zeidan dam chapter 2014

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Design and Analysis of Concrete Gravity Dams

Research · August 2015

DOI: 10.13140/RG.2.1.4676.5289

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Bakenaz A. Zeidan

University of California, Berkeley

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Retrieved on: 30 April 2016

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Dr . Bakenaz A. Zeidan – State of A rt in Design and Analysis of Concrete Gravity Dams 2014

1

Article Review

State of Art

Design and Analysis of Concrete

Gravity Dams

By

Dr Bakenaz A Zeidan

2014


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LIST OF CONTENTS

Summary1.

Introduction to Gravity Dams

2.

About Concrete Gravity Dams3.

Cases of Loading on Gravity Dams

4. Theoretical Approach Gravity Dams

5.

Modeling of Gravity Dam

6. Analysis of Gravity Dams

7.

Safety Criteria for Gravity Dams

8.

Recent Trends in Gravity Dams

9. Summing up


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State of Art in Design and Analysis of

Concrete Gravity Dams

1. INTRODUCTION

Dams have been constructed for millennia, influencing the lives of humans and the

ecosystems they inhabit. Remnants of one such man-made structure dating back 5,000 years are still

standing in northeast Africa (UNESCO-WWAP, 2003). Around 2950-2750 B.C., the first dam

known to exist was built by the ancient Egyptians, measuring 11.3 meters [m] (37 feet [ft]) tall, with

a crest length of 106 m (348 ft) and foundation length of 80.7 m (265 ft) (Yang, et al, 1999). The

dam was composed of 100,000 tons of rubble, gravel, and stone, with an outer shell of limestone.

The immense weight was enough to contain water in a reservoir estimated to have been 570,000

cubic meters [m3] (20 million cubic feet [ft3] or 460 acre-feet) in capacity (Yang, et al, 1999).

Many concrete gravity dams have been in service for over 50 years, and over this period important

advances in the methodologies for evaluation of natural phenomena hazards have caused the design-

basis events for these dams to be revised upwards. Older existing dams may fail to meet revised

safety criteria and structural rehabilitation to meet such criteria may be costly and difficult. The

identified causes of failure, based on a study of over 1600 dams [1] are: foundation problems (40%),

inadequate spillway (23%), poor construction (12%), uneven settlement (10%), and high pore

pressure (5%), acts of war (3%), embankment slips (2%), defective materials (2%), incorrect

operation (2%), and earthquakes (1%). Earthquake is a natural disaster that has claimed so many

lives and destroyed lots of property. Earthquake hazards had caused the collapse and damage to

continual functioning of essential services such as communication and transportation facilities, buildings, dams, electric installations, ports, pipelines, water and waste water systems, electric and

nuclear power plants with severe economic losses. However, the structural response of a material to

different loads determines how it will be economically utilized in the design process. This

necessitates the seismic analysis of concrete gravity dams. Finite element has been widely used in

seismic analysis of concrete gravity dams utilizing the most natural method based on the

Lagrangian – Eulerian formulation.

1.1. Implementation of Concrete Gravity Dams

1.2. Design Procedure of Concrete Gravity Dams

1.3. Safety Criteria of Concrete Gravity Dams

During the recent years, the seismic behaviour of concrete gravity dams was in the centre of

consideration of dam engineers. Numerous researches have been conducted in order to determine

how the dams behave against the seismic loads. Many achievements were obtained in the process of

analysis and design of concrete dams including dam-reservoir-foundation interaction during an

earthquake. The earthquake response of concrete gravity dam-reservoir-foundation system has been

addressed to study the effect of foundation flexibility and reservoir water body on the seismic

response of concrete gravity dams. Safety evaluation of dynamic response of dams is important for


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most of researchers. When such system is

subjected to an earthquake,

hydrodynamic pressures are developed

on upstream face of the dam due to the

vibration of the dam and reservoir water.

Consequently, the prediction of the

dynamic response of dam to earthquake

loadings is a complicated problem and

depends on several factors, such as

interaction of the dam with rock

foundation and reservoir, the computer modelling and material properties used in the analysis.

Gravity dams are very important structures. The collapse of a gravity dam due to earthquake

ground motion may cause an extensive damage to property and life losses. Therefore, the proper

design of gravity dams is an important issue in dam engineering. An integral part of this procedure

is to accurately estimate the dam earthquake response. The prediction of the actual response of a

gravity dam subjected to earthquake is a very complicated problem. It depends on several factorssuch as dam-foundation interaction, dam-water interaction, material model used and the analytical

model employed. In fluid-structure interaction one of the main problems is the identification of the

hydrodynamic pressure applied on the dam body during earthquake excitation. The analysis of dam-

reservoir system is complicated more than that of the dam itself due to the difference between the

characteristics of fluid and dam's concrete on one side and the interaction between reservoir and

dam on the other side.

The earthquake response of concrete gravity dam-reservoir-foundation system has been

addressed to study the effect of foundation flexibility and reservoir water body on the seismic

response of concrete gravity dams. Safety evaluation of dynamic response of dams is important for

most of researchers. When such system is subjected to an earthquake, hydrodynamic pressures are

developed on upstream face of the dam due to the vibration of the dam and reservoir water.

Consequently, the prediction of the dynamic response of dam to earthquake loadings is a

complicated problem and depends on several factors, such as interaction of the dam with rock

foundation and reservoir, the computer modeling and material properties used in the analysis.

For the structure on the rigid foundation, the input seismic acceleration gives rise to an overturning

moment and transverse base shear. As the rock is very stiff, these two stress resultants will not lead

to any (additional) deformation or rocking motion at the base. For the structure founded on flexible

soil, the motion of the base of the structure will be different from the free-field motion because of

the coupling of the structure-soil system. This process, in which the response of the soil influencesthe motion of the structure and response of the structure influences the motion of the soil, is referred

to as soil-structure interaction (SSI) presented by Wolf (1985) [14]. The objective of this paper is to

assess the impact of foundation flexibility and dam-reservoir-foundation interaction on seismic

response of high concrete gravity dams.


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2. ABOUT CONCRETE GRAVITY DAMS

There are two main types of dams, embankment dams and concrete dams. Embankment

dams can either be rock fill or earth fill dams. The method of construction is similar in both cases,

with just the main type of material differentiating. Concrete dams are superior in constructing

massive overflow discharge sections, and are therefore often used in areas where floods are

common. A lot less material is used compared to an embankment dam but concrete is usually morecostly. It is also easier to connect a hydropower station to a concrete dam. Three different height

spans exist when concrete dams are considered and they are defined as: low dams (up to 30 meters),

medium height dams (30-90 meters) and high dams (90 meters and above). This is a measurement

of the difference in elevation between the lowest constructed part of the dam foundation and the

walkway at the dam crest (1).


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Figure 1: Different types of concrete dams (2).

