Guidance on the latest methods of nonlinear seismic analyses of dams is provided by Dr Martin Wieland, Chairman, ICOLD Committee on Seismic Aspects of Dam Design

In the past, the pseudo-static design concept, proposed by Westergaard in the 1930s in connection with the Hoover dam project, was widely used. Several finite element programs were developed in the 1970s for the dynamic earthquake analysis of dams, based on the assumption that mass concrete and foundation rock of concrete dams behave, essentially, in the linear-elastic range. Although the first dynamic analyses of embankment dams were also carried out assuming linear-elastic behaviour, it was recognised that soil behaves inelastically under seismic excitations. Therefore, the equivalent linear method of analysis was proposed in the early 1970s. In this method, the shear strain dependent dynamic shear moduli and damping properties of soils are accounted for in a procedure based on an iterative linear-elastic analysis.

In 1986, a report on the dynamic procedures for the earthquake analysis of dams was prepared on behalf of the Committee on Analysis and Design of Dams of International Committee on Large Dams (icold) by O. C. Zienkiewicz, R. W. Clough, and H. B. Seed[1], the leading experts in the development of the finite element method and the dynamic analysis of concrete and embankment dams. The methods of seismic analysis included in this report, which originated in the 1970s, are still used today.

In 1989, ICOLD Bulletin 72, entitled “Selecting Seismic Parameters for Large Dams”[2] was published. In this document, a two-level seismic design concept for dams was proposed as the state of the practice in dam engineering, ie., the operating basis earthquake (OBE) and the maximum credible earthquake (MCE). Today, “safety evaluation earthquake” (SEE) is the term preferred to MCE. Bulletin 72 states that a dam has to be able to withstand the worst possible ground motion to be expected at the site without catastrophic release of the reservoir. Significant structural damage is, however, accepted as long as the dam can safely retain the reservoir. In today’s terminology, the OBE can be considered as a serviceability limit state whereas the SEE is akin to the ultimate limit state for the earthquake loading.

For the OBE, the linear-elastic analysis methods that were developed in the 1970s are still suitable. The great advantage of linear-elastic dynamic analyses is that two engineers performing the same analysis with two different computer programs will get identical answers if the geometry, material properties and boundary conditions are the same.

In the case of the SEE ground motions, some structural damage is accepted, which is usually characterised by inelastic phenomena, eg., joint opening, formation of cracks, plastification, hysteretic behaviour of materials, sliding and rocking motions, build up of pore pressure, etc. It is obvious that inelastic or nonlinear seismic analysis have to be carried out in view of the performance criteria for dams under the SEE ground motions. The inelastic dynamic analyses are much more demanding and complex than any linear-elastic analysis, and a lot of experience, engineering judgment and a thorough understanding of the main inelastic features governing the dynamic response are needed. In addition, these analyses require an in depth knowledge of the numerical limitations of the nonlinear algorithms involved.

Many engineers still try to avoid nonlinear seismic analyses, for example, by assuming high damping values or considering too low SEE ground motions to reduce the dynamic response substantially, so that a linear-elastic analysis could be justified. It is, however, expected that there will be an increasing demand for nonlinear dynamic analyses of dams in the future.

In the following sections, the different phenomena that may require inelastic earthquake analyses are discussed. The emphasis in this paper is on the dynamic response of a dam to ground shaking. Other seismic safety aspects, such as fault movements in dam foundations or mass movements into the reservoir causing impulsive waves, are outside the scope of this paper.

### Inelastic Deformation of Concrete Dams: Joints and Cracks

Observations of earthquake damage in concrete gravity dams show that ground shaking results in the formation of cracks in the highly stressed, central crest region along some weak planes, such as horizontal lift surfaces and grouted vertical contraction joints.

As no arch dam has so far suffered serious damage during earthquake ground shaking, little experience exists about the possible damage caused in an arch dam by, for example, the SEE. However, linear-elastic dynamic analyses show that tensile stresses exceeding the dynamic tensile strength of mass concrete could occur in an arch dam during a strong earthquake. Therefore, cracks can also be expected to develop in an arch dam during a strong earthquake along the contraction and lift joints, which exhibit a smaller tensile strength than the surrounding mass concrete.

