Investigation of the added mass method for seismic design of lock gates

Engineering Structures xxx (2016) xxx–xxx

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Investigation of the added mass method for seismic design of lock gates Loïc Buldgen a,⇑, Jean-David Caprace c, Philippe Rigo b, Hervé Le Sourne d a

Haute Ecole Libre Mosane, Quai du Condroz 38, 4000 Liège, Belgium University of Liège, Quartier Polytech 1, Allée de la découverte 9, 4000 Liège, Belgium c Federal University of Rio de Janeiro, Ilha Fundão, 21941-972 Rio de Janeiro, Brazil d Institut Catholique d’Arts et Métiers, 35 Avenue du champ de Manoeuvres, Nantes, 44470 Carquefou, France b

a r t i c l e

i n f o

Article history: Received 2 March 2016 Revised 22 October 2016 Accepted 25 October 2016 Available online xxxx Keywords: Added mass method Earthquake Engineering Lock gate Fluid–structure interaction

a b s t r a c t During an earthquake, lock gates are subjected to additional pressure since the water contained in the chamber is put into motion by the earthquake. It is difficult to assess the level of this pressure because the system is affected by a fluid–structure interaction. The gate deformations have an effect on the water pressure, which in turn affects the gate vibrations. A common approach, referred to as the added mass method, consists of simulating the fluid action by distributing lumped masses over the gate. However, this method has been questioned, since the calculation of the lumped masses is usually based on the Westergaard formula, which was derived assuming a perfectly rigid structure. Consequently, fluid–structure interactions may not be captured correctly. This paper proposes to investigate the validity of this approach for such problems and to explain why it might not be conservative. The numerical solutions of an added mass model and a fluid–structure interaction model are confronted. The results indicate that the added mass method may eventually lead to conservative results depending on the type of damping used in the model. Based on these observations, some recommendations are suggested to improve the design of lock gates subjected to earthquakes. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The design of lock gates requires considering different static and dynamic loads, such as the self-weight, the hydrostatic pressure, the wave induced hydrodynamic pressure, temperature variations, ice action, operational forces, and the impact of vessels. A critical issue to be considered during the design of a lock gates concerns the additional pressure generated by earthquakes. However, it is difficult to assess the level of this pressure because the system is affected by a fluid–structure interaction. Indeed, the hydrodynamic pressure induced by an earthquake is affected by the lock gate vibrations, which themselves depend on the surrounding water. A way to deal with this issue is to perform numerical simulations where the fluid domain is explicitly depicted by elastic or acoustic finite elements or even through Lagrangian–Eulerian methods. This can be achieved using commercial software.1 However, numerical models can need a prohibitively large number ⇑ Corresponding author. E-mail addresses: [email protected] (L. Buldgen), [email protected] (J.-D. Caprace), [email protected] (P. Rigo), [email protected] (H. Le Sourne). URLs: http://www.helmo.be (L. Buldgen), http://www.oceanica.ufrj.br/labsen/ en (J.-D. Caprace), http://www.anast.ulg.ac.be (P. Rigo), http://www.icam.fr/ (H. Le Sourne). 1 e.g. LS-DYNA, ABAQUS, MSC NASTRAN, ADINA, ANSYS.

of finite elements due to the large size of the locks, typically exceeding 50 m in length. Moreover, these simulations are time-consuming to set up and computationally intensive. In order to circumvent these drawbacks, civil engineers commonly avoid modeling the fluid domain, instead developing simplified numerical and analytical approaches. Some investigations have been carried out in this manner, and will be briefly reviewed. Due to their economic and strategic importance, gravity dams have been quite thoroughly examined regarding seismic action. This problem was first investigated by [1], who derived an analytical solution for the hydrodynamic pressure generated on the upstream vertical face of a dam during a horizontal harmonic ground motion. These developments were later extended by [2] to take into account both horizontal and vertical arbitrary ground accelerations. However, these solutions were established assuming that the lock gate is perfectly rigid, which is obviously not the case in reality. In order to take into account the fluid–structure interaction in the case of short-length gravity dams, [3] used a thick-plate model in which the modal properties of the coupled system were first calculated by the Rayleigh–Ritz method. Based on these results, a forced vibration analysis was carried out by solving the dynamic equilibrium equations. Besides gravity dams, cylindrical and rectangular containers have also been investigated, in particular for the storage of

http://dx.doi.org/10.1016/j.engstruct.2016.10.047 0141-0296/Ó 2016 Elsevier Ltd. All rights reserved.

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dangerous liquids that might cause severe damages to the environment in the case of failure. In the attempt to include seismic action in the design of such structures, analytical solutions have been developed to assess the hydrodynamic pressure generated by an earthquake. For rigid containers, this has been achieved by [4–9] or [10]. Here again, the structure is assumed to be perfectly rigid, and the fluid–structure interaction is not considered. To take into account this coupling, flexible containers have been examined by [11–13], who postulated a given shape of the vibration modes to calculate the hydrodynamic pressure. In parallel to these analytical developments, numerical investigations were also carried out by [14], who developed a sequential method to take into account the fluid–structure interaction in the 2D analysis of rectangular flexible tanks. This method was later extended by [15] to take into account free surface motions. In order to investigate more deeply the dynamic response of 3D rectangular containers, [16] gave an extensive presentation of the finite element formulation used to model the fluid domain. These developments were first validated by comparing the results with theoretical solutions known for rigid-wall conditions. From these investigations, it was concluded that the structural vibrations have a limited influence on the convective response. Only a slight increase in the pressure was observed, which tends to corroborate the idea that the free surface motions can be evaluated under a rigidity assumption. Similar conclusions were also addressed by [17], who also developed a similar finite element formulation to take into account the fluid–structure interaction. Although there is a great deal of literature related to the effect of seismic loads on dams and storage tanks, not much has been devoted to lock gates subjected to ground motions. In a recent paper, [18] presented a semi-analytical approach to evaluate the seismic hydrodynamic pressure on a lock gate considering the fluid–structure interaction. In this approach, the modal properties of the dry structure were first derived by applying the Rayleigh– Ritz method. The eigenmodes were then used with the principle of virtual work to perform the dynamic analysis. The added mass method recently applied by [19] to design the entrance lock of the Rosyth Royal Dockyard has become popular due to its simplicity. The validity of this method has been under discussion for many years. For example, the question has been raised during the design of the new Panama lock gates, and no consensus has been reached between the specialists. More specifically, engineers are still wondering whether working with lumped mass leads to conservative results. Moreover, the experts [20] do not really explain the origin of the divergences observed between the added mass method and other more elaborate approaches. In a previous publication, [18] demonstrated that the hydrodynamic pressure might not necessarily be conservative. The purpose of the present paper is to go one step further, by explaining the origin of the observed discrepancies. As far as the knowledge of the authors extends, this has never been done before. Finally, the rational arguments exposited in this paper might be useful to change the current engineering practice of lock gate design. This paper is structured as follows. Section 2 describes the lock gate structure used in the discussion. Theoretical and numerical background related to added mass method is then presented in Section 3. The numerical results of an added mass model and a fluid–structure interaction model are compared and discussed in Section 4. Finally, recommendations and conclusions are provided in Section 5.

