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Computer aided stability analysis of gravity dams —CADAM Article in Advances in Engineering Software · July 2003 DOI: 10.1016/S0965-9978(03)00040-1

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Advances in Engineering Software 34 (2003) 403–420 www.elsevier.com/locate/advengsoft

Computer aided stability analysis of gravity dams—CADAM Martin Leclerc, Pierre Le´ger*, Rene´ Tinawi Department of Civil Engineering, E´cole Polytechnique de Montre´al, University of Montreal Campus, P.O. Box 6079, Station CV, Montreal, Que., Canada H3C 3A7 Received 6 March 2002; accepted 3 March 2003

Abstract This paper presents the main features and organisation of CADAM, a computer program, freely available, that has been developed for the static and seismic stability evaluations of concrete gravity dams. CADAM is based on the gravity method using rigid body equilibrium and beam theory to perform stress analyses, compute crack lengths, and safety factors. Seismic analyses could be done using either the pseudostatic or a simplified response spectrum method. CADAM is primarily designed to provide for learning the principles of structural stability evaluation of gravity dams. It could also be used for research and development on stability of gravity dams. In adopting several different world-wide published dam safety guidelines, a large number of modelling options have been implemented. These include (i) crack initiation and propagation, (ii) effects of drainage and cracking under static, seismic, and post-seismic uplift pressure conditions, and (iii) safety evaluation procedures using deterministic, allowable stresses and limit states probabilistic analyses (Monte-Carlo simulations). Structural stability evaluation of a 30 m dam is presented to illustrate the use of CADAM. q 2003 Elsevier Science Ltd. All rights reserved. Keywords: Concrete gravity dams; Stability analysis; Computer aided design; Gravity method; Floods; Seismic response; Monte-carlo simulations

1. Introduction There are over 4800 large concrete gravity dams in existence throughout the world outside China. In North America, in particular, the average age of these dams is about fifty years. The static and seismic safety of existing concrete gravity dams is therefore a continuous concern to dam owners owing to the ageing processes altering their strength and stiffness, as well as revised predictions of the maximum loads associated to severe floods and earthquakes. It is thus required to perform periodic reassessment of their static and seismic structural stability under extreme loads for which these dams were not designed. In addition, owners with a large number of dams need to assess the safety of these structures and prioritise their investment when undertaking expensive rehabilitation works to accommodate a probable maximum flood or a maximum credible earthquake. It is obvious that if under these extreme conditions structural damage can be tolerated, * Corresponding author. Tel.: þ1-514-340-4711x3712; fax: þ 1-514340-5881. E-mail address: [email protected] (P. Le´ger).

the reservoir must be contained to avoid a catastrophe downstream. A progressive methodology is normally adopted starting with the gravity method based on rigid body equilibrium and beam theory before considering linear or nonlinear finite element models, if necessary [1,2]. FERC [3,4], CDA [5], USACE [6], Ancold [7], and USBR [8] present guidelines for dam safety assessment based on the gravity method. Nevertheless, even the gravity method, which is relatively simple to understand and apply, can be lengthy when evaluating crack length, especially for inclined failure planes, or when using a pseudo-dynamic technique. On the other hand, finite elements in the linear or nonlinear range have their share of difficulties related to stress singularities or crack propagation particularly with discrete cracks. Therefore there was a need to develop an interactive friendly software, such as CADAM [9], to assess very quickly for a given dam the safety margins under extreme loads (Fig. 1). Alternatively, CADAM risk analysis capabilities are useful to classify which structures are the most vulnerable within a portfolio of dams. In addition to safety issues, there are several differences among adopted guidelines regarding:

0965-9978/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0965-9978(03)00040-1

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Fig. 1. CADAM loading conditions for static and seismic analyses: (a) basic static analysis conditions; (b) pseudo-static seismic analysis; (c) pseudo-dynamic seismic analysis.

(a) cracking initiation and propagation criteria, (b) static and seismic uplift pressures along ts and cracks, and (c) safety evaluation format (allowable stress, limit state method). Moreover, the response of spillways and gravity dams during the 1996 Saguenay flood in Quebec, Canada emphasised once more the need to consider overtopping and floating debris while performing flood safety assessment [10]. Seismic safety evaluations are very frequently conducted using the pseudo-dynamic (response spectrum) method as presented by Chopra [11] in complement to the pseudo-static seismic coefficient (rigid body) method. Although these calculations are well documented, they are complicated due to the iterative nature of crack length calculation and its consequence on the uplift pressures. As

noted above, there is also a growing interest in performing risk based safety evaluation where the probability of failure of a dam is evaluated considering explicitly uncertainties in strength and loading modelling parameters through suitable probability density functions [12]. In most engineering offices, in-house spreadsheets are developed and adapted on a case by case basis to perform dam stability analysis following particular safety guidelines. This is due to the very lengthy and tedious computations, particularly when pseudo-dynamic seismic analyses are considered. Moreover, there are no widely available computational tools: (a) for learning the principles of stability analysis in the academic or professional environment, and, (b) for performing research and development on the structural safety of gravity dams.

