Simplified Analytical Method for Estimating the Resistance of Lock Gates to Ship Impacts

The present paper is concerned with the design of lock gates submitted to ship impacts. In this paper, a simplified analytical method is presented to evaluate the resistance of such structures under collision. The basic idea is to assume that the resistance is first provided through a local deforming mode, corresponding to a localized crushing of some impacted structural elements. For consecutive larger deformations, the resistance is then mostly provided through a global deforming mode, corresponding to an overall movement of the entire gate. For assessing the resistance in the case of the local deforming mode, the structure is divided into a given number of large structural entities called “superelements.” For each of them, a relation between the resistance of the gate and the penetration of the striking ship is established. However, as some results are already available in the literature, this subject is not treated extensively in this paper. On the contrary, the calculation of the resistance of the gate provided through the global mode is detailed and the strategy to switch from local to global deformation is highlighted. Finally, we propose to validate our developments by making a comparison between results obtained numerically and those predicted by the present analytical approach.

The present paper is concerned w ith the design of lock gates subm itted to ship impacts. In this paper, a sim plified analytical m ethod is presented to evaluate the resistance of such structures u nder collision. The basic idea is to assum e that the resistance is first provided through a local deform ing m ode, corresponding to a localized crushing of some im pacted structural elements. For consecutive larger deform ations, the resistance is then mostly provided through a global deform ing mode, corresponding to an overall m ovem ent of the entire gate. For assessing the resistance in the case of the local deform ing mode, the structure is divided into a given num ber of large structural entities called "superelem ents." For each of them, a relation betw een the resistance of the gate and the penetration of the striking ship is established. However, as some results are already available in the literature, this subject is n ot treated extensively in this paper. On the contrary, the calculation of the resistance of the gate provided through the global m ode is detailed and the strategy to sw itch from local to global deform ation is highlighted. Finally, w e propose to validate our developm ents by m aking a com parison betw een results obtained num erically and those predicted b y the present analytical approach.

Introduction
A m ongst all the loads th at have to be expected for the design of lock gates, the collision of a vessel is one of the m ost difficult to handle.
A collision m ay result in som e m inor dam ages to the plating or to the stiffening system of the gate, producing, for exam ple, a local loss of w ater tightness. How ever, if the initial velocity of the striking ship is large enough, the displacem ents im posed to the structure m ay cause a com plete collapse of the gate. This w ould result in the em ptying of the dam aged reach, w ith probably the com plete sinking of the striking ship.

Journal of A pplied M athem atics
To deal properly w ith ship im pact, it is of course possible to use nonlinear finite elem ent m ethods. N evertheless, at the predesign stage of a gate, such approaches are rather restrictive because of the tim e required to m odel and sim ulate collisions. Therefore, w e p ro pose here to establish an analytical sim plified m ethod in order to verify the resistance of gates subm itted to a ship impact.
For the m om ent, the developm ent of such sim plified m ethods is not really reported in the literature. Some very interesting results have been established for the purpose of analyzing collisions betw een tw o ships. For exam ple, the crushing resistance of w eb girders has already been theoretically and experim entally studied by W ierzbicki and C ulbertson Driscoll [1], W ang and O htsubo [2], Sim onsen [3], Z hang [4], and H ong and A m dahl [5]. Each of them developed analytical form ulations th at m ay be useful for studying locally the contact betw een a ship and a gate.
A dditional results are also available for im pacted panels, w hich have been investi gated in detail by W ang [6 ], W ang and O htsubo [7], and Z hang [8 ]. Some references are also useful for evaluating the resistance of m etal plates after rupture, w hen they are subm itted to tearing and cutting. For example, these phenom ena have been studied by W ang and O htsubo [7], Z hang [8 ], W ierzbicki [9], and Zheng [10]. In the particular case of stiffened panels subjected to lateral loads, the developm ents perform ed by Paik [11], Cho and Tee [12], or U eda et al. [13] constitute a very accurate basis for perform ing analytical estim ation of the resistance of such structural com ponents.
The previous brief literature review show s that som e results are already available to deal w ith a sim plified approach of collisions betw een ships and gates. All these developm ents constitute of course an invaluable help for developing sim plified collision m odels of lock gates, b u t they are not sufficient. The principal reason is th at the behavior of an im pacted gate m ay not directly be assim ilated to the one of an im pacted vessel. Consequently, some researches in this dom ain are still needed.
The aim is to develop som e analysis tools, w hich w ould be tim e and cost-effective in the predesign stage of gates. To achieve this goal, w e w ill follow a sim ilar m ethod to the one proposed by Te Sourne et al. [14], The basic idea is th at the total resistance of the struck gate is provided by tw o deform ing modes: (i) the local one, w hich im plies a local crushing of all the im pacted structural elements; (ii) the global one, w hich supposes an overall deform ation of the gate.
In the present paper, w e try to go further into this philosophy.

General D escription of the Problem
In this paper, w e consider the exceptional situation of ship colliding w ith a lock gate. The collision scenario is depicted in Figure 1, w here the general coordinate system is denoted by (X,Y, Z). For avoiding confusion in the present paper, w e w ill use the term inology "transversal," "vertical," and "longitudinal" in accordance w ith the respective orientation of X, Y, and Z axes.
In our scenario, the vessel is com ing from upstream , and, consequently, the im pact is located on the dow nstream gate of the lock. It is clear that this case is the least desired because the hydrostatic pressure is acting in the sam e direction as the im pact force. O n the contrary, if the collision w as happening in the dow nstream reach of the lock, the resulting hydrostatic pressure w ould act in opposition w ith the im pact force and w ould com pensate it partially. In order to derive an analytic procedure for estim ating the collision resistance of the gate, w e first need to describe the ship and the gate using various param eters. This is the p urpose of the subsequent sections.