2. Description of the lock structure To illustrate the discussion of the reliability of the added mass method, the present paper focuses on the gate depicted in Fig. 1.

Fig. 1. Dimensions of the lock gate (m).

Table 1 Cross-sectional properties. Property Web height Web thickness Flange width Flange thickness

hw tw hf tf

Girders (m)

Frames (m)

Stiffeners (m)

1.000 0.020 0.400 0.025

1.000 0.020 0.400 0.025

0.210 0.006 N/A N/A

The gate is made of a single plating, having a thickness of 0.012 m, a total height h (along the y axis) and a total width l (along the z axis) both equal to 13.1 m. The plating is reinforced by 5 horizontal girders with T-shaped cross-sections (Fig. 1), irregularly placed along the y axis. Six vertical frames are also regularly located along the z axis. Finally, the plating is also reinforced by small horizontal stiffeners with rectangular cross-sections that are distributed between the girders. Their role is mainly to prevent the panels from having buckling instabilities due to compressive stresses. All the cross-sectional dimensions, hw ; tw ; hf , and tf , are listed in Table 1. Regarding the material properties, it is assumed that all the parts of the gate are built using mild steel, with a Young modulus E of 210 GPa, a Poisson ratio m of 0:3 and a mass density q equal to 7850 kg/m3. The lock chamber has a total length L equal to 100 m (Fig. 2). When it is totally filled, the water level hs is equal to 8 m, so that about 60% of the plating surface is in contact with the fluid. Concerning the boundary conditions, the gate is supposed to move freely at the bottom of the chamber, which means that it is not resting against a sill when it is closed. At the lock walls, the leftmost and rightmost vertical frames are placed in a recess that provides contact through a sealing device (Fig. 3): due to the water pressure, the gate is simply pushed against the walls, which produces a deformation of the seal and ensures watertightness. As some space is left between the frames and the lock walls, they do not move exactly synchronously. In this study, this steel– concrete interaction will be neglected in order to avoid a difficult modeling of the contact conditions at the gate boundaries. Consequently, the vertical edges located at z ¼ 0 and z ¼ l are simply

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€ assumed to be subject only to the ground acceleration XðtÞ during an earthquake (Fig. 2). As a final comment, it should also be mentioned that both the lock walls and the lock chamber bottom are supposed to be perfectly rigid and therefore simply follow the seismic acceleration € XðtÞ. In other words, the soil–structure interaction is neglected. 3. The added mass method Fig. 2. Lock chamber configuration (m).

3.1. Theoretical background It is known that the total pressure acting on the gate during an earthquake is the sum of 4 different contributions:  The hydrostatic pressure ps ðyÞ ¼ rf gðhs  yÞ, where r f ¼ 1000 kg/m3 is the fluid mass density, hs is the water level, and g ¼ 9:81 m/s2 is the acceleration due to gravity (Fig. 4).  The rigid impulsive pressure pr ðy; tÞ that is calculated by assuming that the gate is perfectly rigid (Fig. 5) and therefore simply € follows the ground acceleration XðtÞ.  The flexible impulsive pressure pf ðy; z; tÞ that is due to the own€ ðy; z; tÞ of the deformable gate (Fig. 6). accelerations u  The convective pressure pc ðy; z; tÞ that is associated to the sloshing of the free surface. Nevertheless, as mentioned in [21], this phenomenon has very little influence, as the length L and the width l of the lock chamber are usually quite large for most of the lock structures.

Fig. 3. Boundary conditions at lock walls.

where y and z are the horizontal and vertical axes (Fig. 1) and t is time. Neglecting this last contribution, the total pressure pðy; z; tÞ acting on the lock gate may be defined by Eq. (1).

pðy; z; tÞ ¼ ps ðyÞ þ pr ðy; tÞ þ pf ðy; z; tÞ

ð1Þ

In order to assess the hydrodynamic pressures pr ðy; tÞ and pf ðy; z; tÞ in Eq. (1), it is commonly assumed that the lock walls and the bottom of the lock chamber may be considered as rigid bodies. Regarding the derivation of pr ðy; tÞ, this hypothesis is also extended to the upstream and downstream gates. Consequently, for an irrotational, incompressible and inviscid fluid, it is shown by [22] that the rigid pressure may be expressed by Eq. (2) where /ðx; y; z; tÞ is a velocity potential function. This latter is shown to satisfy the Laplace equation (Eq. (3)).

pr ¼ qf

@/ @t

D/ ¼ 0 ()

Fig. 4. Hydrostatic pressure applied on the gate during an earthquake.

ð2Þ @2/ @2/ @2/ þ þ ¼0 @x2 @y2 @z2

ð3Þ

In order to obtain a unique solution, it is required to associate some appropriate conditions to Eq. (3). In accordance with the hypotheses listed previously, as all the boundaries are supposed to be rigid, one may write (Fig. 2): 

@/ @y

¼ 0 for y ¼ 0, the vertical component of the fluid velocity has

to be set to zero at the bottom of the reservoir.  @/ ¼ 0 for y ¼ hs , as sloshing is neglected, the hydrodynamic @t pressure has to be set to zero at the undeformed free surface.  @/ ¼ 0 for z ¼ 0, the horizontal component of the fluid velocity @z along the z axis has to be set to zero at the left lock wall as it is considered to be rigid.  @/ ¼ 0 for z ¼ l, the horizontal component of the fluid velocity @z along the z axis has to be set to zero at the right lock wall as it is considered to be rigid.

Fig. 5. Rigid pressure applied on the gate during an earthquake.

_ ¼ XðtÞ for x ¼ 0, the horizontal fluid velocity along the x axis has to be equal to the rigid upstream gate horizontal velocity. _  @/ ¼ XðtÞ for x ¼ L, the horizontal fluid velocity along the x axis @x has to be equal to the rigid downstream gate horizontal velocity.



@/ @x

At the upstream (x ¼ 0 in Fig. 2) and downstream (x ¼ L in Fig. 2) gates, a solution to Eq. (3) with the previous boundary conditions may be found by adapting the results established by [8]. This leads to Eq. (4) for the rigid pressure where bn ¼ ð2n  1Þp=L.

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Fig. 6. Flexible pressure applied on the gate during an earthquake.

