boundary conditions on beams stuck transversely by a mass. International Journal of Impact Engineering 2011; 38 (5): 384-396.

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