G eo m etrica l D escrip tio n o f th e S trik in g Vessel
The vessel is characterized through the following: m ass Mo and velocity Vo-In other w ords, w e assum e a certain kinetic energy MoVg2/ 2 for the striking ship. These tw o param eters are chosen according to the w aterw ay class, w hich determ ines the m axim al speed as w ell as the allowable shipping of the vessels.
From the geom etric point of view, w e first assum e that the shape of the bow at the u p p erm ost deck m ay be fairly m odeled by a parabola T (see Figure 1) having a transversal radius p and a longitudinal radius q. Consequently, in the local axes (x¡,,~¡,) positioned in point A, the equation of the curve T is given by r-^=<7 ('-pi) < 21) In order to have a global description of the geom etry of the ship, it is also required to introduce the following param eters (see also Figure 1): (i) the height hb betw een the upperm ost deck and the bottom of the ship; (ii) the side angle qi and the stem angle (p, w hich are used to fix the inclination of the bow.
It is im portant to note that all the above m entioned properties are required in p u t data, w hich have to be provided by the user before the beginning of the calculation process.

.. D escrip tio n o f th e G ate
In this paper, w e w ill only focus on gates w ith a single plating. A three-dim ensional picture of such a structure is depicted in Figure 2, w here the notations H and L are used for representing, respectively, the total vertical and the total transversal extension of the gate. In fact, these gates are rather sim ilar to large orthotropic plates, constituted by a plating (for retaining w ater) stiffened by the following elem ents (see Figures 2 and 3) : (i) the transversal fram es, w hich m ay be seen as beam s presenting a T-shaped crosssection; they are placed in the transversal direction (i.e., along the X axis); (ii) the vertical fram es, w hich are also beam s w ith a T-shaped cross-section b u t are arranged in the vertical direction (i.e., along the Y axis); (iii) the stiffeners, w hich are optional reinforcing beam s disposed transversally on the plating in order to avoid instabilities in shells; their cross-section m ay exhibit various shapes.
The geom etric data required for characterizing the stiffening system are m ainly the dim ensions of the different cross-sections. As show n in Figure 3, the needed values are the height hw and thickness tw of the w eb, as w ell as the height htand thickness ttof the flange. For the plating, it is only necessary to precise its thickness tp. W ith all these param eters and know ing the properties of the m aterial constituting the gate, it is possible to derive the m echanical properties of all the stiffening elements.
A nother point that has to be clarified concerns the assum ed su p p o rt conditions of the structure. W hen the gate is closed, the contact is supposed to be established against the su p p o rt denoted by Si, S2 , and S3 in Figure 3, and, consequently, w e m ay adm it that: (i) the gate is sim ply applied against the tw o lock w alls (supports Si and S2 in Figure 3). The translational degree of freedom in the Z direction has therefore to be blocked along the all vertical extensions of the gate in Si and S2 ; (ii) the gate is sim ply applied against the sill located at the bottom of the cham ber (support S3 in Figure 3). Therefore, it seem s to be reasonable to restrain the translational displacem ent in the Z direction along the all transversal extension of the gate in S3.

.3 . D escrip tio n o f th e M a te ria l
The present pap er is concerned w ith the resistance of a lock gate im pacted by a ship. The p rim ary goal is not to assess the dam ages caused to the vessel: w e are m uch m ore interested in the ability aspect of the structural resistance to collisions. As a consequence, w e assum e th at the m aterial constituting of the striking vessel is infinitely rigid. In other w ords, w e will not allow any deform ation in the ship structure, w hich is a conservative approach in the evaluation of the resistance.
O n the contrary, the previous hypothesis is not valid for the gate as it is supposed to be deform able. N ow adays, the m ost com m on m aterial used for such structures is construction steel, so w e w ill only deal w ith this m aterial in the present paper. This kind of steel exhibits 6 Journal of A pplied M athem atics (i) the m axim al elastic stress oti, w ith w hich is associated the m axim al elastic deform a tion c0; (ii) the rupture stress ou, for w hich tearing is observed in the m aterial; the correspond ing deform ation is called eu; (iii) Young's m odulus E characterizing the stiffness of the m aterial during the elastic phase.
In order to sim plify the analytical derivation of the collision resistance, w e will suppose here that the steel has a so-called elastic-perfectly plastic behavior. This m eans that the relation betw een stresses and strains is idealized by curve (2) in Figure 4. Consequently, w e neglect the additional resistance com ing from hardening of steel, w hich is in fact a conservative assum ption.

.4 . G en era l P o sitio n in g in th e Space
To define the collision scenario, it is still necessary to position the resistance elem ent as well as the striking ship w ithin the area of space. To do so, different kinds of in p u t data are still required: (i) the im pact point E, th at is, the point of the gate w here the first contact betw een the bow and the plating w ill be established; this point is located by its coordinate (X f, Y e ) , as show n in Figures 5 and 6 ; (ii) the transversal positions X¿ of the vertical fram es, th at is, the position of each vertical fram e along the X axis (see Figure 5); (iii) the vertical positions Y¿ of the horizontal fram es, th at is, the position of each horizontal fram e along the Y axis (see Figure 5); (iv) the total num ber of stiffeners distributed along the vertical height H of the gate.
W hen all the previous inputs are placed, the three-dim ensional configuration of the gate is com pletely defined.