8 > > > > p ðy; tÞ ¼ qf > < r

L 2



þ1 X n¼1

> > > > > : pr ðy; tÞ ¼ qf

L 2



! 4 cosh ðbn yÞ b2n L cosh ðbn hs Þ

þ1 X n¼1

4 cosh ðbn yÞ b2n L cosh ðbn hs Þ

€ XðtÞ;

if x ¼ 0

!

ð4Þ

€ XðtÞ; if x ¼ L

Therefore, it may be concluded that the rigid pressure has the € same sign as XðtÞ if x ¼ 0 and the opposite sign if x ¼ L. Alternatively, Westergaard’s [1] solution allows avoiding the calculation of the infinite series in Eq. (4). Although this one has been developed for a rigid dam with an infinite reservoir, it can be reasonably used as an approximation for lock gates if the chamber is sufficiently long. The rigid pressure at the upstream gate may then be estimated by Eq. (5).

(

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi € pr ðy; tÞ ¼ qf hs ðhs  yÞ XðtÞ; if x ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi € pr ðy; tÞ ¼  78 qf hs ðhs  yÞ XðtÞ; if x ¼ L 7 8

ð5Þ

For conciseness, in the remaining part of this paper, only the upstream gate will be analyzed, as the developments can be easily extended to the downstream one. As an additional simplification, by considering Eqs. (4) and (5), pr ðy; tÞ may be seen as the product € of a mass mðyÞ per unit of area with the ground acceleration XðtÞ as expressed in Eq. (6).

€ pr ðy; tÞ ¼ mðyÞXðtÞ 8 > > < mðyÞ ¼ q

f

L 2



þ1 X

ð6Þ ! 4 cosh ðbn yÞ b2n L cosh ðbn hs Þ

n¼1 > > : mðyÞ ¼ 7 q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hs ðhs  yÞ; 8 f

; for Laplace

ð7Þ

for Westergaard

Fig. 7. Mass mðyÞ per unit of area according to the Laplace and Westergaard solutions (Eq. (7)).

ing the Laplace equation. Nevertheless, this time the boundary conditions at x ¼ 0 and x ¼ L have to be replaced by expression (9) because the horizontal component of the fluid velocity along the x axis has to be equal to the total velocity of the gate, with due consideration to its flexibility (Fig. 6).

@/ _ _ z; tÞ ¼ XðtÞ þ uðy; @x

ð9Þ

The deformability of the structure explains why the additional _ term uðy; z; tÞ appears now in Eq. (9). [11] provided a closed-form solution to this problem (Eq. (10)).

pf ðy; z; tÞ ¼ 

Z þ1 X þ1 X cmn cosðan yÞ cosðcm zÞ  n¼1 m¼0

0

hs

Z 0

l

€ ðy; z; tÞ cosðan yÞ u

m zÞ dydz;  cosðc cmn ¼ 2qf

1  coshðnmn LÞ hs lm nmn sinhðnmn LÞ

mp l ð2n  1Þp an ¼ 2hs ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nmn ¼ a2n þ c2m

cm ¼

ð10Þ

lm ¼ l if m ¼ 0; lm ¼ l=2 if m > 0

As a matter of comparison, the masses mðyÞ calculated according to the Laplace [8] and to the Westergaard [1] solutions (Eq. (7)) are plotted in Fig. 7 for the lock configuration presented in Fig. 1. By comparing these two curves, it is clear that the Westergaard formula is conservative as it tends to overestimate the hydrodynamic pressure applied on the gate during an earthquake. That is why this solution is often preferred.

In an attempt to calculate numerically the hydrodynamic pressure without modeling explicitly the fluid domain, pr ðy; tÞ is first € substituted by mðyÞXðtÞ in Eq. (1) to give Eq. (8).

Unfortunately, evaluating pf ðy; z; tÞ by Eq. (10) requires knowing € ðy; z; tÞ. Ideally, this should be achieved by the gate accelerations u solving simultaneously the Laplace equation (Eq. (3)) and the dynamic equilibrium of the gate. In this case, the fluid–structure interaction would be correctly captured, but this is definitely not suited for an analytical treatment. Another approach is to use realistic predefined displacement functions to approximate the gate deformation. For example, as suggested by [23], uðy; z; tÞ can be expressed as a linear combination of the gate eigenmodes, which leads to a more versatile form of Eq. (10). However, in the added mass method, a more direct approach is preferred as the flexible pressure pf ðy; z; tÞ is simply expressed by Eq. (11) where mðyÞ is the mass per unit of area given by Eq. (7).

€ þ p ðy; z; tÞ pðy; z; tÞ ¼ ps ðyÞ þ mðyÞXðtÞ f

€ ðy; z; tÞ pf ðy; z; tÞ ¼ mðyÞ u

3.2. Equivalent model

ð8Þ

The last term of Eq. (8), i.e. the flexible pressure pf ðy; z; tÞ, is quite arduous to evaluate because it depends on the gate acceler€ ðy; z; tÞ and therefore entirely captures the fluid–structure ations u interaction. However, an analytical formula can be found by solv-

ð11Þ

The expression for pf ðy; z; tÞ provided by Eq. (11) is quite different from the one given by Eq. (10). Indeed, a rapid observation of Eq. (10) shows that the flexible pressure may be calculated by considering a kind of ’weighted average acceleration’ that is obtained

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through the integral terms. In contrast, in the previous formula, pf ðy; z; tÞ is directly related to the local accelerations, which causes more important variations in the local pressure field. For example, points located at the junction between the plating and the horizontal girders may experience higher accelerations than those located at the middle of the plating. According to Eq. (11), the flexible pres€ ðy; z; tÞ and sure has more or less the same spatial distribution as u shows therefore more important variations than if it were calculated with Eq. (10). By introducing Eq. (11) into Eq. (8), the total pressure applied on the gate according to the added mass method is finally found (Eq. (12)).

will be subjected to a total displacement expressed by Eq. (13) where uðtÞ is the gate displacements vector, XðtÞ is the ground displacement due to the earthquake and r is a vector whose components are equal to unity at x-translational degrees of freedom and zero otherwise.

  € þu € ðy; z; tÞ pðy; z; tÞ ¼ ps ðyÞ þ mðyÞ XðtÞ

Eq. (14) may be further developed by detailing the different contributions that are involved in PðtÞ. In fact, a node i of the plating located at ðyi ; zi Þ may be seen to be bounded by a surface DSi (Fig. 9) over which the hydrodynamic pressure pðy; z; tÞ is roughly uniform and equal to pðyi ; zi ; tÞ. Of course, the precision of this approximation depends on the mesh refinement, but under this assumption, the nodal pressure force may calculated by Eq. (15) where Dyi ¼ ðyiþ1  yi1 Þ=2 and Dzi ¼ ðziþ1  zi1 Þ=2 as depicted in Fig. 9. Instead of using Eq. (15), an improved theoretical solution could be to convert the pressure into nodal forces using the shape functions of the finite elements. However, this approach does not particularly affect the nodal force distribution, so Eq. (15) has been used in this paper for its simplicity.