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We have thus identified needs to develop, and put in the public domain, a comprehensive computer program, CADAM, to perform stability evaluation of gravity dams based on the gravity method. This paper presents the organisation and computational features of CADAM to provide a fully integrated computing environment with output reports and graphic to visualise input parameters and output performance indicators as required in practice (stresses, crack length, resultant position, safety factors). Several modelling options have been included allowing s: † to perform static, pseudo-static, pseudo-dynamic, and probabilistic safety assessment, † to corroborate hand calculations with computer calculations to develop the understanding of the computational procedures, † to conduct parametric analysis on the effects of geometry, strength of materials and load magnitudes on the structural response, † to compare uplift pressures, crack propagation, and shear strength (peak, residual) assumptions from different dam safety guidelines [3 –8], and, † to study different strengthening scenarios including posttensioning. After presenting an overview of CADAM main features and analysis options, specific modelling techniques adopted for basic static analyses, flood, and seismic as well as probabilistic analyses are discussed. Applications related to the structural response of a 30 m high gravity dam are described for illustrative purposes. The paper ends with perspectives for future CADAM developments.

2. Basic principles of the gravity method The evaluation of the structural stability of a dam against sliding, overturning and uplifting along concrete lift ts is performed considering two distinct analyses: † A stress analysis to determine eventual crack length and compressive stresses (Fig. 2). † A stability analysis to determine: (a) the safety margins against sliding along the t considered, and (b) the position of the resultant of all forces acting on the t. The gravity method considers the dam as a cantilevered structure and is based (a) on rigid body equilibrium to determine the internal forces acting on the potential failure plane (ts and concrete-rock interface), and (b) on beam theory to compute stresses. The use of the gravity method requires several simplifying assumptions regarding the structural behaviour of the dam and the application of loads [4]:

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† The dam body is divided into lift ts of homogeneous properties along their length; the mass concrete and lift ts are initially assumed uniformly elastic (uncracked state). † All applied loads are transferred to the foundation by the cantilever action of the dam without interactions with adjacent monoliths. † There is no interaction between the ts, that is, each t is analysed independently from the others. † Normal stresses are linearly distributed along horizontal planes. † The uplift pressures intensity could be modified along a crack, depending upon the drainage condition and rate of crack opening (static vs. seismic conditions). 2.1. Stress analysis CADAM uses an iterative procedure summarised in Fig. 2b to compute the crack length, Lc : Once the crack initiation criterion indicates the formation of a crack, the iterative computation begins. The crack length is computed using a bi-section method and the uplift pressures are updated according to the selected drainage options until the crack propagation criterion indicates stress equilibrium and crack arrest. To consider stress concentration at the crack tip, the criterion for crack initiation may be different than that for crack propagation. Closed form formulations to compute crack length while updating uplift pressures as water penetrates the crack are only available for simple cases where the crack is considered horizontal, a no-tension criterion is used and without drainage. The iterative computation of crack length when concrete tensile strength is nonzero or for an inclined plane requires the computational power of a computer. In most guidelines, uplift pressures are considered as external load acting on the surface of the t. The stress at the crack tip, sn; is computed while including uplift pressures in the force resultant (in the crack propagation iterative procedure) [3,6,8,13]. This calculation produces a linear effective normal stress distribution, sn ; even in the case where a nonlinear uplift pressure distribution is present along the base due to drainage or cracking: P

V ^ sn ¼ A

P

My I

ð1Þ

where P

V ¼ sum of all vertical load including uplift pressures, A P ¼ area of uncracked ligament, M ¼ moment about the centre of gravity of the uncracked ligament of all loads including uplift pressures, I ¼ moment of inertia of the uncracked ligament, y ¼ distance from centre gravity of the uncracked ligament to the location where the stresses are computed.

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Fig. 2. (a) Existing dam vs. idealized structural models. (b) Iterative procedure for crack length computations.