G en era l P rin cip les
W hen a ship collides w ith a gate, its action on the im pacted structure m ay be represented by a force Pt acting in the sam e direction as the indentation 6 of the striking vessel (see Figure 6 ) . By equilibrium , this force m ay be seen as the resistance opposed by the gate to the progression of the ship. Therefore, the goal of our w ork is to assess the value of Pt for a given indentation 6 of the vessel. In other w ords, our aim is to derive the evolution of Pt w ith 6 by m eans of sim plified analytical procedures.
W hen a ship is entering into a lock, it seem s reasonable to adm it that its initial velocity Vo is quite small. Consequently, the dynam ic effects in the gate rem ain m oderate, and w e m ay assum e that the initial kinetic energy MoV0 2/ 2 of the ship is entirely dissipated by deform ation of the im pacted gate Eint, th at is, (3.1) K now ing the relation betw een Pt and 6 , it is possible to calculate Eint sim ply by integration (see Figure 7):

Carnax Eint=
Pt(6)-d6. Jo For a given ship of m ass M o and velocity Vo, (3.1) and (3.2) give the m axim al penetration 6 max, w hich has to be sup p o rted by the gate to w ithstand an im pact w ith such a vessel. A ccording to the m axim al degradation level accepted for the gate, it can be decided if this value of 6 max m ay be applicable or not.

.2 . T h eo retica l B asis
The theoretical basis for deriving Pt(6) is the so-called upper-bound theorem, w hich states that "if the work rate of a system of applied loads during any kinematically admissible collapse of a structure is equated to the corresponding internal energy dissipation rate, then that system of loads will cause collapse, or incipient collapse, of the structure." In the present case, it is obvious that the external dissim ilation rate Êext is entirely produced by the force Pt applied by the ship on the gate. Therefore, w e have Journal of A pplied M athem atics 9 B *X Ui w here (') = d/dt is the derivative w ith respect to time. O n the other hand, if w e neglect the dynam ic effects in the structure, the internal dissipation rate É¡nt is entirely com ing from the deform ation of the gate. If V is the total volum e of the structure, using Einstein's notation we have: w here cri; and are, respectively, the stress and the strain rate tensors defined over the entire volum e V of the gate. By application of the upper-bound theorem , w e have Éext = Éint = t Pt = ^ ƒƒƒ OijéijdXdY dZ. (3.5) Consequently, (3.5) m ay be useful for deriving Pt (6), provided th at w e are able to establish a relation betw een the deform ation rate and the velocity 6. To do so, w e need to define the displacem ents over the entire volum e V. For exam ple, in Figure 8, if w e suppose th at p oint A is m oving to point B for a given value of 6, w e m ay define the three com ponents Ui(X,Y,Z,S), U2(X,Y,Z, 6), and LI3 (X ,Y ,Z ,6 ) of the displacem ent field along axis X, Y, or Z, respectively. N ote th at in the rem aining p art of this paper, w e w ill also use the equivalent notations (U,V,W) and (X i,X 2,X 3) for designating (Lii, 1 1 2 , 1 X3 ) and (X,Y, Z).
Using the G reen-Fagrange tensor, it is finally possible to find a link betw een the deform ation and the penetration of the ship 6: If w e w an t to apply form ula (3.5) to obtain a relation betw een Pt and 6 , it is also required to evaluate the stresses cr¿; as a function of 6. This m ay be achieved using the constitutive laws giving a relation betw een cri; and ei; . As the evolution of w ith 6 is know n by (3.6), w e also have N ote that Lii, Uj, and LI3 are unknow n; for a given value of 6, w e have to postulate a certain displacem ent field. Provided that this displacem ent field is kinem atically adm issible, w e m ay apply the upper-bound theorem and calculate the resistance Pt w ith form ula (3.5). In fact, if w e com bine (3.7) and (3.8) in (3.5), w e obtain Equation (3.9) is the needed relation betw een Pt and 6. H owever, the crucial point in the above-described approach is to define properly a kinem atically adm issible displacem ent field, otherw ise the upper-bound theorem m ay lead to an overestim ation of the crushing resistance.

.. The S u perelem en ts M eth o d
The integration of (3.9) has to be perform ed over the w hole volum e V of the struck structure and is rather im possible to derive analytically. As a consequence, w e need to sim plify the procedure described here over, and this m ay be achieved by splitting the gate into superele m ents.
The basic idea is to divide the gate into different substructures (the so-called superel em ents) that w e assum e w orking independently. For the lock gate depicted in Figure 2, the structure m ay be decom posed into tw o types of superelem ents.
(i) The first superelem ent (SEI) is a rectangular plate sim ply su pported on its four edges and im pacted perpendicularly to its plane, undergoing therefore im portant out-of-plane displacem ents. Such elem ents are typically used for m odeling the plating of the gate.
(ii) The second superelem ent (SE2) is a beam w ith a T-shaped cross-section, im pacted in its plane. This kind of elem ent is therefore quite relevant for m odeling transversal and horizontal frames.
The division of the gate into superelem ents is only based on geom etric considerations. In order to illustrate this process, w e can consider, for exam ple, only a sm all p art of the lock gate represented in Figure 2, for w hich the division principle is show n in Figure 9. As it can be seen, the tw o previous types of elem ents are sufficient for analyzing the structure.  Figure 9: Illustration of the subdivision of a structure into superelem ents.

frames
As long as there is no contact betw een the ship and a given superelem ent, this latter w ill rem ain inactive. This m eans th at it w ill not deform until it has been collided by the bow, w hich is a consequence of the above-m entioned hypothesis that each substructure is w orking independently. A fter being activated, the superelem ent w ill deform and dissipate a certain am ount of energy. If the gate is divided into n superelem ents, as each of them is decoupled from the others, the total internal energy Eint is sim ply obtained by sum m ation of the individual contributions com ing from the n superelem ents, th at is, Emt = X Eínt ^É mt = 2 É : (fc ) int ' k e (1, (3.10) Journal of A pplied M athem atics w here E® is the internal energy dissipated by superelem ent num ber k for a given penetration 6 of the ship. Before using (3.10), it is prelim inary required to know E j^. To do so, form ula (3.4) is still valid, bu t it has to be reequated for the case of superelem ent k : w here w e have introduced the following notations: (i) cr® is the stress tensor defined on the entire volum e of superelem ent k, is the strain rate tensor defined on the entire volum e of superelem ent k, (iii) Vk is the volum e of superelem ent k.
By following a sim ilar reasoning as for relations (3.7) and (3.8), w e get finally the particularization of (3.9): (3.12) w here Pt(fc) m ay be seen as the contribution of superelem ent k to the total resistance of the gate (note th at Einstein's notation has been used for the subscripts i and /). In fact, relation (3.12) is of prim ary im portance because it constitutes the fundam ental basis of the present m ethod. Of course, w e still need to develop adequately the functions involved in this expression. This w ill be done later for SEI and SE2.