The typical blockwise construction of a concrete dam with horizontal lift joints at 2m-3m spacing facilitates the formation of horizontal cracks during a strong earthquake. Most of the deformations of a dam would be confined to these cracks, due to which further cracking is prevented in the dam body. Thus, it can be expected that only a few cracks will be formed in a concrete dam during severe ground shaking.

In order to predict the behaviour of a concrete dam during the SEE, and to check the stability of a cracked dam, nonlinear seismic analyses would be required and the following approaches are used:

(i) smeared crack approach, in which concrete cracking is implemented in the constitutive model of mass concrete (continuum approach);

(ii) the discrete modelling of contraction, base and lift joints in the finite element model of the dam, assuming concrete and rock to be linear-elastic materials; and,

(iii) the discrete crack approach, in which the dynamic behaviour of rigid concrete blocks separated by cracks and/or joints is investigated (rigid body approach).

In approach (i), the smeared crack method requires a concrete model with quite a large number of material parameters, which are difficult to obtain. At present, approaches (ii) and (iii) appear to be better suited for practical applications. For the latter method, knowledge of only the friction coefficient at the base of the detached rigid block is needed. The probable sizes of characteristic concrete blocks, which could form during an earthquake, have to be determined based on practical experience with similar dams, engineering judgment, experimental investigations, and the results of linear-elastic dynamic analyses.

A concrete block separated from the rest of the dam by the formation of cracks behaves essentially as a rigid body, which can experience substantial inelastic (nonlinear) displacements in the form of rocking and sliding without actually leading to a dam failure, owing to the low slenderness ratio of these dams. In fact, the displacements of such detached blocks could even be estimated by means of a simple Newmark-type sliding block analysis. If the cracks were inclined, the sliding movements may, however, be significantly larger than in the case of horizontal cracks [3][4]. Post-earthquake stability analyses should consider the uplift pressure acting on the sliding surface. The dynamic overturning stability is less of a problem as the rocking motion of a detached concrete block is generally a reversible process.

The formation of horizontal cracks during an earthquake in the highly stressed upper portion of a concrete dam is beneficial for the dynamic stability of detached concrete blocks. Joint openings and sliding movements at the cracks are mechanisms that ensure that a massive concrete dam can safely resist ground motions exceeding the original design earthquake. This behaviour is comparable to that of a ductile structure in which large inelastic deformations can take place without causing a disastrous collapse.

### Seismic Aspects of RCC Dams

Most roller compacted concrete (RCC) dams are basically gravity dams and, therefore, their earthquake behaviour is also similar to that of conventional gravity dams. As in conventional concrete dams, cracks can be expected to form along horizontal lift joints. Vertical contraction joints could also open, but this is not a critical safety issue since gravity dams are designed to carry the loads by cantilever action and not by arch action.

The post-cracking dynamic behaviour of blocks separated by cracks and joints in an RCC dam can also be analysed using relatively simple rigid body models, as in the case of a conventional concrete dam. Because of the large thickness of an RCC dam, a sliding movement of several metres would have to occur before a detached concrete block would fall.

### Inelastic deformations of embankment dams

Basically, the seismic safety and performance of embankment dams is assessed by investigating the following aspects:

• permanent (inelastic) deformations experienced during and after an earthquake (e.g., loss of freeboard);

• stability of slopes during and after the earthquake, and dynamic slope movements;

• build-up of excess pore water pressures in embankment and foundation materials (soil liquefaction);

• damage to filter, drainage and transition layers (i.e., whether they will function properly after the earthquake);

• damage to waterproofing elements in dam and foundation (core, upstream concrete face or asphalt membrane, geotextiles, grout curtain, diaphragm walls in foundation, etc.,);

• vulnerability of dam to internal erosion after formation of cracks or formation of loose material zones due to high shear (shear bands);

• vulnerability of hydromechanical equipment to ground displacements and vibrations; and,

• damage to intake and outlet works (safe release of the water from the reservoir may be jeopardised).

Most of the above aspects are directly related to seismic deformations of the dam during strong ground shaking. Therefore, they are governed by the deformational characteristics of the fill materials.

Liquefaction is a major problem for tailings dams and small earth dams constructed of, or founded on, relatively loose cohesionless materials, and used for irrigation and water supply schemes, as in many cases they are not properly designed against earthquakes. In fact, this could be assessed based on relatively simple insitu tests. For example, there exist empirical relationships between SPT blow counts and liquefaction susceptibility to different earthquake ground motions characterised by the number of stress cycles and the peak ground acceleration.