Alternatively, uplift pressures could be considered as an internal load along the t [4]. The stresses at the crack tip are computed from total stresses without uplift pressures. Uplift pressures are then subtracted from total stress to obtain effective stresses, sn to be used for crack initiation (propagation) criterion [4]. These effective stresses may not exhibit a linear distribution along a t. For a stability analysis, the basic shear-friction sliding safety factor (SSF) formula along a horizontal plane is given as: P ð V 2 UÞtan f þ cAC P SSF ¼ ð2Þ H where P

V ¼ Sum of vertical forces excluding uplift pressures, U ¼ uplift pressure force resultant, f ¼ friction angle (peak value or residual value), c ¼ Cohesion (apparent for rough unbonded t or real for bonded t); for apparent cohesion, the may

specify a minimal value of compressive stress, spn ; to determine the compressed area upon which cohesion could be mobilised, AC ¼ area in compression (a function of the crack length) and P H ¼ sum of horizontal forces. Eqs. (1) and (2) have been enhanced in CADAM to consider inclined lift ts and all relevant seismic load components.

3. CADAM—overview of main features and analysis options 3.1. Programming and computing environment Developers tend to divide along language boundaries. Once they know a programming language, they identify themselves by it: ‘a Cþ þ programmer,’ ‘a Delphi

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developer’, etc. The key is applicability and each programming language is as a specialized tool. A hammer specialist does not make a good carpenter. The authors find Cþ þ , Delphi, and Java to all be useful languages, and even a little Visual Basic applies when appropriate.

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Delphi shares the compiler back end with Cþ þ Builder compiler, so the efficiency of the generated codes is comparable. In reliable benchmarks [14], Microsoft Visual Cþ þ rated tops in speed and size efficiency in many cases. Although these small advantages are unnoticeable for

Fig. 3. CADAM overall organization.

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general applications. Visual Basic operates in an interpreted mode and is quite reactive. However, Visual Basic speed rates well behind Delphi and Cþ þ tools. Java-based tools such as Borland JBuilder and Microsoft Visual Jþ þ approach compile times of Delphi. However, Java speed efficiency leaves something to be desired, because Java is an interpreted language. CADAM has been developed using Borland Delphie 6 compiler [15]. Borland Delphi is an object-oriented, visual programming environment to develop 32-bit applications for deployment on Windows and Linux platforms. Using Delphi, the programmer can create highly efficient applications with a minimum of manual coding. Delphi provides all the tools needed to develop, test, debug, and deploy applications, including a large library of reusable components, a suite of design tools, applications and form template, and programming wizards. These tools simplify software coding and shorten development time. Delphi can be used to write Windows and Linux graphical interface (GUI) applications, console applications, service applications, dynamic-link libraries (DLLs), and other programs. Delphi includes features to write easily

distributed applications, including client/server, multitiered, and Web-based systems. Delphi has been adopted at first for its object oriented programming environment. The other reason for this choice is that often the complete design methodology is only worked out in full detail at the programming stage. Therefore understanding of the design engineering problem is more important than knowledge and experience of programming [16]. Finally, the ease of programming, , and suitability for engineering design software, are all additional reasons for adopting this programming language for this particularly complex engineering analysis and design problem. 3.2. Overall program organisation and analysis options Fig. 3 presents CADAM overall organisation. The dam geometry (Fig. 4a,b), material properties (Fig. 4c) the various load conditions, cracking options, and load combinations are first specified as input data for subsequent structural analyses, outlined in Fig. 1. The following analysis options are currently available: (1) static analyses,

Fig. 4. Definition of dam model: (a) dam-foundation-reservoir system and CADAM interface; (b) dam geometry; (c) material properties.

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(2) seismic analyses, (3) post-seismic analyses, (4) incremental load analysis, and (5) probabilistic safety analysis.

4. Static loading conditions The load conditions ed by CADAM are shown in Fig. 1. Some particular features are described in the following. Various dam safety guidelines equations presented to compute the uplift pressures according to the position of the drain from the upstream (u/s) face, the drain effectiveness and the elevation of the drainage gallery have been implemented (Fig. 5). It is interesting to note that Federal agencies in the US (FERC, USACE and USBR) are currently evaluating the need for unified Federal criteria for the calculation of uplift pressures as well as crack initiation and propagation criteria in the stability of concrete gravity dams [17]. It is believed that a computational tool like CADAM could be of great assistance to conduct extensive parametric analyses for various dam geometry and drainage conditions to study the effects of modelling assumptions on computed performance indicators. During a severe flood, it is possible that a section of the dam be overtopped. In this case, water pressure may be considered on the crest surface as well as floating debris.