ð12Þ

This result shows that the hydrodynamic pressure may be modeled by increasing fictitiously the plating mass. Indeed, the second term of Eq. (12) represents the inertia forces produced by a mass € þu € ðy; z; tÞ. As depicted mðyÞ subjected to the total acceleration XðtÞ in Fig. 8, the fluid domain does not have to be explicitly modeled as it can be simply replaced by a mass distributed over the gate surface. 3.3. Numerical model The equivalent representation of the hydrodynamic pressure given by Eq. (12) and illustrated in Fig. 8 can be further formalized to explain how it can be integrated into a numerical model. Let us consider a finite element model of the gate presented in Fig. 1, characterized by its mass, damping and stiffness matrices denoted by ½M; ½C and ½K respectively. During an earthquake, each node

uðtÞ þ rXðtÞ

ð13Þ

If PðtÞ contains the nodal pressure forces, the dynamic equilibrium of the gate may be given by Eq. (14).

  € € ðtÞ þ rXðtÞ _ ½M u þ ½C uðtÞ þ ½K uðtÞ ¼ PðtÞ

Pi ðtÞ ¼ pðyi ; zi ; tÞ DSi ¼ pðyi ; Z i ; tÞ Dyi Dzi

ð14Þ

ð15Þ

Furthermore, taking into account that pðyi ; zi ; tÞ may be directly calculated from Eq. (12) leads to Eq. (16), in which M a;i ¼ mðyi Þ Dyi Dzi may be seen as the added mass (in kg) that has to be lumped at node i to represent the hydrodynamic pres-

Fig. 8. Equivalent model of the hydrodynamic pressure.

Fig. 9. Added mass method.

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sure. Obviously, the first term P s;i ¼ ps ðyi Þ Dyi Dzi is the hydrostatic contribution that can simply be modeled by a constant nodal force for example.

h  i € þu € ðyi ; zi ; tÞ DSi Pi ðtÞ ¼ ps ðyÞ þ mðyi Þ XðtÞ   € þu € ðyi ; zi ; tÞ ¼ Ps;i þ Ma;i XðtÞ

ð16Þ

If the local curvature due to the gate deformation can be neglected, it is worth bearing in mind that the pressure only acts along the horizontal x axis. Consequently, considering the 6 degrees of freedom associated to each node, the force vector that has to be applied at node i is simply given by Eq. (17), which represents the generalization of Eq. (16) to the 6 degrees of freedom of node i.

2 6 6 6 6 Pi ðtÞ ¼ 6 6 6 6 4

P s;i ðtÞ 0 0 0 0 2

6 6 6 6 þ6 6 6 6 4

0 M a;i

3

2

7 6 7 6 7 6 7 6 7¼6 7 6 7 6 7 6 5 4

Ps;i

3

7 7 7 7 7 0 7 7 7 0 5 0 0

Fig. 11. Position of the beam elements.

0 0 0 0 0 0

3

0

0 0 0 0 07 7 7  0 0 0 0 0 7 € 7 ri XðtÞ þ u € ðtÞ ; 0 0 0 0 07 7 7 0 0 0 0 05

0

0 0 0 0 0

0 0 0

2 3 1 607 6 7 6 7 607 7 ri ¼ 6 607 6 7 6 7 405 0 ð17Þ

For convenience, Eq. (17) may be rewritten to obtain Eq. (18), where the vector Ps;i ðtÞ and the matrix ½M a i have been extracted from Eq. (17).

  € þu € ðtÞ Pi ðtÞ ¼ Ps;i ðtÞ þ ½M a i ri XðtÞ

ð18Þ

Finally, the external force vector PðtÞ introduced in Eq. (14) can be obtained by gathering the contributions Pi ðtÞ coming from all the nodes of the plating. This gives Eq. (19), where P s is the global hydrostatic pressure and ½M a  is the added mass matrix.

PðtÞ ¼

 [ [ [ € þu € ðtÞ Pi ðtÞ ¼ Ps;i ðtÞ þ rXðtÞ ½M a i i

i

i

  € þu € ðtÞ ¼ Ps þ ½Ma  rXðtÞ

ð19Þ

This result may be inserted into Eq. (14). Rearranging the terms leads to Eq. (20), which is the dynamic equilibrium equation of the initial lock gate to which additional masses have been lumped to the immersed nodes (Fig. 9).

  € € ðtÞ þ rXðtÞ _ ð½M þ ½M a Þ u þ ½C uðtÞ þ ½K uðtÞ ¼ Ps

ð20Þ

To sum up the previous developments, it may be said that building a numerical model to analyze the seismic response of a lock gate requires first evaluating the added mass M i that is associated to a node i located at position ðyi ; zi Þ over the plating. This may be achieved by using the Westergaard formula as presented in Eq. (21).

(

Mi ¼ 78 qf Mi ¼ 0;

Fig. 10. Numerical modeling of the added mass.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y y hs ðhs  yi Þ iþ1 2 i1

ziþ1 zi1 2

; if yi 6 hs if yi > hs

ð21Þ

Once the added masses M i have been calculated, they may be lumped to the immersed nodes in such a way that they imply inertia forces acting only along the x axis during an earthquake. This restriction corresponds to the expression of the nodal added mass

matrix ½M a i given in Eqs. (17) and (18). While creating the finite element model, it is important that these lumped masses do not create inertia forces along the y and z axes because they are introduced to model an hydrostatic pressure. A proper modeling is therefore required, for example by adding the lumped mass to fictitious nodes that are connected to the plating through very stiff springs (Fig. 10). 4. Numerical investigation of the added mass method 4.1. Description of the numerical models In order to investigate the validity of the added mass method, this section focuses on the lock configuration presented in Fig. 1. The purpose is to perform dynamic analyses using the finite element software LS-DYNA and to compare the results obtained by explicitly modeling the fluid domain with the ones obtained from using the lumped masses distribution. To do so, a finite element mesh of the gate has been built using Belytschko–Tsay shells [24] for the plating and Hughes–Liu beams [24] for the horizontal girders, the vertical frames and the stiffeners (Fig. 1). These beam elements have a T-shaped cross-section with the dimensions listed in Table 1. As shown in Fig. 11, they are defined with an appropriate offset of t p =2 in order to properly take into account the plating thickness. The gate mesh is then entirely associated to an isotropic elastic material law2 characterized by a Young modulus E, a Poisson ratio m and a mass density q having the values mentioned in Section 2. Regarding the boundary conditions, the structure is supposed to be simply-supported at the lock walls but free at the bottom. Consequently, to simulate an earthquake, all the nodes located on the bold lines depicted in Figs. 12 and 13 are subjected to the horizontal seis€ mic acceleration XðtÞ, which is consistent with the hypothesis of neglecting the steel–concrete interaction, as mentioned in Section 2. Finally, the mesh of the structure is roughly regular, with a size of 20  20 cm. As to the FSI-Model (i.e. fluid–structure interaction model), the surrounding water is meshed using constant stress solid elements [24] associated to a particular material law.3 In fact, the fluid is seen 2 3

Refer to MAT-ELASTIC in the LS-DYNA manual [24]. Refer to MAT-ELASTIC-FLUID in the LS-DYNA manual [24].