.4 . G lo b a l a n d Local D efo rm in g M o d es
We previously assum ed that each superelem ent w as w orking independently from the others. This hypothesis rem ains valid as long as the penetration 6 is reasonably minor. However, w hen the penetration 6 of the ship is increasing, deform ations w ill occur in superelem ents th at still have not been undertaken by the bow. Consequently, the internal energy rate for superelem ent k m ay not be equal to zero, although it has not been activated. This m ay be seen on Figure 10, w here out-of-plane displacem ents occur in the entire gate, even if some regions have not been im pacted by the striking ship bow. In order to take this coupling into account, let us introduce the concept of local and global deform ing m odes.
(i) We say that the structure exhibits a local deform ing m ode (see Figure 10) w hen the developm ents perform ed in Section 3.3 m ay be applied. In other w ords, w e suppose here that the penetration of the vessel into the gate is only allow ed by the local deform ations of the activated superelem ents. O nly the area im pacted by the ship contributes to the energy dissipation; the other parts of the gate rem ain undeform ed. Of course, it m ay be easily understood that the local m ode is only valid for quite sm all values of 6.
(ii) O n the contrary, w e say th at the structure exhibits a global deform ing m ode (see Figure 10) w hen the displacem ents are not confined in a sm all area located around the im pact point. In this case, the entire gate is involved in the energy dissipation process and w e m ay no longer assum e that it behaves like a set of independent substructures activated progressively Consequently, the superelem ents m ethod is not valid anym ore and the resisting force P t has to be evaluated by another w ay th an the one discussed in Section 3.3. This is precisely the topic of Section 5.
In order to m odel the phenom ena depicted in Figure 10, w e suppose that there is a su d d en sw itch betw een the tw o m odes. A t the beginning, w hen the striking ship starts m oving into the gate, the resistance P t is essentially provided by the local deform ing m ode. This statem ent rem ains valid as long as the penetration 6 does not exceed a transition value 6t, for w hich the global m ode is activated. In fact, the sw itch betw een the tw o m odes occurs w h en the force P t applied by the ship on the gate is sufficient to cause an overall displacem ent of the w hole structure. As soon as 6 < 6 t tw o different values for P t are com puted: (i) the value of P t obtained by supposing a local deform ing m ode; it is denoted by P \oc; (ii) the value of P t obtained by supposing a global deform ing m ode; it is denoted by -Pglob-For a given penetration 6 , P \oc and P gi0b are then com pared. As long as P \oc < P gi0b / the force exerted locally by the ship is not sufficient to cause an overall displacem ent of the gate, so the ship continues penetrating into the structure by local indentation. However, as soon as P\oc = Pgiob/ the force becom es sufficient and the sw itch from the local m ode to the global one is obtained. The corresponding value of 6 is the required 6t (see Figure 11). After that, for the values of 6 greater th an 6 t , the resistance P t is evaluated using equations specially developed for the global m ode (see Section 5).

Evaluation of the Resistance in the Local D eform ing M ode
In the local deform ing m ode, the resistance of the gate is given by (3.12), w here w e assum e th at the total resisting force is sim ply obtained by adding the individual contributions of all the activated substructures. In this section, the law s governing the behavior of the tw o types of superelem ents introduced in Section 3.3 are detailed. How ever, as this topic is already well treated in the literature (see e.g., [4]), in order to avoid any redundancy, w e have m ade a quite concise presentation of our approach.

Note
In the tw o follow ing sections, w e w ill use the superscript (fc) for characterizing any property of the superelem ent num ber k.

S u p erelem en t Type 1 (SEI)
The first superelem ent is used for m odeling the plating of the lock gate. Its boundaries are defined by the surrounding transversal and vertical fram es, as show n in Figure 12. C onsid ering the location of the im pact p oint E, it is possible to fix the four param eters \ a ® , bfK and b^. The thickness of the plate is equal to the thickness tv of the plating. How ever, a correction is needed for taking into account the horizontal stiffeners placed in the transversal direction. D uring the collision, the stiffeners are m ainly subm itted to an axial extension; they w ill deform along the X direction by exhibiting a m em brane behavior. Consequently, the plate thickness has to be m odified for taking these effects into account. If A s is the total area of all the stiffeners connected to the superelem ent k (see Figure 13), then w e obtain Figure 12: Definition of the m ain dim ensions of SEI.
Transverse frame bj + b2 ? This correction has to be applied for the calculation of m em brane effects in the X direction. H owever, if w e consider m em brane effects in the Y direction, the stiffeners have no influence and they do not need to be considered. Consequently, w e have = tp and the plate becom es orthotropic.