5. Seismic and post-seismic safety analysis Some original features that have been included for seismic and post-seismic safety analyses are presented below. Existing cracks computed from the initial static conditions may close depending on the intensity and orientation of the earthquake forces. Separate analyses could be performed successively with the base acceleration

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pointing u/s and d/s to estimate the cumulative damage by reducing the cohesion that could be mobilised along the t considered. The cohesion is considered null along the seismically induced crack length to compute the SSFs in seismic and post-seismic conditions. Since the pseudo-static method does not recognise the oscillatory nature of earthquake loads CADAM performs the safety evaluation in two phases: (a) a stress analysis using peak ground acceleration (or spectral acceleration) values to compute the crack length, and (b) a stability analysis using sustained acceleration values to compute SSFs. A single acceleration peak might be sufficient to induce a crack but it may not be of sufficient duration to induce significant sliding displacement. For stability evaluation an ‘effective’ acceleration equals to 0.67 –0.5 the peak value has often been used in practice [18]. The stress analysis is therefore used to determine the length over which cohesion will be applied in the stability analysis. 5.1. Pseudo-static analysis In a pseudo-static seismic analysis, the inertia forces induced by the earthquake are computed from the product of the mass and the acceleration. The dynamic amplification of inertia forces along the height of the dam due to its flexibility is neglected. In a pseudo-static analysis, it is required to specify the peak ground horizontal and vertical accelerations as well as the sustained accelerations. Westergaard added mass [6] is used to represent the hydrodynamic effects of the reservoir on the dam. Options

Fig. 5. Uplift pressures and drainage system options.

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are provided to for water compressibility effects and inclination of the u/s and d/s faces. 5.2. Pseudo-dynamic analysis A pseudo-dynamic seismic analysis is based on the response spectrum method. A pseudo-dynamic analysis is conceptually similar to a pseudo-static analysis except that it recognises the dynamic amplification of the inertia forces along the height of the dam. However, the oscillatory nature of the amplified inertia forces is not considered. That is the stress and stability analyses are performed with the inertia forces continuously applied in the same direction. The basic input data required to perform a pseudo-dynamic analysis, using the simplified response spectrum method proposed by Chopra [11], are: (a) peak ground and spectral accelerations (peak and effective values), (b) dam and foundation stiffness and damping properties, (c) reservoir bottom damping properties and velocity of an impulsive pressure wave in water, (d) modal summation rule. In a pseudo-dynamic analysis, the moment and axial force acting on the lift t considered are computed from the selected modal combination rule. The resulting moment and axial force are then used to compute the related stresses and crack length. This approach is generally conservative. In linear (uncracked) analysis, it is more appropriate to compute stresses separately for the first mode and the higher

modes and then apply the modal combination rule to stresses. However, this approach, adopted in linear analysis, is not suitable to estimate crack length, especially if uplift pressures are to be varied within the seismic crack (e.g. with no uplift pressure in an opened crack). Moreover, it is assumed that the period of vibration of the dam is unaffected by cracking which is obviously an approximation that might be overcome only if transient nonlinear dynamic analyses are considered [1]. However, consistency in modelling assumptions, as implemented in CADAM, ensures that the results of a pseudo-dynamic analysis converge towards the results of a pseudo-static analysis as the period of vibration tends to zero in the pseudo-dynamic analysis. This novel consistent approach is made possible and simple to use within CADAM.

6. Cracking options and evolution of uplift pressures in cracks CADAM provides various options for specification of: (a) tensile strengths for crack initiation and propagation, (b) dynamic amplification factor for the tensile strength, (c) incidence of cracking on static uplift pressure distributions and drain effectiveness (Fig. 6), (d) effect of cracking on the transient evolution of uplift pressures during earthquakes (Fig. 7), (e) evolution of uplift pressures in the post-seismic conditions.

Fig. 6. Cracking options: effect of cracking on the drainage system.

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† USACE [3] and FERC [3,4] assume that uplift pressures are unchanged by earthquake load (i.e. at the preearthquake intensity during the earthquake). † USBR [8] mentions: ‘When a crack develops during an earthquake event, uplift pressures within the crack are assumed to be zero’. This is based on the assumption that a rapid crack opening reduces the uplift pressures and that the cyclic crack motions are too fast to allow reservoir water to penetrate and build-up the pressure. † CDSA [13] mentions: ‘In areas of low seismicity, uplift pressures prior to the seismic event are normally assumed to be maintained during the earthquake even if cracking occurs. In areas of high seismicity, the assumption is frequently made that uplift pressures on the crack surface are zero during the earthquake when the seismic force are tending to open the crack’. Fig. 7. Transient evolutions of uplift pressures in seismically induced crack.