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7

Fig. 12. Schematic representation of the FSI-Model.

for both entities in order to ensure good contact conditions. Consequently, a more or less regular mesh size of 19  19  19 cm has been chosen for the solid elements. Concerning the other boundaries in Fig. 12, it may be observed that the lock walls are not modeled explicitly using rigid elements. However, the z-translational degree of freedom of all fluid nodes located in vertical planes z ¼ 0 and z ¼ l is directly constrained. Therefore, those nodes remain free to move along the x and y directions, as if the water was simply sliding on the walls. Similarly, modeling the bottom of the lock is avoided by constraining the y-translational degree of freedom of all fluid nodes located in the horizontal plane y ¼ 0. As to the AM-Model (i.e. added mass model), the method exposited in Section 3.3 is used to represent the fluid. The added masses are calculated using the Westergaard formula and lumped with the gate immersed nodes (Fig. 13). The structure has the same properties as detailed previously, but this time the finite element model is easier to build, as only the upstream gate has to be meshed. € Finally, it is worth mentioning that the accelerogram XðtÞ

Fig. 13. Schematic representation of the AM-Model.

Fig. 14. Pressure diagram at the middle of the lock gate.

as an elastic medium with a mass density qf and a bulk modulus K f , for which the stress and strain rates are related by Eq. (22).

p_ ¼ r_ xx ¼ r_ yy ¼ r_ zz ¼ K f ðe_ xx þ e_ yy þ e_ zz Þr_ xy ¼ r_ xz ¼ r_ yz ¼ 0

ð22Þ

Eq. (22) shows that no shearing is supposed to occur in this material. Moreover, the pressure increment p_ inside the solid elements is directly related to the change of volume. This law is adequate to model the behavior of an inviscid fluid in small displacements (no turbulence), which should be appropriate for seismic analyses. The parameters r f and K f are taken equal to 1000 kg/m3 and 2.25 GPa respectively. At the boundaries with the upstream and downstream gates, the interaction is modeled by using distinct nodes (Fig. 12). The automatic surface to surface penalty algorithm4 is chosen to prevent the water from passing through the structure. Doing so, the liquid is only allowed to slide on the flexible gate without friction. As depicted in Fig. 12, it is important to use non-facing nodes 4

Refer to GENERAL-AUTOMATIC-SURFACE-TO-SURFACE in the LS-DYNA manual [24].

depicted in Fig. 15 is employed for all numerical simulations. As mentioned above, this horizontal acceleration is imposed on the nodes located on the bold lines depicted in Figs. 12 and 13. From the Fourier transform shown in Fig. 16, the main part of the seismic excitation is seen to be located below 15 Hz. This synthetic accelerogram has been calibrated from data recorded in France. It is only used for application purposes. Of course, in a more severe seismic zone, the acceleration levels would be higher. As a final comment, it is interesting to compare the size of the AM and FSI models. More than 400,000 nodes are required for the FSI-Model, while only 5000 are used for the AM-Model. As a consequence, the calculation time are drastically different. In the first case, three days are required to run a simulation but only 5 min are needed with the added mass method.

4.2. Validity of the FSI-Model In the remaining part of this paper, the FSI-Model presented in the previous section (Fig. 12) will be used as reference to corroborate the results obtained using the added mass method. As a consequence, it is important to validate the use of an elastic medium for the fluid domain. The first validation step is to associate both the upstream and downstream gates to a rigid material law, which forces them to move as perfectly rigid bodies. The pressure computed should thus be close to the solution of the Laplace Eq. (4), € in which the accelerogram of Fig. 15 is used for XðtÞ. In order to compare the numerical and analytical solutions, the total resulting hydrodynamic force has been calculated with Eq. (23).

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Fig. 15. Time evolution of considered seismic acceleration.

Fig. 16. Fourier transform of considered seismic acceleration.

Fig. 17. Evolution of the resulting pressure force FðtÞ in MN with the time t in seconds obtained numerically and semi-analytically.

Z

l

Z

FðtÞ ¼

hs

pðy; z; tÞ dy dz 0

ð23Þ

0

By comparing the analytical and numerical results, it is found that the relative error on the extreme value of FðtÞ does not exceed 7%. This discrepancy is quite satisfactory and gives a first argument for using the FSI-Model as a reference. In order to consider the ability of the FSI-Model to correctly capture the fluid–structure interaction, numerical simulations have also been performed with flexible upstream and downstream gates having a structure similar to Fig. 1 and material properties close to those listed in Section 4.1. The hydrodynamic pressure provided by the numerical model is then compared with the solutions obtained through a semi-analytical method developed by [18]. The purpose of that semi-analytical approach is to evaluate the hydrodynamic pressure on plane lock gates by considering the fluid–structure coupling. In this case, the modal properties is first derived by using the Rayleigh–Ritz method. The dynamic analysis is then performed by applying the principle of virtual work, in which the

eigenmodes were used to approximate the shape of the displacement field. For illustrative purposes, Fig. 17 compares the total hydrodynamic pressure force FðtÞ obtained by the numerical model and the semi-analytical method mentioned previously. These results, compiled from [23], show a quite satisfactory agreement, although some discrepancies can be observed. Considering the time evolution of FðtÞ depicted in Fig. 17, it is interesting to compare the extreme value reached by FðtÞ during an earthquake. The values predicted by the two methods and listed in Table 2 show that the semi-analytical method underestimates by 11% the force post-processed from LS-DYNA simulations. In addition, it can be observed that the calculation time of the simpliTable 2 Numerical and semi-analytical extreme values of FðtÞ that occur around t ¼ 7:4 s. FSI-Model (MN)

Semi-analytical method (MN)

Ratio

1.42

1.26

0.89

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Fig. 18. Evolution of the resulting pressure PðtÞ in kN/m with time t in seconds using the same mass damping a ¼ 8:85 for the FSI and AM models.

fied method is only equal to two minutes, while it lasts for a couple of days with the numerical approach. As a conclusion, due to the quite good agreement between the results of the FSI-Model and those provided by the analytical and semi-analytical methods, the FSI-Model has been kept as a reference to investigate the validity of the added mass method. 4.3. Numerical simulations with mass-proportional damping Regarding the modeling of the damping in the numerical simulations, a damping part mass is first used.5 Eq. (24) is employed to calculate the damping forces, where f 1 is the vibration frequency of the first mode and n is the critical damping coefficient.