■ Total section = A s
In the present approach, w e suppose that the im pacted plate is com pletely in d ep en dent from the surrounding other superelem ents. Therefore, it is acceptable to consider the plate as sim ply su pported on its four edges. For a given indentation 6 , the superelem ent will undergo m ostly a m em brane deform ation; the effects of bending rem ain negligible.
W hen superelem ent SEI is im pacted by the bow of the vessel, for a given value of 6 , w e m ay deduct the deform ation pattern show n in Figure 14. W ith this displacem ents field, it is possible to evaluate the internal energy rate, w hich has already been done by Z hang [4].
For the situation illustrated in Figure 14, Z hang [4] found the following crushing resistance for superelem ent k:

.2 . S u p erelem en t Type 2 (SE2)
The second superelem ent th at w e w ill consider is used for m odeling the transversal and vertical frames. The boundaries of a horizontal superelem ent are defined by the tw o adjacent vertical fram es (and inversely for a vertical superelem ent). The principal dim ensions and (see Figure 15) of superelem ent k are positioned in accordance w ith the location of the im pact point. The resisting cross-section has a T-shape, w hose properties are defined in the general geom etry of the gate.
W hen this superelem ent is im pacted, w e suppose that it w ill deform like a concertina. To do so, three plastic hinges are form ed. They are designated by ABF, ACF, and ADF in Figure 16. These lines allow for relative rotation betw een the triangular surfaces ABC, ACD, BCF, and FCD. B ending effects are therefore preponderant along these lines.
H ow ever, the rotational m ovem ent of the triangular surfaces is not free because it m ust respect the com patibility betw een surfaces ABD and BFD along their com m on line BD. Therefore, surfaces ABD and BFD are subm itted to an axial extension im plying m ainly m em brane effects. Consequently, for a given indentation 6, the w eb w ill be folded as represented in Figure 16, w here 2H is the total height of one fold. A ccording to the previous hypothesis, during this m otion, the energy is absorbed by m em brane extension of the triangular regions ABD and BFD, b u t also by bending effects in the three plastic hinges ABF, ACF, and ADF. The phenom enon of concertina folding has already been studied by a great num ber of authors. For exam ple, it w as theoretically and experim entally studied by W ierzbicki and C ulbertson Driscoll [l],W an g and O htsubo [2], Sim onsen [3], and Z hang [4], H ong and A m dahl [5] com pared all these various approaches and also developed their ow n model.
A ccording to the developm ents perform ed by Z hang [4], for a given penetration 6 , the resistance of superelem ent num ber k is to be taken as w here is the w eb thickness of superelem ent k. In this form ula, H is a p aram eter fixed by m inim izing the m ean crushing resistance over one fold. By so doing, Zhang [4] found that

.3 . Total R esista n ce in Local D efo rm in g M o d e
The total resistance of the gate in the local deform ing m ode is sim ply obtained by sum m ing the individual contributions of the n superelem ents: Pío c = X 4 fc)-(4-5) k= 1 Of course, for a given value of 6 , if a superelem ent has not been activated, it w ill not provide any resistance to the total resistance, and so w e have Plk) = 0.

D isp la cem en ts F ields
W hen the global m ode is activated, the gate is assum ed to undergo a global m otion involving the entire structure. The displacem ents field obtained in this case is plotted in Figure 17(a). As m entioned earlier, the first contact betw een the bow and the plating is located in point E, w ith coordinates (XE,YE, 0). In the vertical plan passing through this point (i.e., the plan w ith equation X = XE), w e suppose th at the displacem ents are distributed along the vertical Y axis as show n in Figure 17(b). The m athem atical form ulation of this profile is as follows: w here WE indicates that w e consider the displacem ent in the plan X = XE. The tw o previous form ulae are only valid as long as there is no other contact betw een the ship and the gate. H owever, as the vessel is m oving forw ard, another contact w ill appear betw een the plating and the low erm ost deck (see Figure 18). The particular value of 6 for w hich this situation w ill occur is denoted by 6C , and w e have 6c = Ye • cot<J>, (5.2) w here < j> is the stem angle. W hen 6 > 6C , the contact betw een the bow and the plating is com pletely developed along the height hj, (see Figure 18). Consequently, it is required to ad ap t the previous displacem ents fields in order to account for this phenom enon. Then, for 6 > 6C , w e will use the following equations: if Ye -f a < Y< YE, 6 if Ye < Y < H.

.2 . M ech a n ica l M o d e l o f th e G a te
In the previous section, w e have postulated a kinem atically adm issible displacem ents field. In accordance w ith the upper-bound m ethod, it is now possible to use the principle of virtual velocities in order to estim ate the resistance of the structure deform ed in the global mode. U nfortunately, it is rather difficult to derive analytically the resistance of a gate subm itted to the displacem ents given by (5.1), (5.3). In order to sim plify the problem , we m ake the assum ption th at the main contribution to the resistance is coming from the bending of the gate between the two lock walls. This hypothesis seems to be reasonable for a global m ode, especially w hen the ratio H /L is w ide, b u t w e have to bear in m ind th at it m ay not rem ain valid in som e special cases. As a consequence, the resistance in the global deform ing m ode is m ostly provided by transversal fram es, the stiffeners. Therefore, the gate m ay be seen as a set of in dependent beam s subjected to a given displacem ents field. The contribution of the vertical fram es is only to apply the expected displacem ents to these beams; w e suppose that they do not take p art m echanically in the resistance. A ccording to these hypotheses, w e obtain the equivalent m odel of the gate depicted in Figure 19.
The previously m entioned beam s have a cross-section obtained by taking the gross cross-section of the transversal fram es, to w hich the collaborating p art of the plating is added (see the picture at the top of Figure 19). The values of hf, tf, hw, tw, and tp are defined as The calculation of the effective w id th bea (see Figure 20) on both sides of each tran s versal fram e can be achieved by applying the rules provided by Eurocode 3 for longitudinally stiffened plates.
Finally, in order to achieve the m echanical m odeling of the gate, w e still have to give som e details about the su p p o rt conditions of the beam s. As they are connected to the vertical fram es, they will be m ostly restrained at tw o levels: (i) a rotational restriction along the transversal X axis, w hich will hinder the torsional effects in the beams; (ii) a translational restriction along the longitudinal Z axis, w hich w ill hinder out-ofplane shearing and bending of the beams.
Of course, it is rather difficult to precisely account for these effects in an analytical procedure. As w e are not trying to have an accurate resistance of the gate (but a good approxim ation), it is adm issible to consider that each beam is sim ply sup p o rted at both ends. By so doing, w e com pletely om it the additional restrains provided by the vertical frames, w hich is a conservative hypothesis for evaluating globally the im pact resistance.