When cracking is allowed, a distinction is made between the criteria for crack initiation and crack propagation. After crack initiation, say at the u/s end of a t where stress concentration is minimal, it is likely that stress concentration will occur near the tip of the propagating crack [4]. For example the crack initiation criterion could be set to a tensile strength of 1000 kPa but once the crack is initiated, it should be propagated to a length sufficient to develop compression at the crack tip (no-tension condition for crack propagation). The allowable tensile strengths for crack initiation and propagation are specified for different usual, flood, seismic and post-seismic load combinations. Allowable tensile strengths for crack initiation and propagation are specified as the tensile strength divided by appropriate coefficients. Upon cracking when drainage is considered, four options are offered (Fig. 6): (1) no drain effectiveness under any cracking condition, (2) no drain effectiveness when the crack reaches the drain line, (3) full drain effectiveness, but with full uplift pressures applied between the reservoir and the drain line, (4) full drain effectiveness with a linear decrement in uplift pressures starting from full reservoir pressure at the reservoir level to the drainage pressure at the drain line. In a design office, the consideration of these options is often studied to obtain upper and lower bounds values due to uncertainties related to drainage conditions. Due to the lack of historical and experimental evidences, there is still a poor knowledge of the transient evolution of uplift pressures in a crack due to its cyclic movements during earthquakes (Fig. 7). † ICOLD [19] states: ‘The assumption that pore pressure equal to the reservoir head is instantly attained in cracks is probably adequate and safe’.

To model these various assumptions CADAM allows three options for transient evolution of uplift pressures in seismic cracks: (i) no change from pre-earthquake pressure conditions, (ii) full pressure, and (iii) zero pressure. To reach equilibrium in the post-earthquake condition two options are provided. First, a conservative assumption for post-seismic uplift pressures is to use the full reservoir pressure in earthquake-induced cracks in the post-seismic safety assessment. However, if seismic cracks are closed after the earthquake the uplift pressures to be used for the post-seismic condition could be relieved by the drainage system. In this case, the preearthquake uplift pressure intensity is used immediately after the earthquake.

7. Load combinations and safety evaluation format Five load combinations are ed by CADAM (Fig. 8): (1) (2) (3) (4) (5)

normal operating, flood seismic 1 (e.g. Design Base Earthquake), seismic 2 (e.g. Maximum Credible Earthquake), and post-seismic.

For each load combination, multiplication factors could be specified for each basic load conditions. This option is very useful when an applied load is increased until a safety factor equal to 1 is reached and thus determines the ultimate strength of the dam. For each load combination, the required safety factors to ensure an adequate safety margin for structural stability are specified. These values are not used in the computational algorithm of the program. They are

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Fig. 8. Load combinations, SSF computation and required safety factors.

reported in the output results to facilitate the interpretation of the computed safety factors in comparison with the corresponding allowable values. Different strategies have been adopted to study the safety margin of concrete dams as a function of the uncertainties in the applied loading and material strength parameters. By proper definition of basic loading condition parameters and multiplication factors to form load combinations, a variety of loading scenarios could be defined to assess the safety of a dam-foundation-reservoir system. In some cases, the applied loads are increased to induce failure (e.g. u/s, d/s water levels are increased, ice loads, water density, etc.). The safety margin is then assessed by comparing the magnitude of the load inducing failure with that of the applied load for the combination under study. CADAM can be used effectively to perform this type of study using a series of analyses while increasing the applied loads either through the basic loading input parameters, by applying appropriate load condition multiplication factors while forming the load combinations, or by using the incremental load analysis option. In a different approach, the specified material strengths are reduced while inputting basic data (friction coefficient ðtan fÞ; cohesion, tensile strength, etc.). Series of analyses are then performed until a safety factor of 1 is reached for a particular failure mechanism. Comparing the material strength inducing failure to the expected material strength could then assess the safety margin. The Australian National Committee on Large Dams [7] presented a dam safety evaluation format based on a limit state approach. Various magnification and reduction factors are applied to basic load conditions and material

strength parameters to reflect related uncertainties. By adjusting the input material parameters, and applying the specified load multiplication factors, CADAM could be used to perform limit analysis of gravity dams as described by ANCOLD [7].