A gravity dam is a solid structure, made of concrete or masonry, constructed across a river to

create a reservoir on its upstream. The section of the gravity dam is approximately triangular in

shape, with its apex at its top and maximum width at bottom. The section is so proportioned that it

resists the various forces acting on it by its own weight. Most of the gravity dams are solid, so that

no bending stress is introduced at any point and hence, they are sometimes known as solid gravity

dams to distinguish them from hollow gravity dams in those hollow spaces are kept to reduce the

weight. Early gravity dams were built of masonry, but nowadays with improved methods of

construction, quality control and curing, concrete is most commonly used for the construction of

modern gravity dams. A gravity dam (Figure.1) is generally straight in plan and, therefore, it is also

called straight gravity dam. The upstream face is vertical or slightly inclined. The slope of the

downstream face usually varies between "0.7: 1‖ to ―0.8: 1‖. There are different types of concretedams based on the principal for the transfer of the hydrostatic pressure.

dams, Figure 2

Figure 3

Figure 4

Figure 2: Concrete gravity dam north of Irkutsk, Russian

(www.hydroelecritc.energy.blogspot.com).

http://www.hydroelecritc.energy.blogspot.com/http://www.hydroelecritc.energy.blogspot.com/http://www.hydroelecritc.energy.blogspot.com/http://www.hydroelecritc.energy.blogspot.com/http://www.hydroelecritc.energy.blogspot.com/


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The theory behind gravity dams is that their own weight should be sufficient to withstand

the hydrostatic pressures affecting them. This means that gravity dams are usually massive and

therefore require a lot of construction material. With the amount of concrete required, this dam type

may be somewhat expensive but on the other hand, it is very versatile. Another advantage is that it

can possess substantial overflow discharge capacity (1).

Buttress dams, shown in figure 8, are similar to gravity dams with the distinction that they also use

the gravity of the reservoir water instead of only the gravity of the dam itself. Because of this, the

dam body does not need to be as massive and use buttresses instead of a solid downstream part of

the dam. Being less solid on the downstream side, buttress dams have the advantage of being a lot

less affected by the water uplift force (1).

Arch dams, shown in figure 9, are curved around a vertical cord to resist the hydrostatic

pressure by arching thus transferring the pressure into the canyon walls. For this transfer to be

possible and cost effective the width to height ratio should not exceed 5:1, although in some cases

arch dams has been built with a ratio as high as 10:1. Another criteria which is important for arch

dams is the shape of the canyon, if it is symmetrical an arch dam is often very suitable. If the canyon

is a little less symmetrical, an arch dam with influences of a gravity dam may be constructed. If thecanyon is extremely asymmetric, another dam type may be preferred (1).

2.1. Basic Definitions

Axis of the dam

The axis of the gravity dam is the line of the

upstream edge of the top (or crown) of the

dam. If the upstream face of the dam is

vertical, the axis of the dam coincides with

the plan of the upstream edge. In plan, the

axis of the dam indicates the horizontal traceof the upstream edge of the top of the dam.

The axis of the dam in plan is also called the

base line of the dam. The axis of the dam in

plan is usually straight. However, in some

special cases, it may be slightly curved

upstream, or it may consist of a combination

of slightly curved right portions at ends and a central abutment straight portion to take the best

advantages of the topography of the site.

Length of the dam

The length of the dam is the distance from one abutment to the other, measured along the axis of the

dam at the level of the top of the dam. It is the

usual practice to mark the distance from the left

abutment to the right abutment. The left

abutment is one which is to the left of the

person moving along with the current of water.

Structural height of the dam:

The structural height of the dam is the

difference in elevations of the top of the dam

and the lowest point in the excavated

foundation. It, however, does not include the

depth of special geological features offoundations such as narrow fault zones below

the foundation. In general, the height of the dam

Figure 3: One of the buttresses in the Manic-Ceng buttress

dam in Québec, Canada (www.dappolonia.com).

Figure 4: Arch dam in Zernez, Switzerland, view from side,

(www.commandatastorage.googleapsis.com).

http://www.dappolonia.com/http://www.dappolonia.com/http://www.dappolonia.com/http://www.commandatastorage.googleapsis.com/http://www.commandatastorage.googleapsis.com/http://www.commandatastorage.googleapsis.com/http://www.commandatastorage.googleapsis.com/http://www.dappolonia.com/


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means its structural height.

Maximum base width of the dam:

The maximum base width of the dam is the maximum horizontal distance between the heel and the

toe of the maximum section of the dam in the middle of the valley.

Toe and Heel:

The toe of the dam is the downstream edge of the base, and the heel is the upstream edge of the

base. When a person moves along with water current, his toe comes first and heel comes later.

Hydraulic height of the dam

The hydraulic height of the dam is equal to the difference in elevations of the highest controlled

water surface on the upstream of the dam (i. e. FRL) and the lowest point in the river bed (3).

2.2. Dam Concrete Static Properties

2.2.1. Strength

A gravity dam should be constructed of concrete that will meet the design criteria forstrength, durability, permeability, and other required properties. Because of the sustained loading

generally associated with them, the concrete properties used for the analyses of static loading

conditions should include the effect of creep. Properties of concrete vary with age, the type of

cement, aggregates, and other ingredients as well as their proportions in the mix. Since different

concretes gain strength at different rates, measurements must be made of specimens of sufficient

age to permit evaluation of ultimate strengths. Although the concrete mix is usually designed for

only compressive strength, appropriate tests should be made to determine the tensile and shear

strength values (4).

2.2.2. Elastic Properties

Poisson’s ratio, the sustained modulus of elasticity of the concrete, and the latter’s ratio tothe deformation modulus of the foundation have significant effects on stress distribution in thestructure. Values of the modulus of elasticity, although not directly proportional to concrete

strength, do follow the same trend, with the higher strength concretes having a higher value for

modulus of elasticity. As with the strength properties, the elastic modulus is influenced by mix

proportions, cement, aggregate, admixtures, and age. The deformation that occurs immediately with

application of load depends on the instantaneous elastic modulus. The increase in deformation

which occurs over a period of time with a constant load is the result of creep or plastic flow in the

concrete. The effects of creep are generally accounted for by determining a sustained modulus of

elasticity of the concrete for use in the analyses of static loadings. Instantaneous moduli of elasticity

and Poisson’s ratios should be determined for the different ages of concrete when the cylinders areinitially loaded. The sustained modulus of elasticity should be determined from these cylinders after

specific periods of time under constant sustained load. These periods of loading are often 365 and

730 days. The cylinders to be tested should be of the same size and cured in the same manner as

those used for the compressive strength tests. The values of instantaneous modulus of elasticity,

Poisson’s ratio, and sustained modulus of elasticity used in the analyses should be the average of alltest cylinder values (4).

2.2.3. Thermal PropertiesThe effects of temperature change in gravity dams are not as important in the design as those

in arch dams. However, during construction, the temperature change of the concrete in the dam

should be controlled to avoid undesirable cracking. Thermal properties necessary for the evaluation

of temperature changes are the coefficient of thermal expansion, thermal conductivity, specific

heat, and diffusivity.The coefficient of thermal expansion is the length change per unit length for 1

degree temperature change. Thermal conductivity is the rate of heat conduction through a unit


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thickness over a unit area of the material subjected to a unit temperature difference between faces.

The specific heat is defined as the amount of heat required to raise the temperature of a unit mass of

the material 1 degree. Diffusivity of concrete is an index of the facility with which concrete will

undergo temperature change. The diffusivity is calculated from the values of specific heat, thermal

conductivity, and density. Appropriate laboratory tests should be made of the design mix to

determine all concrete properties (4).

2.3. Dam Concrete Dynamic Properties (USBR)

2.3.1. Strength No data are yet available to indicate what the strength characteristics are under dynamic

loading.

2.3.2. Elastic PropertiesUntil dynamic modulus information is available, the instantaneous modulus of elasticity

determined for concrete specimens at the time of initial loading should be the value used foranalyses of dynamic effects.