E lastic R esista n ce
The m echanical m odel presented here is a set of n beam s subm itted to the displacem ents fields detailed in Section 5.1. In this section, w e use the superscript (k) in order to refer to a particular beam , w ith k e ( 1 , 2 A t the beginning of the im pact, a beam located at any given vertical position Y(fc) is deform ed according to the classical bending theory (see Figure 21). The deflected shape is then given by a parabola: if 0 < X < XE, Ak) XE 2 X e (X e -L ) X -L X 2 + X 2 -2LXe X e -L 2Xe(Xe -L) (

5.4)
• W f (6 ) = / 2 (X) ■Wlk> (6) if XE < X < L, w here WF \S ) = WeÇY^Kô) and has been defined in Section 5.1. The curvature and the bending m om ents M (fc) are obtained using the tw o following w ell-know n relations: w here E is the elastic m odulus and I (fc) is the bending inertia of beam k.
If w e only consider the deform ation energy arising from the bending of beam k, the internal pow er defined by (3.4) can be calculated by

P la stic R esista n ce
Of course, (5.7) rem ains valid as long as there is no plastic effect in beam k. H owever, w hen it is bent beyond its elastic limit, the transversal fram e exhibits another kind of behavior, w hich m ay be described by using the tw o following properties: (i) Mpfc) : the plastic bending m om ent of beam k, corresponding to a com plete plastic cross-section in bending (see Figure 22(a)); (ii) Npk): the norm al plastic force of beam k, corresponding to a com plete plastic crosssection in traction or com pression (see Figure 22(b)).
W ith these properties, a classical plastic analysis m ay be perform ed. As soon as Mpk) is reached, the section located in X = XE behaves like a plastic hinge and the structure becom es a m echanism . A t this m om ent, the yield locus characterizing the cross-section is reached. How ever, it does not m ean th at the resistance is not increasing anym ore. As the deform ations are increasing, tensile stresses appear inside the beam k, and the cross-section is subm itted to both a norm al force N (fc) and a b ending m om ent As they are linked by the equation of the yield locus, these tw o actions are not independent.
In order to evaluate M (fc) and N (fc) for each of the n beam s representing the gate, w e need m ore inform ation about plastic interaction. Yukio and R ashed [15] have elaborated a very refined description of the yield locus for the cross-section depicted in Figure 22. How ever, as suggested by Paik [11], it is easier to adopt a parabolic interaction criterion for beam num ber k (see Figure 23 If w e note A(fc) and the axial extension and the rotation in beam k, the required condition of norm ality is verified for the present yield locus if w e have (see Figure 23)  By introducing (5.10) into (5.9), w e finally get a second relation betw een M (fc) and N (fc). We then obtain the classical form ula giving the m em brane force in an axially restrained beam:

d M {k) _ dA {k) _ A( f c) ^ 2 M {k)N {k) _ A( f c)
A t this stage, it is im portant to note th at this result im plies th at the beam is perfectly restrained in the axial direction. This hypothesis im plies that no transversal m otions (along direction X) occur at the supports w here X = 0 and X = L. This seems quite reasonable for the gate u n d e r consideration because of the action of vertical frames. H owever, it is im portant to keep in m ind that w e have form ulated such an assum ption because even sm all displacem ents m ay reduce considerably the present foreseen resistance. (5.14) By introducing (5.10), (5.11), and (5.12) in (5.14), w e finally get the individual contribution of beam k to the plastic resistance of the gate in the global deform ing mode:

Total R esista n ce in G lo b a l D efo rm in g M o d e
In supposed to happen w hen the elastic bending m om ent in section X = Xe reaches its m axim al value Mpk> .

R esista n ce o f th e B eam s A lre a d y Im p a cted d u rin g th e Local P h ase
The transition betw een local and global deform ing m ode has already been discussed in Section 3.4, w here a su d d en sw itch is assum ed to occur w hen 6 = 6t (the so-called transition v alu e). In the present section, w e give m ore precision on the w ay to com bine tw o different m odes. For a given value of 6 , P\oc is evaluated by (4.5) and Pgi0b by (5.16). Two cases are then possible.
(i) If P io c > P g io b / then the force applied by the ship on the gate is not sufficient for activating the global deform ing m ode. Consequently, w e have 6 < 6t and Pt = P i o c, w ith P\oc given by (4.5).
(ii) If P g io b = P io c , then the global bending m ode is activated and the gate starts to resist by an overall m ovem ent. So w e have 6 = 6t; the transition value is reached.
For 6 > St, w e know th at the global m ode is valid, b u t the resistance Pt m ay no longer be evaluated by relation (5.16). If w e exam ine Figure 26, for example, w e see that w hen the transition occurs at 6t, the third transversal fram e has already been crushed over a certain length St -So, w here 6 0 is the initial distance betw een the bow and the frame. As a consequence, for beam 3, w e m ay not assum e that (5.7) and (5.15) are still valid.
It is too conservative to suppose th at a beam that has already been crushed during the local phase does not provide any resistance during the global one. On the contrary, the uncrushed p art of the cross-section is still able to develop a certain resistance by acting like a m em brane. This is illustrated in Figure 25, w here, for beam k, w e see that the total area of the uncrushed section is A ® = I, « ( « , (6 .1 ) w here , and are the cross-sectional dim ensions for beam k, is the initial distance betw een the bow and beam k. Of course, this form ula has only to be applied if St > 6 gfc), otherw ise beam k is not im pacted during the local m ode and the classic form ulae of Section 5 rem ains valid. However, a correction is still needed to take into account the beginning of a new phase of m otion. In fact, in (5.7) and (5.15), w e have to evaluate W¿fc) and dW ¿fc)/öS for the actual global displacem ent, th at is, 6 -St and not for the total displacem ent 6, w hich also includes the displacem ents du rin g the local phase. This concept is illustrated in Figure 27.