8. Probabilistic and risk analyses The objectives of CADAM probabilistic analysis module is to compute the probability of failure of a gravity dam as a function of the uncertainties in loading and strength parameters that are considered as random variables (Fig. 9). A probabilistic analysis requires more information than a deterministic analysis. For example, probability density functions (PDF) (uniform, normal, log-normal or defined) are to be selected for the friction coefficient and cohesion; the mean values, and the standard deviation must then be specified. CADAM probabilistic analysis module could be used: † For educational purpose to develop a basic understanding of the concepts and procedure required to perform a risk analysis, where risk is evaluated as the product of the probability of failure ðpf Þ and the related consequences. † To actually perform probabilistic (risk) analysis for a particular dam. It is then possible to construct a fragility curve, that defines the probability of failure as a function of an applied load level and compute reliability indices (as a function of ð1 2 pf Þ). † To perform R&D in risk based dam safety assessment such as calibration of nominal strength (resistance R),

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Fig. 9. Probabilistic analysis input data.

reduction factor, f; and load ðLÞ factor, g; to develop limit state based safety evaluation of the form: fR $ gL: † To study different safety approaches (e.g. strength requirements to ensure uniform risk during the service life of a dam). Due to concrete cracking, and related modifications in uplift pressures, the stress and stability analysis of a dam is in general a nonlinear process. Monte-Carlo simulation is used as the computational procedure to perform the probabilistic ‘nonlinear’ analysis in CADAM. MonteCarlo simulation technique ‘involve sampling at random to simulate artificially a large number of experiments and to observe the results’ [20]: (1) a large number (up to 250,000) of loading and strength parameters are ‘sampled’ at random within bounds of specified PDF to perform a large number of possible strength-loading scenarios. For the reason that Monte-Carlo simulations require thousands of nonlinear analysis, a program like CADAM is almost the only and the most effective tool to execute such computations. For example, performing 250,000 nonlinear analyses with three independent variables take about 65 s on a 1.7 GHz Pentium 4; (2) stress and stability analyses are performed; (3) statistics are performed on the results (e.g. sliding safety factors, SSF) to determine the probability of failure, pf : pf ¼

nf nðSSF , 1Þ ¼ N N

ð3Þ

where N ¼ total number of simulations and nf ¼ ¼ number of failures. The output results can also be analysed statistically to define the mean, the variance, the PDF and the cumulative density function (CDF).

9. s and comments During the development process of CADAM, developers consulted dam engineers to ensure that appropriate technical were used and that every feature of the software is in respect with the state-of-the-practice. On the other hand, practicing engineers were invited to comment on the overall organisation and characteristics of the software as well as on the ’s manual. Practicing engineers suggested us to improve the software in a hierarchical way, similar to a wizard window that is asking the to progressively define its problem. This suggestion was implemented in the software by ordering efficiently the menu items and shortcut buttons on the tool bars, thus leading to a more efficient definition of the input data. Moreover, ’s leaded to a better organization in the software and a more detailed illustration in CADAM dialog windows of the uplift pressure distributions of the most important North American dam safety guidelines. Even so, all cracking related options were restructured and regrouped to ensure a precise definition of the nonlinearities to be considered in the analysis. CADAM was enabled to provide detailed output results of all intermediate computational steps. The output reports were

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also designed to allow s to trace clearly all input features that were activated in a particular analysis. In addition, direct connection to Microsoft Excel spreadsheet was developed to allow s to further perform complementary post-processing of CADAM results. Another feature of the program is the instantaneous verification in consistency of the input data. At any time in the definition of the problem, the ’s input data consistency is verified by CADAM. Upon inconsistency, CADAM generates a warning message explaining how the ’s inputs are in conflict. For example, if the selects a certain guideline to activate drainage and select another guideline to model the effects of cracking on drainage efficiency, CADAM warns the without forcing him to change his choice. Finally, CADAM website (http://www. struc.polymtl.ca/cadam) offers an electronic form allowing s to comment on the software.

10. Application examples 10.1. System analysed The 30.48 m (100 ft) high gravity dam to illustrate some of CADAM potentials is shown in Fig. 4a. This dam was used in USACE [12] to evaluate and compare stability analysis and uplifting criteria for gravity dams by three US Federal agencies. The analyses are performed considering the material properties shown in Fig. 4b and c. The usual u/s and d/s reservoir elevations are set to 27.432 m (90 ft) and 1.524 m (5 ft), respectively. Lift ts are spaced at every 3.048 m (10 ft) in elevation from the base. The drainage system is initially considered according to USACE [3] guideline, the drain position, efficiency and the elevation of the drainage gallery are given in Fig. 5.