2.3.3. Average Properties Necessary values of concrete properties may be estimated from published data for

preliminary studies until laboratory test data are available. Until long-term tests are made to

determine the effects of creep, the sustained modulus of elasticity should be taken as 60 to 70

percent of the laboratory value for the instantaneous modulus of elasticity. Criteria-If no tests or

published data are available, the following average values for concrete properties may be used for

preliminary designs until test data are available for better results (USBR):

Compressive strength-3,000 to 5,000 Ibs/in2 (20.7 to 34.5 MPa)

Tensile strength-5 to 6 percent of the compressive strength

Shear strength: Cohesion-about 10 percent of the compressive strength

Coefficient of internal friction- l.0

Poisson’s ratio- 0.2

Instantaneous modulus of elasticity- 5.0 x 106 lbs/in2 (34.5 GPa)

Sustained modulus of elasticity- 3.0 x 106 lbs/in2 (20.7 GPa)

Coefficient of thermal expansion- 5.0 x 10-6 /“F (9.0 x l0-6 PC)

Unit weight- 150 Ibs/ft 3 (2402.8 kg/m3 )

2.4. Foundation Properties

2.4.1. Deformation ModulusFoundation deformations caused by loads from the dam affect the stress distributions within

the dam. Conversely, response of the dam to external loading and foundation deformability

determines the stresses within the foundation. Proper evaluation of the dam and foundation

interaction requires as accurate a determination of foundation deformation characteristics as

possible. Although the dam is considered to be homogeneous, elastic, and isotropic, its foundation

is generally heterogeneous, inelastic, and anisotropic. These characteristics of the foundation have

significant effects on the deformation moduli of the foundation. The analysis of a gravity dam

should include the effective deformation modulus and its variation over the entire contact area of

the dam with the foundation. The deformation modulus is defined as the ratio of applied stress to

elastic strain plus inelastic strain and should be determined for each foundation material. Theeffective deformation modulus is a composite of deformation moduli for all materials within a

particular segment of the foundation. Good compositional description of the zone tested for


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deformation modulus and adequate geologic logging of the drill cores permit extrapolation of results

to untested zones of similar material.

Criteria - The following foundation data should be obtained for the analysis of a gravity dam (4):

The deformation modulus of each type of material within the loaded area of the foundation.

The effects of joints, shears, and faults obtained by direct (testing) or indirect (reduction

factor) methods.

An effective deformation modulus, as determined when more than one type of material is

present in a foundation.

The effective deformation moduli, as determined at enough locations along the foundation

contact to provide adequate definition of the variation in deformability and to permit

extrapolation to untested areas when necessary.

2.4.2. Shear StrengthResistance to shear within the foundation and between the dam and its foundation depends

upon the cohesion and internal friction inherent in the foundation materials and in the bond between

concrete and rock at the contact with the dam. These properties are determined from laboratory and

in situ tests. The results of laboratory triaxial and direct shear tests, as well as in situ shear tests, are

generally reported in the form of the Coulomb equation:

R = C.A + N. tan φ or

(Shear resistance) = (unit cohesion times area) +

(effective normal force times coefficient of internal friction)

which defines a linear relationship between shear resistance and normal load. The value of shear

resistance obtained as above should be limited to use for the range of normal loads used for the

tests. Although this assumption of linearity is usually realistic for the shear resistance of intact rock

over the range of normal loads tested, a curve of shear resistance versus normal load should be usedfor materials other than intact rock. The shear resistance versus normal load relationship is

determined from a number of tests at different normal loads. The individual tests give the

relationship of shear resistance to displacement for a particular normal load. The results of these

individual tests are used to obtain a shear resistance versus normal load curve. The displacement

used to determine the shear resistance is the maximum displacement that can be allowed on the

possible sliding plane without causing unacceptable stress concentrations within the dam. Since

specimens tested in the laboratory or in situ are small compared to the foundation, the scale effect

should be carefully considered in determining the values of shear resistance to be used.

When a foundation is nonhomogeneous, the possible sliding surface may consist of several

different materials, some intact and some fractured. Intact rock reaches its maximum break bond

resistance with less deformation than is necessary for fractured materials to develop their maximum

frictional resistances. Therefore, the shear resistance developed by each fractured material depends

upon the displacement of the intact rock part of the surface. If the intact rock shears, the shear

resistance of the entire plane is equal to the combined sliding frictional resistance for all materials

along the plane (4).

2.4.3. Pore Pressure and PermeabilityAnalysis of a dam foundation requires a knowledge of the hydrostatic pressure distribution

in the foundation. Permeability is controlled by the characteristics of the rock type, the jointing

systems, the shears and fissures, fault zones, and, at some dam sites, by solution cavities in the rock.

The exit gradient for shear zone materials that surface near the downstream toe of the dam should

also be determined to check against the possibility of piping. Laboratory values for permeability of

sample specimens are applicable only to the portion or portions of the foundation that they


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represent. Permeability of the aforementioned geologic features can best be determined by in situ

testing. The permeability obtained are used in the determination of pore pressures for analyses of

stresses, stability, and piping. Such a determination may be made by several methods including two-

and three dimensional physical models, two- and three-dimensional finite element models, and

electric analogs. If foundation grouting and drainage or other treatment are to be used, their effects

on the pore pressures should be included (4).

2.4.4. TreatmentFoundation treatment is used to correct deficiencies and improve physical properties by

grouting, drainage, excavation of inadequate materials, reinforcement and backfill with concrete.

Some reasons for foundation treatment are: (1) improvement of deformation moduli, (2) prevention

of sliding of foundation blocks, (3) prevention of relative displacement of foundation blocks, (4)

prevention of piping and reduction of pore pressures, and (5) provision of an artificial foundation in

the absence of adequate materials. Regardless of the reason for the foundation treatment, its effects

on the other foundation properties should be considered in the analyses (4).

2.4.5. Compressive and Tensile StrengthCompressive strength of the foundation rock can be an important factor in determining

thickness requirements for a dam at its contact with the foundation. Where the foundation rock is

nonhomogeneous, tests to obtain compressive strength values should be made for each type of rock

in the loaded portion of the foundation. A determination of tensile strength of the rock is seldom

required because unhealed joints, shears, etc., cannot transmit tensile stress within the foundation.

3. GRAVITY DAM LOAD COMBINATION

Factors to be considered as contributing to the loading combinations for a gravity dam are:

(1) reservoir and tail water loads, (2) temperature, (3) internal hydrostatic pressure, (4) dead weight,

(5) ice, (6) silt, and (7) earthquake. Such factors as dead weight and static water loads can be

calculated accurately. Others such as earthquake, temperature, ice, silt, and internal hydrostatic

pressure must be predicted on the basis of assumptions of varying reliability (USBR).

3.1. Static Loads:

There are several types of forces acting on dams, the eight that are commonly used are listed

below. In figure (5) the forces acting on a gravity dam are shown (1):

Dead weight Hydrostatic pressure from reservoir

Hydrostatic pressure from tail water

Internal hydrostatic pressure (uplift pressure)

Sand and silt

Ice

Temperature

Earthquake

Figure (5)


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Dead weight is the gravity affecting the dam itself, since we are dealing with compact, heavy

structures this is a major factor. This is a static load as well as the others, except for earthquake.