For large storage dams, the earthquake-induced permanent deformations must be calculated. Damage categories are, for example, expressed in terms of the ratio of crest settlement to dam height. Calculations of the permanent settlements of large rockfill or concrete face rockfill dams (CFRDs) based on dynamic analyses are still very approximate as most of the dynamic soil tests are usually carried out on samples with a maximum aggregate size of less than 5cm. To estimate representative dynamic material properties, dynamic direct shear or triaxial tests on large samples are needed. These tests are, however, too costly for most rockfill dams. As information on the dynamic behaviour of rockfill published in the literature is also scarce, the settlement prediction involves sensitivity analyses and engineering judgment.

Substantial seismic settlements could occur in rockfill dams and other dams with large rock aggregates, especially if the fill materials have not been adequately compacted at the time of construction. In spite of large settlements, a rockfill dam could still safely withstand a strong earthquake.

Cracks may cause failure of an embankment dam under the following circumstances[4]:

• filter, drain and transition zones are missing;

• filter, drain and transition zones do not extend above the reservoir level; or,

• modern filter criteria were not used to design the dam.

Transverse cracking as a result of deformations could also be an important issue.

Seismic aspects of CFRDs

rockfill dams

The seismic safety of a CFRD is often assumed to be superior to that of a conventional rockfill dam with an impervious core. However, the crucial element in CFRDs is the behaviour and performance of the concrete slab during and after an earthquake.

The settlements of a rockfill dam caused by the MCE or SEE are rather difficult to predict and depend on the type of rockfill and the compaction of the rockfill during dam construction. Depending on the valley shape, dam deformations will also be non-uniform over the upstream face, causing differential movements of the concrete face, local buckling in the compression zones, etc.

In many cases, embankment dams are analysed by the equivalent linear method using a two-dimensional model of the highest dam section. In such a seismic analysis, only reversible elastic deformations and stresses are calculated, which are relatively small and do not produce high dynamic stresses in the concrete face slab. These simple models have to be complemented by models, which also include the cross-canyon component of the earthquake ground motion as well as the inelastic deformations of the dam body. For such a dynamic analysis, a three-dimensional dam model has to be used and the interface between the concrete face and the soil transition zones must be modelled properly.

The concrete slab acts as a rigid diaphragm and has a deformational behaviour that is very different from that of the rockfill and transition zone materials. This may result in high in-plane stresses in the concrete slab, especially as the cross-canyon response of the dam may be restrained by the relatively rigid concrete slab. The seismic forces that can be transferred from the rockfill to the concrete slab are limited by the friction forces between the transition zone and the concrete slab. Since the whole water load is supported by the concrete slab, these friction forces are quite high and, therefore, the in-plane stresses in the concrete slab may become sufficiently large to cause local buckling, shearing off of the slab along the joints, or to damage the plinth.

As experience with the seismic behaviour of CFRDs is still very limited, more efforts have to be undertaken to study the seismic behaviour of these dams[4].

### Special features of seismic analysis of concrete and embankment dams

For the dynamic analysis and seismic safety assessment of concrete and embankment dams, various features have to be considered, such as:

• occurrence of earthquake (return period of different design earthquake ground motions);

• characteristics of strong ground shaking (peak ground acceleration, frequency content, duration of strong ground shaking);

• spatial variation of ground motion at dam site;

• superposition of static and dynamic load cases;

• dynamic soil-structure interaction effects;

• dam-reservoir interaction effects (shape of reservoir, compressibility of water, wave absorption in reservoir bottom, wave height effects in reservoir, etc.,);

• dynamic material properties of concrete, soil, rockfill and foundation rock;

• dynamic (tensile) strength properties of concrete, soil, rockfill and foundation rock;

• joints in concrete and rock;

• effect of uplift or pore pressure in joints;

• pore pressure build-up in soils;

• structural damping;

• corner stress concentrations;

• type of numerical analysis (time domain analysis, response spectrum analysis, linear analysis, nonlinear analysis, etc.,);

• compilation of results of time history analyses (use of maximum response quantities for design and/or safety assessment); and,

• performance criteria (allowable stresses, stability safety factors, etc.,) for assessing the results of dynamic (and static) analyses.