PGA). It is then possible to evaluate for which loading intensity, safety factors will fall below allowable values such that proper action could be planned. CADAM allows proceeding with an incremental load analysis, which will be illustrated here to evaluate the evolution of cracking and safety factors in function of the flood upstream reservoir elevation for the 30 m dam shown in Fig. 4a. The effect of reservoir overtopping on the applied crest vertical water pressure has been taken into . Floating debris are not considered in this incremental flood analysis. Fig. 12 presents the results from an analysis where the flood reservoir elevation was incrementally raised from elevation 27.432 m (90 ft) to 33.528 m (110 ft) by increment of 0.001 m (0.003 ft). Fig. 12 shows the evolution of both peak and residual SSFs as well as cracking at the base as a function of the flood reservoir elevation. Cracking initiates at a reservoir elevation of 30.166 m (98.97 ft). This crack reaches the drain line for a reservoir elevation of 30.546 m (100.22 ft). As soon as the crack reaches the drain, the drain effectiveness is completely lost [3]. This results in an increase in the uplift force for finally propagating the crack to 43.7% of the base length. At the same time, SSFs are dramatically reduced. Shear failure (sliding) of the dam at the base occurs for a reservoir elevation of 31.4 m (103 ft), when considering the peak shear strength. The structural analysis results along a particular t could be displayed providing a visual of interrelations between the position of the resultant, cracking, uplift pressures, and normal stresses distributions. In an incremental flood analysis, Fig. 13 shows the results corresponding to a full reservoir elevation of 30.48 m. The resultant force in this case is outside the third median. Individual force components (normal, moments, and shear) are indicated to allow independent validation of results.

10.2. Usual load combination 10.4. Probabilistic safety analysis Figs. 10 and 11 show the global stability drawings, generated by CADAM, for the usual load conditions. Fig. 10 is related to the stress analysis indicating normal and principal stresses on the u/s and d/s faces as well as uplift pressures. Fig. 11 is related to the stability analysis presenting resultant forces and safety factors. 10.3. Incremental flood safety analysis In dam safety evaluation there is most often high uncertainties with the loading intensity associated with extreme flood and earthquake events with very long return periods such as the reservoir elevation corresponding to the 10,000 years event (or Probable Maximum Flood (PMF)). It is very useful to know the evolution of typical SSFs (for peak and residual strengths) as well as other performance indicators (e.g. crack length) as a function of a progressive increase in the applied loading (i.e. reservoir elevation or

In dam safety guidelines, it is customary to define safety factors in of allowable stresses (forces). The calculations are performed using a deterministic model of the dam assuming specific numerical values for the loads and the strength parameters. A probabilistic safety analysis considers explicitly the uncertainties in the loading and strength parameters that are treated as random variables. The uncertainties in input parameters are then transformed in probability of failure of a dam. As an illustrative example, Fig. 14 shows the results of probabilistic safety analyses where the peak shear strength parameters (cohesion and friction angle) are considered as random variables following a normal PDF with means and variances indicated in Fig. 9. To construct the fragility curve, shown in Fig. 14, twenty probabilistic analyses were performed by increasing the flood reservoir elevation. A probability of failure is obtained for each reservoir

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Fig. 10. Stability drawing: stress analysis.

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Fig. 11. Stability drawing: stability analysis.

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Fig. 12. Incremental load analysis (Flood analysis).

elevation, thus defining a point of the curve. By definition, the dam failure occurs when the peak SSF is less than one. Annual exceedance frequency of particular reservoir elevations are also plotted in Fig. 14 for illustrative purposes, allowing to compute the frequency of dam failure per year due to hydrological events [21].

10.5. Seismic safety analysis Fig. 15 shows the results from a pseudo-static analysis (Fig. 15a), and a pseudo-dynamic analysis (Fig. 15b). Reservoir u/s and d/s elevations are set to the same normal operating levels as the usual combination. Uplift pressures

Fig. 13. Normal stress and uplift distribution at the base (reservoir at 30.48 m).

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forces near the crest is around 3.8 times that of the pseudostatic analysis. This could lead to cracking of ts near the crest as it has been observed on several occasions in the field [22]. CADAM provides three options to consider the transient evolution of uplift pressures in cracks (Fig. 7) during earthquakes. Fig. 17 shows the effect of seismic water pressure assumptions in cracks. Based on the chosen uplift assumption, the following results are obtained:

Fig. 14. Probabilistic safety analysis.

are assumed to remain unchanged during seismic analyses. The peak ground horizontal acceleration is set to 0.15 g. The peak ground vertical acceleration is set 0.1 g. The peak horizontal spectral acceleration is set to 0.2 g at the fundamental period of the dam ðT1 ¼ 0:1 sÞ: Sustained accelerations are taken as half of the peak acceleration values. As shown in Fig. 15, both seismic analyses indicate almost the same crack length along the base, while the pseudo-dynamic analysis indicates more cracking in the upper ts of the dam. This is due to the dynamic amplification of inertia forces along the height of the dam in the pseudo-dynamic analysis. Fig. 16 presents the amplification of the seismic loads (shear forces and moments) of the pseudo-dynamic analysis over the pseudo-static analysis. For this dam, the dynamic amplification of seismic

(a) insignificant cracking (Fig. 17a) occurs when there is no uplift pressures in the opened crack, (b) slightly longer crack (Fig. 17b) occurs when uplift pressures remain unchanged, and (c) longer crack with a small SSF (Fig. 17c) is obtained when full uplift pressures are applied to the crack section. This example shows the versatility of CADAM to answer typical engineering question in seismic safety analysis: What if?