Hydrostatic pressure from reservoir is the

pressure created by the upstream water.

Hydrostatic pressure from tail water affects the

dam the same way as the hydrostatic pressure

caused by reservoir water does, with the only

difference that it is the pressure from the water

downstream of the dam.

Internal hydrostatic pressure is what we also

call uplift pressure. This is the pressure from the

water in the foundation and in the dam body that

will push the dam body upward, causing an

enhanced risk of sliding. This is especially

problematic for gravity dams since they cover a

greater area than other types of concrete dams,Figure (6).

Sand and silt is the load applied as an earth

pressure from eroded material at the upstream face

of the dam. It is usually reasonably small

compared to the hydrostatic reservoir pressure.

Ice load affects the upstream face of the dam, as the surface of the water freezes. If the ice gets thick

enough it will cause a significant load. load is concentrated to a small surface where the dam body is

thinnest.

Temperature is a concern both during the construction phase and the entire lifespan of the dam.

The hardening of the concrete causes severe temperature variations that will lead to strains.

Depending on where in the world the dam is located the temperature changes may continue, during

the entire lifespan of the dam, to be an important issue.

Earthquake is a dynamic load. This load is, unlike the other seven, very hard to predict but is very

important to consider in earthquake affected regions.

3.2. Seismic Loads

Figure (7)


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Earthquake or seismic loads are the major dynamic loads (4) being considered in the analysis

and design of dams especially in earthquake prone areas. The seismic coefficient method is used in

determining the resultant location and sliding stability of dams. Seismic analysis of dams is

performed for the most unfavourable direction, despite the fact that earthquake acceleration might

take place in any direction. Figure (7) shows the seismic coefficient α for dynamic loads on agravity dam. There are different ways of computing earthquake loads on dams. The deterministic

approach may be employed where the ground acceleration in terms of g (acceleration due to gravity)

is specified for the region where the dam will be constructed. Hence, the exciting force on the

structure is (3), Figure (8),

P(t) = Max (1)

and

ax = αg(2)

where ax, α, g are the ground

acceleration, seismic coefficient

and acceleration due to

gravity respectively. From

Fig.10, the equilibrium system

is expressed as:

Pex=Max=Wαg/g=Wα (3a)The hydrodynamic pressure exerted along dam-reservoir interface is given by:

Pew =(2 * Ce * α * y * √ (h *y)) / 3 (4a)where

Ce = 51 / √ (1 – 0.72 * (h / (1000te))2) (4b)where Pex, M, ax, W, α, g are the horizontal earthquake force on the dam, mass horizontalearthquake acceleration, weight, acceleration due to gravity and seismic coefficient respectively.

Also Pew, h, te are the additional total water load down to depth y, total height of reservoir, and

period of vibration respectively (3).

3.3. Load Combinations

Gravity dams should be designed for all appropriate load combinations, using the proper

safety factor for each. Combinations of transitory loads, each of which has only a remote probability

of occurrence at any given time, have negligible probability of simultaneous occurrence and should

not be considered as an appropriate load combination. Temperature loadings should be included

when applicable, as previously discussed in this monograph, Figure (9).

(1) Usual loading combinations.-Normal design reservoir elevation with appropriate dead loads,

uplift, silt, ice, and tail water. If temperature loads are applicable, use minimum usual temperatures

occurring at that time.

(2) Unusual loading combinations .-Maximum design reservoir elevation with appropriate dead

loads, silt, tail water, uplift, and minimum usual temperatures occurring at that time, if applicable.

(3) Extreme loading combinations.-The usual loading plus effects of the ―Maximum CredibleEarthquake.‖

Figure (8)


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14

(4) Other loadings and investigations.-(a) The usual or unusual loading combination with drains

inoperative. (b) Dead load. (c) Any other

loading combination which, in the

designer’s opinion, should be analyzed fora particular dam.

From the loads mentioned above it is

possible to create load cases as:

A usual load case occurs often, oreven all the time. For example the

combination of dead weight,

sand and silt load, uplift pressure,

hydraulic pressures from reservoir

and tail water at a normal level.

An unusual load case may be the

usual case from above with added

ice load and lowest possible

temperature.

An extreme load case could be a

combination of the worst scenario in all eight load-types, including a nearby earthquake. The reasonto use these load cases is to be able to estimate and calculate safety factors, the more usual a load

case is, the higher the safety factor should be. This is just an example of how different scenarios can

be predicted, in reality these different cases are very thoroughly evaluated, with a lot of different

combinations (1).

3.4. Stability criteria

The loads listed in section 2.1 will create different types of stresses in the dam body.Although every dam project is unique, problems with these stresses will often occur in the same

areas. Figure 10 shows these general critical areas. To create a clear overview of the figure none of

Figure (9)


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15

the applied loads are displayed (1). To evaluate shear stress in the different, carefully chosen, areas

in and below the dam foundation, information about the friction coefficient both in concrete, rock

and between the two is needed. The cohesion in concrete and rock is also needed. With the vertical

stress, the shear stress, the friction coefficient and the cohesion a safety factor, K can be calculated.

This safety factor is calculated

according to Mohr-coulomb failure criterion.

Unstable sliding surfaces can occur in numerous places in the dam and the foundation. Therefore it

is important to single out the areas where the greatest risk of damage exists. For example such

surfaces could be cracks in the ground, where a change of rock material occurs and in various places

in the dam, the two most obvious of such being in the foundation plane just in the contact surface

with the underlying rock and the horizontal plane where the slope of the downstream side of the

dam body starts to flatten out shown in figure 10 (1).

Stability criteria for concrete gravity dams accounts for four types of controls to be

considered as follows:

Sliding stability; To make sure the calculations are accurate, the element standards, described later

in this chapter, will be considered when creating elements inside and around the critical areas,

especially close to the dam foundation and in the batter (5).

Tension stress often occurs, in the region around the dam heel. When analyzing the tension stress

we use a simplification for FEM modeling that states that the number of elements with tension in

the bottom layer of the dam cannot exceed seven per cent of the total amount of elements in that

layer. The reason that we can use this simplification is that the elements in the foundation have

roughly the same size, which leads to that the percentage of tension elements is considered the sameas the percentage.

Compression stresses are handled by looking at the whole model and then determine where the

greatest risk for compressive failure appears. Material characteristics have to be evaluated and

compared to the computed compressive stresses (5).

Displacement control is based on the entire model. Since for example the displacement in the top

of the dam depends on the displacement in the bottom part of the dam there is not really one area to

focus on to receive good displacement results.

3.5. Factors of Safety

All loads to be used in design should be chosen to represent, as nearly as can be determined,the actual loads that will occur on the structure during operation, in accordance with the criteria

under ―Load Combinations.‖ Methods of determining load-resisting capacity of the dam should bethe most accurate available. All uncertainties regarding loads or load-carrying capacity should be-

resolved as far as practicable by field or laboratory tests and by thorough exploration and inspection

of the foundation. Thus, the factor of safety should be as accurate an evaluation as possible of the

capacity of the structure to resist applied loads. All safety factors listed are minimum values. Like

other important structures, dams should be regularly and frequently inspected. Adequate

observations and measurements should be made of the structural behavior of the dam and its

foundation to assure that the structure is functioning as designed.