½C ¼ a½M;

with a ¼ 4pf 1 n

ð24Þ

When considering steel structures, it is commonly recommended [25] to choose a value close to 4% for this parameter. From a modal analysis performed with the finite element software NASTRAN, f 1 is found to be equal to 17:6 Hz for the dry gate, which leads to a ¼ 8:85. For the purposes of comparing the AM-Model and the FSI-Model, the results for their hydrodynamic pressures are post-processed at mid-gate (i.e. for z ¼ l=2 in Figs. 1 and 14) because the gate displacements are maximal at this location, which means that the fluid–structure interaction has a large influence on the water pressure pðy; z; tÞ. For this reason, the resulting pressure force PðtÞ acting at mid-gate is used to corroborate the results of the addedmass method. As z ¼ l=2, Eq. (25) defines PðtÞ.

Z PðtÞ ¼

hs

0

  l p y; ; t dy 2

ð25Þ

The time evolution of PðtÞ is presented in Fig. 18 for the seismic acceleration depicted in Fig. 15. These curves clearly show that the added-mass model strongly underestimates the water pressure. This observation may be reinforced by considering the extreme value P u of PðtÞ given by Eq. (26), where T is the duration of the earthquake (15 s in the present simulation).

Pu ¼ max j PðtÞ j 06t6T

ð26Þ

The values of Pu are presented in Table 3 for the FSI and AM models. It is found that the added mass method underestimates P u by a factor equal to 3:25. 5

Refer to DAMPING-PART-MASS of the LS-DYNA manual [24].

Table 3 Extreme pressure force Pu at the middle of the gate (a ¼ 8:85 for both the FSI and AM models). FSI-Model (kN/m)

AM-Model (kN/m)

Ratio

134.3

41.3

3.25

A reason that could be immediately invoked to explain such a discrepancy is the fact that the added mass method is not able to correctly capture the vibrational properties of the coupled system made of the gate and the fluid domain. Indeed, considering the dry gate (i.e. without surrounding water), a modal analysis performed by NASTRAN shows that the fundamental frequency is equal to 17:6 Hz. However, if the fluid is taken into account, the vibrational properties can be considerably modified. This is the case for the FSI-Model, for which the fundamental frequency falls to 5.56 Hz. On the other hand, a value of 5.13 Hz is found for the AM-Model. From these modal analyses, it appears that the added mass approach does not lead to the correct fundamental frequency. This is partly visible on the zoom presented in Fig. 18, where it can be observed that the two curves are slightly out of phase. However, the difference between the fundamental frequencies obtained from AM and FSI models is very small (only 0:43 Hz) and is probably not sufficient to justify the divergences noted in Table 3 and Fig. 18. This argument may be reinforced by considering the acceleration spectrum of Fig. 16 together with the frequencies obtained from the AM and FSI models. This is done in Fig. 19, where it can be seen that the acceleration amplitude associated to the AMModel frequency is greater than that corresponding to the FSIModel. Consequently, regarding the spectral distribution only, the AM-Model should provide conservative results as the dominant mode is more activated than in the FSI-Model. This is clearly not in line with previous observations made in Table 3 and Fig. 18. As a conclusion, the fact that the added mass is unable to correctly capture the vibrational properties of the coupled system cannot explain the drastic underestimation of the resulting pressure force at mid-gate.

4.4. Numerical simulation with a modified mass-proportional damping A better reason that could be invoked to justify the discrepancies mentioned in the previous section is the definition of the damping forces. Indeed, in Eq. (24), the matrix ½C is directly pro-

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Fig. 19. Comparison of the fundamental frequencies with the acceleration spectrum for a mass-proportional damping (a ¼ 8:85).

Fig. 20. Single degree of freedom systems representing the lock gate with or without added mass.

portional to the mass matrix ½M of the dry structure. However, in the AM-Model, this is no longer the case because the lumped masses are also taken into account. This means that Eq. (27) is verified.

½C ¼ að½M þ ½M a Þ

ð27Þ

However, considering a mass-proportional damping, it is clear that the damping forces should only be proportional to the mass of the structure, and not to the added mass of the water. In other words, the damping matrix ½C has to be identical for the dry structure, the AM-Model, and the FSI-Model, because the damping forces are only related to the mass of the gate, which remains unchanged in these three models. This is objectively the case in the FSI-Model, because solid elements are used for the fluid domain, so there is no additional mass distributed over the gate. In contrast, Eq. (27) clearly shows that the damping forces are overestimated by the AM-Model. This is particularly evident knowing that the mass of the gate only represents 5% of the total added mass of water. To ^ keep the matrix ½C unchanged in the AM-Model, the coefficient a should be defined via Eq. (28), which is obviously impossible for a system with multiple degrees of freedom.

a½M ¼ a^ ð½M þ ½Ma Þ

ð28Þ

An approximate solution to Eq. (28) may be found by first replacing the dry gate by a system with only one degree of freedom (Fig. 20). In this case, the matrices ½M and ½M a  in Eq. (28) can be directly replaced by the total mass M of the dry gate and the total ^ ¼ 0:46, added mass M a , leading to Eq. (29). The last relation gives a which is much smaller than the previous value of 8:85 calculated for a.