11. Perspectives for future developments There are almost endless possibilities for further developments of a computer program like CADAM for structural safety assessment of gravity dams. Currently, work is progress to add the following features: † From a pseudo-static or a pseudo-dynamic seismic analysis, the lift t most susceptible to cracking can be easily obtained using CADAM. Calculation of seismic sliding displacements and rocking response of cracked dam components using transient dynamic analysis of rigid body is envisaged.

Fig. 15. Seismic analyses (cracking and uplift pressures): (a) pseudo-static analysis; (b) pseudo-dynamic analysis.

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Fig. 16. Amplification of pseudo-dynamic over pseudo-static analysis.

Fig. 17. Incidence of uplift pressures in cracks during pseudo-static analysis.

† Computation of displacements using beam theory for the dams and Boussinesq coefficients for the semiinfinite elastic foundation. † Thermal analysis will be performed along lift ts using finite differences to evaluate the thermal field required for thermal displacement and stress computations. The displacement response of a 2D model could be calibrated against that of a preliminary 3D finite element model to determine the fraction of the hydrostatic load that is resisted in a pure cantilever mode. Unit thermal loads could also be used for calibration purposes. The computation of displacement using beam theory will allow simple and effective coupled thermo-mechanical analyses to link the deterministic model of a dam with its statistical model derived from field measurements of pendulum displacements. This can be viewed as an intermediate step before undertaking detailed coupled thermomechanical finite element analyses, which requires large resources.

† Definition of more complex 2D geometry, spillway and water intake sections, eventually 3D sections. † Arbitrary defined uplift pressure distributions. † Link with finite element programs: automatic transfer of models data to finite element programs for detailed static, thermal, seepage and seismic analyses.

12. Conclusions CADAM provides a very versatile computing environment to learn or investigate modelling assumptions and computational processes related to the static and seismic structural stability of gravity dams based on the gravity method. It has been shown in this paper that several assumptions related to load conditions, cracking criteria, uplift pressures intensities and analysis procedure could be used for static, seismic, and post-seismic safety assessments. In general, the computations are complex to perform due to the coupling between uplift pressures and crack

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length. In an actual situation, parametric analyses are most often performed to cover uncertainties in strength and loading parameters to take appropriate decision concerning a particular structure. The authors have successfully used CADAM as a computational laboratory in seminars, to engineers from practice, involved in dam safety evaluation. CADAM is also used for industrial applications and R&D in dam engineering and has been extensively validated, using extensive lengthy manual calculations or linear and nonlinear Finite Element Methods, during the past years. The organisation of the program and the particular features that have been presented herein are useful for those interested in the development and application of computer aided stability analysis of gravity dams.

Acknowledgements The development of the computer program CADAM was funded by NSERC (Natural Sciences and Engineering Research Council of Canada), Hydro-Que´bec and Alcan. The of these organisations is gratefully acknowledged.

References [1] Ghrib F, Le´ger P, Tinawi R, Lupien R, Veilleux M. Seismic safety evaluation of gravity dams. Int J Hydropower Dams 1997;4(2): 126–38. [2] Le´ger P, Tinawi R, Rheault S, Leclerc M. A progressive methodology for structural safety evaluation of gravity dams subjected to floods. Proceedings of Canadian Dam Safety Conference, Niagara Falls, Ontario; 1996. p. 2–16 [3] FERC (Federal Energy Regulatory Commission). Engineering guidelines for evaluation of hydropower projects—Draft Chapter III Gravity Dams. Federal Energy Regulatory Commission, Office of Energy Projects, Division of Dam Safety and Inspections, Washington DC, USA; 2000 [4] FERC (Federal Energy Regulatory Commission). Engineering guidelines for evaluation of hydropower projects—Chapter III Gravity Dams. Federal Energy Regulatory Commission, Office of Hydropower Licensing, Report No. FERC 0119-2, Washington DC, USA; 1991 [5] Canadian Dam Association (CDA). Dam safety guidelines. Edmonton, Alberta; 1999

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