Although somewhat lower safety factors may be permitted for limited local areas within the

foundation, overall safety factors for the dam and its foundation after beneficiation should meet

requirements for the loading combination being analyzed. For other loading combinations where

safety factors are not specified, the designer is responsible for selection of safety factors consistent


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with those for loading combination categories previously discussed. Somewhat higher safety factors

should be used for foundation studies because of the greater amount of uncertainty involved in

assessing foundation load-resisting capacity. Safety factors for gravity dams are based on the use of

the gravity method of analysis and those for foundation sliding stability are based on an assumption

of uniform stress distribution on the plane being analyzed.

Cri ter ia (USBR)

(I) Compressive stress.-The maximum allowable compressive stress for concrete in a gravity dam

subjected to any of the ―Usual Loading Combinations‖ should not be greater than the specifiedcompressive strength divided by a safety factor of 3.0. Under no circumstance should the allowable

compressive stress for the ―Usual Loading Combinations‖ exceed 1,500 lbs/in2 (10.3 MPa). Asafety factor of 2.0 should be used in determining the allowable compressive stress for the ―UnusualLoading Combinations.‖ The maximum allowable compressive stress for the ―Unusual LoadingCombinations‖ should in no case exceed 2,250 lbs/in2 (15.5 MPa). The maximum allowable

compressive stress for the ―Extreme Loading Combinations‖ should be determined in the same wayusing a safety factor greater than 1.0. Safety factors of 4.0, 2.7, and 1.3 should be used indetermining allowable compressive stresses in the foundation for ―Usual,‖ ―Unusual,‖ and―Extreme Loading Combinations,‖ respectively, Figure (11).(2) Tensile stress.-In order not to exceed the allowable tensile stress, the minimum allowable

compressive stress computed without internal hydrostatic pressure should be determined from the

following

expression which takes into account the tensile strength of the concrete at the lift surfaces:

σ z = p. γ. h – (f t /s)

where:

σ z = minimum allowable stress at the face

p = a reduction factor to account for drains

γ = unit weight of water

h = depth below water surface

f t = tensile strength of concrete at lift surfaces

s = safety factor.

All parameters must be specified using consistent units. The value of p should be 1.0 ifdrains are not present or if cracking occurs at the downstream face and 0.4 if drains are used. A

Figure (11)


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safety factor of 3.0 should be used for ―Usual‖ and 2.0 for ―Unusual Loading Combinations.‖ Theallowable value of 0 σ z for ―Usual Loading Combinations‖ should never be less than 0. Crackingshould be assumed to occur if the stress at the upstream face is less than σ z , computed from theabove equation with a safety factor of 1.0 for the ―Extreme Loading Combinations.‖ The structure

should be deemed safe for this loading if, after cracking has been included, stresses in the structuredo not exceed the specified strengths and sliding stability is maintained.

4. THEORETICAL APPROACH

In both the Eulerian and Lagrangian methods, the governing fluid-structure system equation

is solved using wave propagation through the fluid by assuming linear incompressibility and

inviscousity (6), Figure (12).

4.1. Governing EquationsAssuming that water is linearly compressible and neglecting its viscosity, the small

amplitude irrotational motion of the water is governed by the two-dimensional wave equation(7),(10):

Ω (1)

where is the acoustic hydrodynamic pressure; t is time and is the two-dimensional Laplace

operator and C is the speed of pressure wave given by:

4.2. Dam-Reservoir Boundary Condition:In the common boundary between the reservoir and the dam body, an interaction betweenthese two boundaries occurs which is the result of an inertia force caused by the movement of the

reservoir wall. At the surface of fluid-structure, there must be no flow across the interface. This is

based on the fact that face of concrete dams is impermeable. Hence, the applied pressure on the

reservoir wall caused by the inertial force is as follow:

(3)

Figure (12)


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where is normal acceleration of the dam body on the upstream face and n is normal vector on the

interface of the dam-reservoir outwards the dam body and is the mass density of the reservoir

water.

4.3. Reservoir-Foundation Boundary Condition:According to the rigidity of the reservoir bottom, by assuming the horizontal movement of

the earth, the pressure gradient is neglected. Reservoir bottom absorption effect is implemented as:

(4)

where is the damping coefficient characterizing the effects of absorption of hydrodynamic

pressure waves at the reservoir boundary and is the wave reflection coefficient, which represents

the ratio of the amplitude of the reflected wave to that of the normally incident pressure wave at the

reservoir is related to by the following expressions:

It is believed that a value from 1 to 0 would cover the wide range of materials encountered at

the boundary of actual reservoirs. The value of the wave reflection coefficient that characterizes

the reservoir bottom materials should be selected based on their actual properties, not on properties

of the foundation rock. Materials on the reservoir bottom has great influence in absorbing of

earthquake waves and decreases the system response under the vertical component of the earthquake

and this effect is also important for horizontal component.

4.4. Reservoir-Far-End Boundary Condition

With the vibration of the dam, volumetric hydrodynamic pressure waves are created in thereservoir and propagate toward the upstream. If the length of the dam is assumed to be infinity, then

these waves would approach to vanish. It should be noted that the length of reservoir is assumed as

a finite length, in numerical modeling. Hence, an artificial boundary is applied to simulate effect of

an infinite reservoir. For modeling far-end truncated boundary, viscous boundary condition (called

as Sommerfeld boundary condition) is utilized to absorb completely the outgoing pressure waves

given as Somerfield-type radiation boundary condition may be implemented namely:

(5)

4.5. Free-Surface Boundary Condition In high dams, surface waves are negligible and hydrodynamic pressure on the free surface is

set to be zero, the boundary condition is easily defined as:

P(x, y, z, t) = 0 (6)

The dam-reservoir interaction is solved by coupled solution procedure while the boundary

condition is applied at the reservoir’s far -end truncated boundary. The foundation is defined as adifferent part from the structure with different modulus of elasticity. An efficient coupling

procedure is formulated by using the cou pling coincide nodes method. Summerfield’s boundary

condition at the far end of the infinite fluid domain is implemented. Figure 13 shows the coupleddam-reservoir- foundation problem idealization.


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Figure13: Dam-Reservoir-Foundation System

5. FINITE ELEMENT MODELING

FEM modeling of concrete gravity dams is a method with a lot of advantages compared to

traditional structural dynamics and scale modeling. Compared to scale modeling the time and cost

issue is the main factor, it is a lot cheaper to construct a virtual model than a physical one. Also the

convenience of computer based models compared to the location and rarity of scale models provide

a significant advantage. Compared to structural mechanics FEM has a big advantage in the

alteration of both construction and external loads. Once a dam has been modeled in FEM it is

possible to experiment and change details about it without the need to restart the whole process.

This is still just an analysis of a single section in a static state of a dam; a lot of aspect is

because of that limitation not dealt with at all. Examples of these aspects are: discharge capacity,

temperature changes, cracks, earthquakes and fatigue of the concrete. FEM means Finite Element

Method and it is a way of turning real life objects, such as a dam construction, to a computable

model. In the FEM the object is divided into smaller elements which are calculated separately,

preferably by a computer. It is the density and shapes of these elements that determines the accuracy

of the FEM-model. From advanced mathematical models to simple models made of for exampleclay, they are still just models. Models can be more or less accurate, but they will never behave

exactly as reality would, (2).