Total R esista n ce
In the previous sections, w e have established all the required form ulas for assessing the resistance in the local and global deform ing m odes. For clarity, w e w ill now m ake a short sum m ary of the results.

28
Journal of A pplied M athem atics D uring the local phase, the total resistance of the gate Pt is equal to the local one, w hich m eans th at Pt(6) = P (6 ).
We assum e that deform ations rem ain local as long as Tgiob(ö) > Pioc(S). The transition betw een the local and the global m ode occurs at the particular value 6t, for w hich w e have Pgi0b(<5f) = P 0c(<5f).
(2) For the values of 6 greater than 6t, the global m ode is activated. The total resistance of the gate is still given by

Numerical Validation
In order to validate all the developm ents described in the previous sections, w e com pare them to the results obtained by num erical sim ulations on tw o different gates. For each studied lock gate, tw o situations of collision have been considered: in the first case, the im pact point E is located in the u p p e r p art of the gate; in the second case, it is positioned in the low er part.

N u m erica l M o d e l o f th e S trik in g Vessel
As m entioned above, w e are only interested by the w o rst dam ages that m ay be caused to the gate during the collision. So far, w e are not concerned by the destruction of the striking vessel. Therefore, w e conservatively assum e th at the ship is perfectly rigid and w ill not deform over the total im pact duration. For the num erical sim ulations, it is useless to deal w ith the entire ship. We only need to have a quite refined m odel of the bow. As explained in Section 2, the geom etry of the ship is fixed w ith help of the five param eters p, q, ip, qi, and hj, (see Figure 28). Its m ass Mo is noted and its initial velocity Vo-For the present example, w e have chosen the num erical values listed in Table 1. These param eters have been chosen in order to represent a classical ship for the inland w aterw ays.
The num erical m odel of the vessel is show n in Figure 28. It is com posed of 6955 Belytschko-Tsai shell elem ents, w hich are described in the TS-DYNA theoretical m anual by H allquist [17]. As it can be seen in Figure 28, the m esh is m ore refined in the central zone of the ship, w here the contact w ith the gate is likely to occur. In this region, the m esh size is about 1 cm X 1 cm. In the rem aining parts of the m odel, as they are not supposed to develop any contact w ith the im pacted structure, the m esh is coarser.   The m aterial used for m odeling the bow is assum ed to be rigid. It is defined w ith help of the classical properties of steel recalled in Table 2. These param eters are only required for defining the contact conditions betw een the ship and the gate. They are not used for calculating any deform ation in the vessel, as the m aterial is infinitely rigid.

N u m erica l M o d el o f G a te 1
The m ain dim ensions of the structure are plotted in Figure  (ii) six vertical fram es, w hich are regularly placed over the length L of the gate; their locations along the transversal X axis are plotted in Figure 29(c); (iii) tw enty stiffeners distributed over the height H of the gate w ith an average space of 6 6 cm.
O ther geom etrical data are listed in Table 3. The corresponding notations are defined in accordance w ith the sym bols introduced in Figure 3. N ote that the transversal and vertical fram es have a T-shaped cross-section, w hile the stiffeners sim ply have a rectangular one. The gate is m odeled w ith help of 204226 Belytschko-Tsai shell elem ents. The m esh is quite refined, w ith a m esh size of 5 cm x 5 cm. Of course, it m ay appear excessive to use such a regular m esh over the entire structure, b u t it w as required because w e did not know in advance w hich p art   The m aterial used for gate 1 is defined to represent m ore or less the behaviour of steel. The elastic-plastic stress-strain curve m ay be divided in tw o distinct portions (see Figure 30). The first p art of the curve corresponds to the elastic phase. The stress-strain curve is linear, w ith an inclination corresponding to Young's M odulus Ey. W hen the yield stress a Figure 30: Stress-strain relation of steel. cro is reached, the plastic phase begins. The stress-strain curve is still linear, b u t the slope has changed and is given by the tangent m odulus Er-In the present low velocity im pact m odel, the strain-rate effect is not taken into account. The values of the different param eters are listed in Table 4. The su p p o rt conditions are the ones described in Section 2, that is, displacem ents in direction Z are blocked in X = 0, X = L, and Y = 0.

N u m erica l M o d e l o f G a te 2
The second gate is w id er than the first one. Its total height and length are H = 15 m and L = 17.1m. The m ain dim ensions are plotted in Figure 31. (Please note that the origin of the axes (X ,Y ,Z) is correctly positioned, regarding all the previous figures.) This tim e, the stiffening system is m ore com pact and m ade of (i) five transversal fram es, w hose vertical positions along the Y axis are show n on Figure 31; (ii) six vertical fram es, regularly separated by a distance of 1.9 m; (iii) tw enty-six stiffeners, regularly separated by a distance of 50 cm.
O ther geom etrical data are listed in Table 6 . The gate is m odeled by 92671 Belytschko-Tsai shell elem ents. The regular m esh size is 10 cm x 10 cm. The m aterial m odel and the su p p o rt conditions are the sam e as for gate 1 .

N u m erica l S im u la tio n s
Four num erical sim ulations have been perform ed by using the finite-elem ents softw are LS-DYNA. Two sim ulations are required for each gate, according to the position of the im pact p oint E. The transversal and vertical positions X E and Ye of point E are listed in Table 5 for the different collision cases considered here. C oncerning the resulting crushing force curves com pared in Figures 32 and 33, sim ulation 1 corresponds to a ship im pact h appening in the low er p art of the structure and sim ulation 2 to an im pact in the u p p er one.