Many of these features call for nonlinear analyses. Due to lack of information, these features may involve considerable uncertainties, which have to be accounted for by performing sensitivity analyses. To avoid a large number of analyses, engineering judgment and conservative assumptions are needed and often used in practice.

The seismic behaviour of dams under strong ground shaking is hard to predict, as each large dam is a prototype located at a unique site. Generalisation of results is often not possible or questionable.

### Nonlinear analysis aspects

There are a number of general-purpose computer programs (ABAQUS, ADINA, ANSYS, FLAC, etc.) that can be used for nonlinear seismic analyses of concrete and embankment dams. However, there is a lack of information on the material properties to be used in the available nonlinear constitutive models. Therefore, it is very important that the engineer first clearly formulates the problem to be analysed taking into account the available information. A simplified nonlinear analysis may often be superior to a sophisticated analysis for which some basic information is not available.

The objectives of the nonlinear analysis are: (i) to predict the earthquake behaviour of a dam as realistically as possible; (ii) to assess the deformations of the dam; (iii) to assess the damage the dam will experience; and most importantly, (iv) to assess the safety of the dam.

The following stepwise approach towards nonlinear seismic analyses is recommended (direct time history analysis is preferred in all cases):

Concrete dams:

• Linear-elastic analysis for OBE;

• Newmark-type sliding block analysis of whole gravity dam structure or detached blocks in a concrete dam;

• Rigid body analysis of cracked concrete (gravity, arch-gravity or arch) dam assuming that all deformations occur along cracks or joints, whereby cracks form along lift joints or the dam-foundation contact (combined rocking and sliding motion of two-dimensional model); and,

• Analysis of arch-gravity and arch dams with contraction joint opening, or opening of dam-foundation contact or peripheral joint (if provided).

A concrete damage model with tension failure criterion may be suitable for monolithic dams, but it does not account for reduced strength properties of contraction and lift joints and, therefore, may not be better than the simple models listed above when the behaviour of the dam under the SEE ground motions has to be analysed.

Embankment dams:

• Equivalent linear dynamic analysis of dam:

Advantages: (i) substantial amount of information on shear strain dependent material properties exists, and (ii) computer programs, such as FLUSH, QUAD4M, etc., are readily available for analysis of two-dimensional dam sections;

Disadvantages: (i) method is almost 40 years old and does not properly represent the nonlinear behaviour of soil, (ii) it is cumbersome for three-dimensional analysis, and (iii) inelastic deformations are difficult to estimate based on results of dynamic analysis;

• Newmark sliding block analysis (simple method for estimating sliding movements of slopes);

• Analysis using Coulomb friction elements to model interface between transition/filter materials and impervious core, or interface between concrete face and supporting layer in the case of a CFRD (analysis mainly for static effects); and,

• Elastoplastic soil models (preferably using a constitutive model with only a few parameters).

### Conclusions

A thorough understanding of the inelastic and nonlinear seismic phenomena, which are expected during strong ground shaking, is prerequisite for nonlinear seismic analysis of dams.

In a concrete dam, strong ground shaking could lead to opening of contraction joints and formation of horizontal cracks along lift joints, as a result of which high dynamic stresses are prevented in other parts of the dam. Similarly, an embankment dam may undergo significant permanent deformations during a severe earthquake. Some structural damage is accepted in a dam as long as its water retaining function is ensured. For the seismic safety and damage assessment of concrete and embankment dams, nonlinear dynamic analyses are often needed to determine the expected inelastic deformations under the SEE.

Simple nonlinear analyses methods are still widely used for the seismic analysis of dams, such as the Newmark sliding block method and the equivalent linear method for the analysis of the embankment dams. These methods are, however, nearly 40 years old. In view of an increasing demand for nonlinear methods of analysis for the safety evaluation of existing and new dams according to the current seismic design criteria, it is recommended to update ICOLD Bulletin 52 (1986), which is still based on linear-elastic concepts.

The methods for nonlinear dynamic analysis of dams are, however, still under development. Nonlinear seismic analyses need substantial engineering judgment. The proper formulation of the goals of the seismic analysis is probably the most difficult task required to ensure that such an analysis can actually ‘succeed’. Relatively simple models should be preferred to complex models employing nonlinear constitutive laws using parameters that are either not available or very hard to determine.