The size and shape of elements is utterly important, and some basic standards have been set

up to make it easier to create well-functioning elements. The two most common types of two-

dimensional elements are quad and tri elements. Tri elements are made from three different nodes

and contain only one integration point while quads, as implied by their name, are made of four

nodes and contain four integration points. Therefore quad elements are more accurate and are to be

preferred. In our models we use only quad elements as displayed in Figure 14. The height-width

relation should not exceed three to one, for the quad elements, and no interior angle should be less

than 45 degrees (7). In practice the limit may be reduced to 30 degrees and the result will still be

acceptable. According to the two shape standards above, the most precise element has the height-

width ratio of one to one and all interior angles 90 degrees, the perfect square. Usually, fitting all

elements into these standards is impossible and not even that important, a small percentage of the


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elements may remain distorted. The element standards in critical regions of the model where the

mesh may also possibly be denser. Minor alteration in the model’s geometry can be made, in such away that they generate no significant difference in the results. These alterations should be done so

the elements could form a better shape, given height-width relation and their interior angle (2).

Figure 14: A screenshot from a typical dam model displayed in ANSYS (8).

5.1. Strain and Stress in FEM

The elements used in FEM processes can either be plane stress or plane strain elements. In

both plane stress and plane strain analyses there are three components that need consideration in the

xy-plane. Two of these components are normal stresses, one horizontal and one vertical, the last of

the components is shear stress. However there are differences between the stress and strain. In plane

stress analysis all components of stress, except the three mentioned, are zero which leads to no

addition to the internal work in the elements. In the case of plane strain analysis, the stress

orthogonal the xy-plane can vary. However because of the definition of plane strain, the strain in

that perpendicular direction to the plane does not exist. Given this correlation of these contributions

to the internal work in the elements will not be affected. This makes it comparable to plane stress.

benefit of this is that the xy-plane can be evaluated with the three main stress components at the end

of all computations (7).

5.2. FEM Formulation

Referring to the total hydrodynamic pressure during the earthquake against the upstream

face, it has been shown that during the initial earthquake phases, the hydrodynamic pressure is

higher at the upper part of the dam because of the prevailing effect of water compressibility. If the

dominating period of earthquake is long, the increase of the hydrodynamic pressure is negligible.

Under the same condition, however, earthquake can also generate overall oscillation of the fluid

mass, because of the inertia forces developed in the fluid body (7). In the present study, the standard

Finite Element technique is adopted utilizing Galerkin’s method in which the structure displacementvector is discretized as

u= Nu (6a)

and the fluid is similarly discretized as

p= Np (6b)


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where and are the nodal parameters of each field and Nu and Np are appropriate shape functions.

The discrete equations of the structure dynamic response following Galerkin method reads (7):

M + C +K - Q + f =0 (7)

In which M, C, K and f refer to mass matrix, damping matrix, stiffness matrix of the structure and

prescribed force vector respectively, where , and are displacement, velocity and acceleration

vectors respectively. The coupling term in Equation 7 arises due to the pressures specified on the

boundary reads [7]

= (8)

Matrix Q shown in equation (8) transforms the accelerations of the structure to fluid pressure and

also transforms the hydrodynamic pressure into applied loads on the structure to simulate fluid

structure interaction. In Equation 8 is the direction vector of the normal to the interface. StandardGalerkin’s discretization applied to the fluid Equation (1) and its boundary conditions leads to [7]

S + + H + QT

+ q = 0 (9)

in which S, H and q are pseudo fluid mass matrix, pseudo fluid damping matrix, pseudo fluid

stiffness matrix and prescribed flux vector respectively which are given by

S = - (10a)

= (10a)

H = (10a)

where Q is identical to that of Equation 8 and and are nodal pressure vector, the first and

second order derivatives of nodal pressure vector with respect to time, respectively. Hence, the

coupled equation of the fluid-structure system based on Equations (7) and (9) subjected to

earthquake ground motion can be presented as follows (8):

+ (11)

In which represents the nodal ground acceleration vector.

5.3. Fluid-Structure Interaction

Earthquake-induced hydrodynamic pressures on upstream face of a dam are important

factors in design consideration. Assuming that the fluid is incompressible, Westergaard [9] was the

first who derived an analytical solution for the hydrodynamic pressure acting on a rigid dam with a

vertical upstream face as a result of horizontal harmonic ground motion. In this method for the

analysis of concrete dams, fluid is treated as an added mass to the body of the dam. In the last fiftyyears, many researchers have extended the Westergaard’s (9) classical work to include more

physical parameters such as the compressibility of the fluid in the reservoir, the flexibility of the


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dam, and reservoir bottom absorption. However, analytical solutions are rare and available only for

a reservoir with a simple geometry and boundary conditions. Numerical models proved to be

powerful techniques that can be used for dam-reservoir interaction problems with complex two and

three dimensional geometries as well as complicated boundary and initial conditions. Studies of

Chopra [10]-[11] showed that fluid incompressibility assumption does not predict correctly the

applied hydrodynamic pressure on the dam body. The early studies on 2-dimentional gravity dam

roots back to late 1970s, in which interaction effects were considered through the exact and non-

numerical solutions of the governing equations, Chopra et al. [12]-[13]. Zienkiewicz and Taylor

[14] solved the coupled governing fluid-structure equations utilizing the finite element method. For

modeling the upstream boundary of the reservoir, they used radiative boundaries of thermal

analysis. AKK¨OSE [15] presented the linear and nonlinear responses of a selected arch subjected

to earthquake ground motion. The hydrodynamic effects on the dynamic response of arch dams

were investigated using step-by-step integration by the Lagrangian approach. Aznarez et al. [16]

studied the effects of reservoir bottom absorbent materials, on the dynamic analysis of fluid-

structure interaction problem through the use of the boundary element method in the frequency

domain. Du et al. [17] studied the nonlinear seismic response analysis of a foundation-arch damsystem. They found that the maximum dynamic response was obtained from their method is lower

than that of the common methods. Akkose et al. [18] have studied the effect of sloshing on the

nonlinear dynamic response of the arch dam. Seghir et al. [19] used the coupled finite element and

symmetric boundary element to model the interaction problem. Bonnet et al. [20] used the

combination of the finite element and boundary element methods to simulate the dam-reservoir

interaction in the frequency domain. They considered an elastic material behavior and a rigid

reservoir bottom. The results of the analysis showed good agreement between theoretical and

numerical methods. Shariatmadar et al. [21] studied hydrodynamic pressures induced due to seismic

forces and Fluid-Structure Interaction. The interaction of reservoir water-dam structure and

foundation bed rock are modeled using the ANSYS code. The analytical results obtained from over

twenty 2D finite element modal analysis of concrete gravity dam show that the accurate modeling of

dam-reservoir-foundation and their interaction considerably affects the modal periods, mode shapes

and modal hydrodynamic pressure distribution. Seleemah and co-authors, [22] studied the seismic

response of base isolated liquid storage ground and elevated tanks employing coupled fluid-

structure system via ANSYS code and a good agreement was attained between analytical and

numerical results. Akhaveissy et al. [23] studied the linear dynamic behavior of the Pine Flat

concrete dam in the time domain to analyze reservoir – dam interaction. Also the effect of fluidcompressibility and the Sommerfeld boundary are used to determine the hydrodynamic pressure.

Hamidian et al. [24] studied dam-water-foundation rock interaction effects on linear and nonlinear

earthquake response of arch dams system subjected to earthquake ground motion using the finite

element method involving the materially and geometrically nonlinear effects. Zeidan [25] studiedthe seismic dam-reservoir interaction of concrete gravity dams using ANSYS software with

coupling coincide nodes on the interface, Fig. (14).