C om parison o f N u m erica l a n d A n a ly tic a l R esu lts
In order to validate the analytical developm ents established in the previous section, w e will m ake a com parison betw een the results provided by LS-DYNA and the ones predicted by our sim plified m ethod. The curves of interest are those show ing the evolution of the crushing force Pt w ith the total penetration 5. The com parisons are plotted in Figures 32 and 33. The curves referenced as "num erical results" are those obtained by LS-DYNA, w hile the "analytical results" are derived by the present sim plified approach.
As it can be seen, the agreem ent betw een the curves is quite good. In m ost cases, the analytical curves provide slightly conservative results. How ever, for gate 2, the first sim ulation (see Figure 33(a)) exhibits a m ore im portant divergence: our sim plified m ethod tends to underestim ate the crushing resistance, especially for the great values of 6. This observation is due to a quite conservative approach in the global m ode. This m ay be explained by the tw o following reasons.
(i) The resisting cross-sections are determ ined according to the recom m endations of Eurocode 3. It seem s th at these rules are quite severe in the present case as the num erical sim ulations show that a greater p art of the plating is actually collaborating to the resistance of the transversal frames. This last point is confirm ed by the curves plotted on Figure 34, w here w e com pare the energy dissipated by the different structural com ponents of the gate. The num erical results are those given by TS-DYNA, w hile the analytical results are those predicted by the theoretical m odel of this paper. As it can be seen on this picture, the discrepancy is satisfactory for the plating, the stiffeners, and the transverse fram es, b u t it is not really the case for the vertical frames. O ur m ethod underestim ates the energy that these elem ents really dissipate, b u t this approxim ation rem ains conservative.

Conclusion
In this paper, w e exposed a sim plified procedure for assessing the resistance of a gate sub m itted to a ship impact.
It is im portant to bear in m ind the hypotheses that w e have form ulated for m odeling the struck gate and the striking vessel. C oncerning the struck gate, our approach is devoted to (i) gates w ith single plating; structures w ith double plating or caissons are not covered by the present developm ents; (ii) gates w ith a classical orthogonal stiffening system , that is, stiffeners and fram es in the transversal direction and fram es in the vertical direction; (iii) gates su pported on both sides by the lock w alls and by a sill at the bottom of the lock.
The three form er conditions have to be fulfilled for applying the m ethodology exposed previously. C oncerning the striking vessel, it is m odeled by using a certain num ber of param eters. The global bow shape is assum ed to be a parabola, w ith given radii p and q, decreasing according to given stem and side angles (p and qi.
W hen the penetration 6 becom es w ider, the gate w ithstands through an overall m ovem ent im plying the entire structure. In this case, w e say that the resistance is provided th rough a global deform ing m ode. For the su p p o rt conditions assum ed presently, the required theoretical displacem ents fields have been exposed in detail. We also derived an equivalent m echanical m odel for evaluating the resistance in such a situation.
The transition betw een the local and the global deform ing m odes is assum ed to occur at a given penetration 6t, for w hich the collision force exerted by the ship during the local phase is sufficient to produce an overall displacem ent of the gate.
Finally, our presentation ends w ith a com parison betw een som e finite elem ents num erical results and those obtained by the present sim plified approach. In m ost cases, the 36 Journal of A pplied M athem atics procedure exposed here leads to a quite satisfactory estim ation of the collision resistance. The predicted results are conservative, w ith o u t underestim ating too m uch the num erical values. The m ain advantage of the m ethodology exposed here is to provide rapidly an evaluation of the collision resistance. The analytical curves plotted in Figures 32 and 33 are useful for know ing if a lock gate is able to behave satisfactorily to an im pact of a ship w ith given initial velocity Vo and m ass Mo-W ith these curves, it is in fact possible to know the total needed indentation 6 max required for dissipating the initial kinetic energy MoV02 /2 . If this value of 6 max exceeds a given criterion, then w e m ay suppose that the structure w ill not be able to w ithstand satisfactorily a collision w ith a vessel of m ass Mo and velocity Vo.
Of course, w e have to be conscious that our sim plified analytical m ethod is only applicable at the predesign stage of a lock. For m ore advanced stages of a project, it is still necessary to resort to m ore advanced tools, like finite elem ents software. Flange w id th of a stiffener or a transversal fram e hf.

List of N otations
Vertical distance betw een the low erm ost and up p erm o st decks of the striking ship hp: C ollaborating w id th of the plating in the global deform ing m ode hw\ Web height of a stiffener or a transversal frame k: Superscript used for referencing a particular property of superelem ent n°k n: Total num ber of superelem ents used for m odeling the entire gate (p,q). Param eters defining the parabolic dim ensions of the striking ship upperm ost deck tf.
Flange thickness of a stiffener or a transversal beam tf.
Thickness of the plating tw: Web thickness of stiffener or a transversal fram e tx-Equivalent plating thickness after sm earing all the transversal stiffeners ty.
Equivalent plating thickness after sm earing all the vertical stiffeners.

E:
Young's m odulus of the steel m aterial constituting the gate Text* External energy Eint Internal energy Ej'.
Tangent m odulus of steel Ey'.
Young's m odulus of steel H : Total vertical height of the gate I: Bending inertia of a transversal beam L: Total transversal length of the gate M : Bending m om ent Mf.
Plastic bending resistance of a transversal beam M 0: Mass of the striking ship N: Axial force N p: Plastic axial resistance of a transversal beam Pglob: Total resistance of the gate in the global deform ing m ode Ploc'.
Total resistance of the gate in the local deform ing m ode Pf-Total collision resistance of the im pacted gate