a^ ¼ a

M M þ Ma

ð29Þ

The time evolution of the resulting pressure PðtÞ at mid-gate ^ ¼ 0:46 is depicted in Fig. 21, from which it can be obtained for a observed that the amplitudes are now in better agreement. This is confirmed by the resulting extreme values presented in the first line of Table 4 (i.e. for E ¼ 210 GPa), where it is found that the

added mass method provides conservative results. This may be explained for two reasons. The first one is that the AM-Model is not able to correctly capture the vibrational properties of the gate surrounded by water. As a consequence, the fundamental frequencies may be favorably or unfavorably positioned on the acceleration spectrum (Fig. 19). In the present case, the AM-Model fundamental frequency matches a peak of the excitation spectrum, which leads to a safe evaluation of the pressure. The second reason is directly related to the previous one: the results of the AM-Model are conservative, provided that the damping forces are not excessively evaluated. To do so, the correction suggested in Eq. (29) has been applied. However, this correction is known to be too optimistic in the sense that only a fraction of the collaborating mass is mobilized by the first mode. Although the correction given by Eq. (29) drastically reduces the damping forces, it does not necessarily imply that the added mass approach is conservative in all cases. Indeed, a wrong evaluation of the fundamental frequencies may be favorable or not. In order to illustrate this, other simulations have been run using both AM and FSI models in which Young modulus has been increased up to 300 GPa. This causes the fundamental frequencies to also increase. From a modal analysis, they are found to be equal to 6:64 Hz and 6:02 Hz for the FSI and AM model respectively. From Fig. 22, it is observed that the situation is exactly the opposite as the one presented in Fig. 19 for E ¼ 210 GPa. This time, the AM-Model fundamental frequency does not coincide with a peak of the acceleration spectrum. This explains why the difference between the extreme values of AM and FSI models listed in the second line of Table 4 (i.e. for E ¼ 300 GPa) is now reduced. To conclude, it can be said that implementing the added mass method with the correction in Eq. (29) does not necessarily lead to conservative results. The results might be conservative, depending on the excitation spectrum and the ability of the AM-Model to correctly capture the vibrational properties of the coupled system consisting of the gate and the surrounding water. However, even if the correction provided by Eq. (29) is quite simplistic, it contributes to improve the results provided by the added mass method. As a final comment, it should be mentioned that if a response spectrum analysis is used in conjunction with

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Fig. 21. Evolution of the resulting pressure PðtÞ in kN/m with time t in seconds using a modified mass damping a ¼ 0:46 for the AM-Model.

Table 4 Extreme pressure force Pu at the middle of the gate (a ¼ 0:46 for the AM-Model). Young modulus E (GPa)

FSI-Model (kN/m)

AM-Model (kN/m)

Ratio

210 300

134.3 193.5

186.4 200.4

0.72 0.96

lumped masses, the Rayleigh coefficient a is not directly involved, but the damping coefficient n that is used to calibrate the response spectrum must be adapted. Indeed, considering the two systems of pffiffiffiffiffiffiffiffiffiffiffi Fig. 20, Eq. (30) may be written where x ¼ K=M and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ¼ K=ðM þ M a Þ. x

C ¼ 2xn M;

b ¼ 2x ^ ^n ðM þ M a Þ C

ð30Þ

As mentioned previously, the added mass of the water should not modify the damping forces, which means that Eq. (31) can be deduced.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M þ Ma

b ) ^n ¼ n C¼C

ð31Þ

Consequently, the critical damping coefficient ^ n may no longer be equal to 4% but has to be reduced to take into account the lower ^ of the system with the added mass. Assuming circular frequency x that the gate may be approximately represented by a one degree of freedom system (which is practically the case if there is only one significant mode), Eq. (31) leads to ^ n ¼ 1%. This means that the response spectrum should be calibrated with a damping coefficient of 1% instead of 4%. Of course, this provides higher spectral accelerations. 4.5. Numerical simulation with a stiffness-proportional damping From the previous simulations, it results that the added mass approach fails to correctly evaluate the hydrodynamic pressure

because the damping forces are overestimated when using massproportional damping. In order to study this explanation and to investigate more deeply the validity of the added mass method, another option is to work with a stiffness-proportional damping. In this case, the damping matrix ½C is calculated following Eq. (32), where f 1 is the fundamental frequency of the dry gate, which was found to be equal to 17:6 Hz.

½C ¼ b ½K;

b¼

n

pf 1

ð32Þ

Considering, as previously, a damping coefficient n ¼ 4%, according to Eq. (32), b is found to be equal to 7:44  104 . The main advantage of using stiffness-proportional damping is that the matrix ½K is the same in the FSI and AM models, which was not the case in a mass-proportional damping where the matrices ½M and ½M þ ½M a  were clearly distinct. Consequently, a correction similar to Eq. (29) is not required here and the same value for b can be used in both models. The numerical results are depicted in Fig. 23, where the AMModel is shown to provide an overly conservative estimate of the resulting pressure force PðtÞ. The same conclusion may be drawn from Table 5, from which the extreme value provided by the AMModel is found to be twice that of the FSI-Model. Once again, this difference may be explained by the same reason as invoked in Section 4.4, i.e. the fact that the fundamental frequency of the AMModel matches a peak value of the acceleration spectrum (Fig. 19). However, considering a Young modulus of 300 GPa, the situation is inverted (Fig. 22) and as shown in the second line of Table 5, working with a lumped mass is not conservative in this case. This example reinforces the conclusion of Section 4.4: the success of the added mass method depends on its ability to correctly evaluate the vibrational properties of the coupled system. As a final comment, it can be mentioned that if a stiffnessproportional damping is used in a response spectrum analysis, a modified value has to be used for the damping coefficient n. Indeed,

Fig. 22. Comparison of the fundamental frequencies with the acceleration spectrum (E ¼ 300 GPa) for a modified mass-proportional damping (a ¼ 0:46).

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Fig. 23. Evolution of the resulting pressure PðtÞ in kN/m with time t in seconds using the same stiffness-proportional damping for both the FSI and AM models.

Table 5 Extreme pressure force Pu at the middle of the gate (b ¼ 7:44104 for both the FSI and AM model). Young modulus E (GPa)

FSI-Model (kN/m)

AM-Model (kN/m)

Ratio

210 300

132.3 433

282.1 313

0.47 1.36

doing the same developments as in Section 4.4, Eq. (33) might be ^ have already been defined in Section 4.4. obtained, where x and x

C¼

2n

b ¼ 2n K C ^ x

Rayleigh-type damping described in Eq. (35), which is a mix of the two situations considered earlier.

½C ¼ a½M þ b½K

ð35Þ

To determine the coefficients a and b in (35), engineers may decide to have the same damping coefficient n for the two first modes. Eq. (36) is derived, denoting the frequencies of these two modes by f 1 and f 2 , respectively.

a¼

4pf 1 f 2 ; f1 þ f2

b¼

n

pðf 1 þ f 2 Þ

ð36Þ

ð33Þ

From a modal analysis of the dry gate, employing a Young modulus E ¼ 210 GPa, it is found that f 1 ¼ 17:6 Hz and f 2 ¼ 22:5 Hz,

As the added mass of water should not modify the damping forces, Eq. (34) may be written, which is similar to Eq. (31).