Damdomain

Reservoir

Domain Ω HB

LC

HC

HF

heel toex

y

ГR

ГF

ГI

ГB


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Figure 14: Dam-Reservoir coupled system (25).

5.4. Fluid-Structure-Foundation Interaction

Studies that present a method called Soil Structure Interaction Method (SSI) for estimation

of free field earthquake motions at the site of dams. This method neglects the presence of structure

(Dam) during the earthquake and assumes that the relative displacement at the truncated boundary is

zero, and shows that under these circumstances, the foundation just bears the inertia force and does

not bear the earthquake force. Recent studies on dam-reservoir-foundation interaction deal with the

boundary conditions of the foundation

and foundation interaction with both

reservoir and dam bodies. However,

the problem of fluid-soil-structureinteraction is a complicated problem

and need more interest from

researchers. More efficient methods

are required to properly assess the

safety of concrete gravity dams

located in regions with significant

seismicity (8). The methods used for

the analysis of concrete dams under

earthquake loading range from the

simple pseudo-static method initially

proposed by Westergaard (1933) toadvanced numerical methods that

include the well-known FEM.

Westergaard [9] introduced an

approach to determine approximately

the linear response of the dam-reservoir system by a number of masses that are added to the dam

body. The method proposed by Westergaard assumes that the hydrodynamic effect on a rigid dam is

equivalent to the inertial force resulting from a mass distribution added on the dam body. The dam-

reservoir system can be categorized as a coupled field system in a way that these two physical

domains interact only at their interface [26]. To simplify and economize the finite element modeling

of an infinite reservoir, the far-end boundary of the reservoir has to be truncated. Sommerfeld boundary condition [27] is an appropriate boundary condition for the truncated part of the reservoir.

Hydrodynamic pressure in seismic response of dam-reservoir interaction in time domain has been

investigated [28]. Preliminary design and evaluation of concrete gravity sections is usually

performed using the simplified response spectrum method proposed by Fenves and Chopra [10-13].

A standard fundamental mode of vibration, representative of typical sections, is used in this method.

This mode shape does not take into account the foundation flexibility since it is representative of a

standard concrete gravity section on rigid foundation.

As an alternative, the first mode of vibration of the concrete section could be estimated

using a finite element model with massless foundation. Fenves and Chopra [26], [27] studied the

dam-reservoir-foundation rock interaction in a frequency domain linear analysis. In the work

presented by Gaun et. al [28], an efficient numerical procedure has been described to study thedynamic response of a reservoir-dam-foundation system directly in the time domain. Ghaemian et.

al [29] showed that the effects of foundation’s shape and mass on the linear response of arch dams

0.5 5 50 500

0.00

0.02

0.04

0.06

0.08

0.10

crest

dam toe

dam heel

M a x .

h z .

d i s p l a c e m e n t m

Ef / E

c ratio

LF LB

Figure 15


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are considerable. The dam – foundation interaction effects are typically presented by a ―standard‖mass-less foundation model [30]. In this case, it is assumed that the displacement at the bottom of

the foundation vanishes and roller supports is placed at the vertical sides of the foundation. The

most widely used model for soil radiation damping is the one of Lysmer and Kuhlemeyer [31]. In

this model the foundation is wrapped by dashpots tuned to absorb the S and P waves. In this model,

modeling the radiation damping on the far – end boundary of the massed foundation, 2- nodeelements as boundary elements are used to apply the lumped dashpot on the far – end nodes of themassed foundation model. The viscous boundary condition is applied on the far – end boundary ofthe foundation to prevent the wave reflection form the artificial boundary of the infinite media in

finite element analysis (32). The most common soil – structure interaction (SSI) approach is based onthe ―added motion‖ formulation. This formulation is valid for free– field motions caused byearthquake waves generated from all sources. The method requires that the free – field motions at the

base of the structure be calculated prior to the soil – structure interaction analysis [33]. Zeidan [34]studied the seismic dam-reservoir-foundation interaction of concrete gravity dams using ANSYS

software with different ratios for Ec/Ef representing foundation flexibility, Fig. (15).

5.5. FEM Modeling Assumptions

In FEM simulation, in order to satisfy the continuity conditions between the fluid and solid

media at the boundaries. The nodes at the common lines of the fluid and the plane elements are

constrained to be coupled in the direction normal to the interface. Relative movements are allowed

to occur in the tangential directions. This is implemented by attaching the coincident nodes at the

common lines of the fluid and the plane elements in the normal direction. At the interface of the

fluid-structure system, only the displacements in the direction normal to the interface are assumed to

be compatible in the structure as well as the fluid. The fluid is generally assumed to be linear-

elastic, incompressible, irrotational and nonviscous. 2-D finite element model is implemented.

Absorption is considered at reservoir bottom. Since the extent of the reservoir is large, it isnecessary to truncate the reservoir at a sufficiently large distance from the dam. A length of

reservoir equivalent to two to three times its depth is chosen for adequate representation of

hydrodynamic effects on the dam body (7). The depth of foundation is taken as 1.5 the dam base

width into account in the calculations. The dam and foundation materials are assumed to be linear-

elastic, homogeneous and isotropic. The effect of foundation flexibility is considered for dam-

foundation rock interaction ratios i.e. modulus of elasticity of foundation to modulus of elasticity of

dam E f /E c.

6. ANALYSIS OF CONRETE GRAVITY DAMS

Selection of the method of analysis should be governed by the type and configuration of the

structure being considered. The gravity method will generally be sufficient for the analysis of most

structures, however, more sophisticated methods may be required for structures that are curved in

plan, or structures with unusual configurations.

5.1. Gravity Method

The gravity method assumes that the dam is a 2 dimensional rigid block. The foundation

pressure distribution is assumed to be linear. It is usually prudent to perform gravity analysis before

doing more rigorous studies. In most cases, if gravity analysis indicates that the dam is stable, no

further analyses need be done. A Stability criteria and required factors of safety for sliding arerequired.


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5.2. Finite Element Method

In most cases, the gravity analysis

method discussed above will be sufficient

for the determination of stability.

However, dams with irregular geometriesor spillway sections with long aprons may

require more rigorous analysis. The Finite

Element Method (FEM) permits the

engineer to closely model the actual

geometry of the structure and account for

its interaction with the foundation. For

example, consider the dam in Figure 16.

Note that the thinning spillway that forms

the toe of the dam is not stiff enough to

produce the foundation stress distributionassumed in the gravity method. In this case, gravity analysis alone would have under-predicted base

cracking.

Finite element analysis allows not only modeling of the dam, but also the foundation rock

below the dam. One of the most important parameters in dam/foundation interaction is the ratio of

the modulus of deformation of the rock to the modulus of elasticity of the dam concrete. Figure 17

illustrates the effect that this ratio has on predicted crack length. As the modular ratio varies, the

amount of predicted base cracking varies also. As can be seen in Figure 17, assuming a low

deformation modulus (Er ), is not necessarily conservative.

However, it is implicitly assumed that shear stress is distributed uniformly across the base.