leading to a ¼ 4:694 and b ¼ 3:17  104 . As previously, it may be interesting to consider the time evolution of the pressure PðtÞ at the middle of the gate where the fluid– structure interaction is maximal. The numerical results are depicted in Fig. 24, from which it can be observed that the AMModel is not conservative. This may be quantified by considering the extreme value Pu of PðtÞ defined by Eq. (26) obtained in each case. For the FSI and AM-Models, it is found that P u is equal to 144:9 kN/m and 91:1 kN/m, respectively, leading to a ratio of 1:6. In other words, the AM-Model underestimates the dynamic pressure by 60%. This conclusion may be investigated by comparing the fundamental frequencies of the AM and FSI-Models. This is carried out in Fig. 25, from which it can be seen that in the two cases, the fundamental frequencies coincide with a peak value of the acceleration spectrum. This is particularly true for the AM-Model. Therefore, the underestimation of the water pressure in the AMModel is much more related to an overestimation of the damping forces due to the use of lumped masses. Indeed, the damping matrix in this case is ½C ¼ að½M þ ½M a Þ þ b½K, while it is still given by Eq. (35) in the FSI-Model.

x

K;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M M þ Ma

b ) ^n ¼ n C¼C

ð34Þ

Consequently, if a mass or stiffness-proportional damping is used, the same correction has to be applied to the damping coefficient used to calibrate the response spectrum. This implies that the same correction is also valid if a mixed Rayleigh-type damping a½M þ b½K is used within the added mass method. 4.6. Numerical simulation with a Rayleigh-type damping In the previous sections, the effects of mass- and stiffnessproportional damping were discussed separately. However, in the current applications of finite element analysis, these models are not often used. Indeed, in the first case, Eq. (24) shows that there is an important damping of the lower modes, while in the second case, according to Eq. (32), there is a strong damping of the higher modes. To avoid these drawbacks, current practice is to use the

Fig. 24. Evolution of the resulting pressure PðtÞ in kN/m with time t in seconds using the same Rayleigh-type damping for both the FSI and AM-Models.

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Fig. 25. Comparison of the fundamental frequencies with the acceleration spectrum for a Rayleigh-type damping.

As a final comment, noting that the corrected damping coefficients ^ n given by Eqs. (31) and (34) are obtained by multiplying pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the initial coefficient n by the same factor M=ðM þ M a Þ for the mass- and stiffness-proportional damping, the same correction is also valid if a mixed Rayleigh-type damping a½M þ b½K is used within the added mass method.

5. Discussion and conclusions In this paper, the theoretical bases of the added mass method have been presented extensively to point out one of the main limitation of this approach: the lumped masses are derived from an analytical solution developed for perfectly rigid structures. Therefore, two major consequences can be deduced:  The added mass method fails to correctly capture the vibrational properties of the coupled system made of the gate and the surrounding water. In particular, the fundamental frequency is not correctly evaluated, which leads to a design situation that may be safe or not. Indeed, if the fundamental frequency according to the AM-Model matches a peak value of the excitation spectrum, the added mass approach may turn out to be conservative. However, this is not always the case and it is impossible to predict whether the results of the AMModel are reliable or not unless a dynamic analysis is performed with an extensive modeling of the fluid domain.  The added mass method fails to correctly model the fluid–structure interaction, with the direct consequence that the pressure field is not correctly evaluated. In the AM-Model, both the magnitude and time oscillations of the local pressure are not in line with the predictions of the FSI-Model. This implies that the total force applied on the gate during an earthquake is not properly calculated, which also leads to questionable design situations. The added mass method may be quite appropriate for performing modal analyses, provided that the added mass is properly calculated. However, another problem has also been underlined in this paper, to explain why the method is not suited to perform dynamic analysis. Numerical simulations based on the AM-Model have shown that the influence of damping may significantly reduce the forces applied on the gate. More specifically, if a Rayleigh-type damping is used, care should be taken to properly define the damping coefficients. Indeed, as lumped masses are distributed over the gate, the mass matrix is considerably modified. However, this modification is purely virtual and is only introduced to model the effect of the surrounding water. As a consequence, by comparison with the dry structure, there should be no additional damping associated to the added mass. Unfortunately, the classical definition of the Rayleigh-type damping considers the total mass matrix

of the structure to calculate the damping forces, which means that these latter could be drastically overestimated. To avoid this unsafe situation, it is suggested to reduce the critical damping coefficient n in accordance with Eqs. (31) or (34). Therefore, the evaluation of the pressure forces can be improved, but this does not necessarily lead to a conservative design situation, as the fundamental frequency remains incorrectly estimated by this approach. Adding dashpots to all the nodes of gate could be an option, but these dashpots should be calibrated with the appropriate damping coefficients, which are not easy to determine. Furthermore, on a practical engineering level, it is desirable to keep the finite element model as simple as possible. As a final comment, it can be said that the added mass approach should be carefully used as an approximate method to carry out the seismic design of lock gates, particularly if it is based on theoretical solutions that are derived for perfectly rigid structures. Its main advantage is avoiding the explicit modeling of the fluid domain, leading to fewer pre- and post-processing operations. Indeed, building the AM-Model, running a simulation, and postprocessing the results requires more or less 3–5 h, while 2 or 3 days are needed for the FSI-Model. However, the results of the AM-Model are not necessarily conservative, and their validity should be more thoroughly investigated, particularly at the final design stages of a lock gate. In order to circumvent the difficulties of the AM and FSI models, another possibility is to use the semianalytical procedure suggested by [18], which allows achieving a better evaluation of the water pressure (with due consideration for the fluid–structure interaction) without having to model the entire fluid domain. Acknowledgements This research was partially supported by the Government of the Walloon Region of Belgium [Grant No. 1410027]. References [1] Westergaard HM. Water pressures on dams during earthquakes. Trans Am Soc Civil Eng 1933;98(2):418–33. [2] Chopra A. Hydrodynamic pressures on dams during earthquakes. J Eng Mech Div 1967;93:205–23. [3] Rashed AA, Iwan WD. Dynamic analysis of short-length gravity dams. J Eng Mech 1985;111(8):1067–83. doi: http://dx.doi.org/10.1061/(asce)0733-9399 (1985)111:8(1067). [4] Graham E, Rodriguez A. The characteristics of fuel motion which affect airplane dynamics. J Appl Mech 1952;19:381–8. [5] Wendel K. Hydrodynamic masses and hydrodynamic moments of inertia. Tech. rep., David and Taylor Model Basin Translation 260, Washington; 1956. [6] Housner G. Dynamic pressures on accelerated fluid containers. Bull Seismol Soc Am 1957;47:15–37. [7] Epstein H. Seismic design of liquid storage tanks. J Struct Div 1976;102:1659–73. [8] Abramson H. The dynamic behavior of liquids in moving containers. Tech. rep., National Aeronautics and Space Administration, Washington; 1966.

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[17]

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Please cite this article in press as: Buldgen L et al. Investigation of the added mass method for seismic design of lock gates. Eng Struct (2016), http://dx.doi. org/10.1016/j.engstruct.2016.10.047