This assumption is arbitrary and not very accurate. Finite element modeling can give some insight

into the distribution of base contact stress. As can be seen in figure 17, shear stress is at a maximumat the tip of the propagating base crack. In this area, normal stress is zero, thus all shear resistance

must come from cohesion. Also, the peak shear stress is about twice the average shear stress. An un-

zipping failure mode can be seen here, as local shear strength is exceeded near the crack tip, the

crack propagates causing shear stress to increase in the area still in contact.

6.3. Modal Analysis and Natural Response

The structural response of a material to different loads determines how it will be economically

utilized in the design process. Earthquake is a natural disaster that has claimed so many lives and

destroyed lots of property. Earthquake hazards had caused

the collapse and damage to continual functioning ofessential services such as communication and

transportation facilities, buildings, dams, electric

installations, ports, pipelines, water and waste water

systems, electric and nuclear power plants with severe

economic losses. Earthquake is a major source of seismic

forces that impinge on structures others are Tsunami,

seethe etc. Earth wall is chosen as a material for the dam

since its major constituent earth is abundantly available and

provides a sustainable solution. This necessitates the

seismic analysis of concrete gravity dam. Earthquakes hadcaused severe damages and consequently huge economic

losses including losses of lives (3). Figure (18) shows mode

Figure (16)

Figure (17)


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shapes for a gravity dam with empty reservoir while Figure (19) shows mode shapes for a gravity dam with

full reservoir.

6.4. Dynamic Equilibrium

Figure (18)

Figure (19)


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DAM EQUATION OF MOTION (Zeidan 2013)

RESERVOIR EQUATION OF MOTION

6.5. Dynamic AnalysisDynamic analysis refers to analysis of loads whose duration is short with the first period of

vibration of the structure. Such loads include seismic, blast, and impact. Dynamic methods are

appropriate to seismic loading. Because of the oscillatory nature of earthquakes, and the

subsequent structural responses, conventional moment equilibrium and sliding stability

criteria are not valid when dynamic and pseudo dynamic methods are used. The purpose of

these investigations is not to determine dam stability in a conventional sense, but rather to

determine what damage will be caused during the earthquake, and then to determine if the dam

can continue to resist the applied static loads in a damaged condition with possible loading

changes due to increased uplift or silt liquefaction. It is usually preferable to use simple dynamic

analysis methods such as the pseudo dynamic methodor the response spectrum method(described below), rather than the more rigorous sophisticated methods. The procedure for

performing a dynamic analysis includes the following (3):

1. Review the geology, seismology, and contemporary tectonic setting.

2. Determine the earthquake sources.

3. Select the candidate maximum credible and operating basis earthquake magnitudes and

locations.

4. Select the attenuation relationships for the candidate earthquakes.

5. Select the controlling maximum credible and operating basis earthquakes from the candidate

earthquakes based on the most severe ground motions at the site.

6. Select the design response spectra for the controlling earthquakes.

7.

Select the appropriate acceleration-time records that are compatible with the design responsespectra if acceleration-time history analyses are needed.

8. Select the dynamic material properties for the concrete and foundation.


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9. Select the dynamic methods of analysis to be used.

10.

Perform the dynamic analysis.

11. Evaluate the stresses from the dynamic

analysis.

6.5.1. Pseudo Dynamic MethodThis procedure was developed by

Professor Anil Chopra as a hand calculated

alternative to the more general analytical

procedures which require computer programs. It

is a simplified response spectrum analysis which

determines the structural response, in the

fundamental mode of vibration, to only the

horizontal component of ground motion. This

method can be used to evaluate the compressive

and tensile stresses at locations above the baseof the dam. Using this information, degree of

damage can be estimated and factored into a post earthquake stability analysis.

6.5.2. Modal Dynamic MethodDynamic response analysis is typically performed using finite element modal analysis. The

major modes of vibration are calculated, and the response of the structure to the earthquake is

expressed as a combination of individual modal responses. There are 2 acceptable techniques for

modal analysis, Response Spectrum Analysis and Time History Analysis, Figure (20).

4.3.2.1 Response Spectrum Method

In the response spectrum method, the modes of vibration determined from finite element

modeling are amplitude weighted by a response spectrum curve which relates the maximum

acceleration induced in a single degree of freedom mechanical oscillator to the oscillator's natural

period. A typical response spectrum curve is shown in figure 12. Because the timing of the peaks of

individual modal responses is not taken into account, and because peaks of all modes will not occur

simultaneously, modal responses are not combined algebraically. Modal responses are combined

using the SRSS (square root of sum of squares) or the CQC (complete quadratic combination)

methods, Figure (21).

4.3.2.2 Time History MethodThe time history method is a more

rigorous solution technique. The response

of each mode of vibration to a specific

acceleration record is calculated at each

point in time using the Duhamel integral.

All modal responses are then added

together algebraically for each time step

throughout the earthquake event. While

this method is more precise than the

response spectrum method for a givenacceleration record, its results are

contingent upon the particulars of the

Figure (20)

Figure (21)

Figure (20)


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acceleration record used. For this

reason, time history analysis should

consider several accelerograms.

4.4.3. Direct Solution MethodThe modal superposition

methods described above require the

assumption of material linearity. Direct

solution techniques solve the differential

equations of motion in small time steps

subject to material stress strain

relationships which can be arbitrary, and

therefore the development of damage

can be accounted for. Their results are

also highly affected by the particular

accelerogram used.

4.4.4 Block Rocking AnalysisWhen dynamic analysis techniques such as those discussed above indicate that concrete

cracking will occur, a block rocking analysis can be done. This type of analysis is useful to

determine the stability of gravity structures or portions thereof, when it is determined that cracking

will progress to the extent that free blocks will be formed. The dynamic behavior of free blocks can

be determined by summing moments about the pivot point of rocking.

4.4.6. Reservoir Added MassDuring seismic excitation the motion of the dam causes a portion of the water in the

reservoir to move also. Acceleration of this added mass of water produces pressures on the dam thatmust be taken into account in dynamic analysis. Westergaard derived a pressure distribution

assuming that the dam would move upstream and downstream as a rigid body, in other words, the

base and crest accelerations of the dam are assumed to be identical. This pressure distribution is

accurate to the extent that the rigid body motion assumption is valid. The dam's structural response

to the earthquake will cause additional pressure. Figure 22 shows the difference in pressure

distributions resulting from rigid body motion and modal vibration. Westergaard's theory (9) is

based on expressing the motion of the dam face in terms of a fourier series. If the acceleration of the

upstream face of the dam can be expressed as:

where α is the ground acceleration, then the resulting pressure is given by :

While, Westergaard assumed a rigid body acceleration, the above equations can be generalized to

accommodate any mode shape. As with the application of finite element techniques for static

analysis, the reviewer must not lose sight of the purpose of the analysis, ie to determine whether or

not a given failure mode is possible. Finite element techniques assume linear stress strain

characteristics in the materials, and almost always ignore the effect of cracking in the dam. These

assumptions can constitute rather gross errors. For this reason when reviewing the finite elementresults, the stress output should be viewed qualitatively rather than quantitatively. Finite element

Figure 13

Figure (22)


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dynamic output can show where the structure is most highly stressed, but the stress values should

not be considered absolute.

4.5 Cracked Base AnalysisThe dam/foundation interface shall be assumed to crack whenever tensile stress normal to

the interface is indicated. This assumption is independent of the analysis procedure used. The practical implementation of this requirement is illustrated in the gravity analysis shown below. All

forces, including uplift are applied to the structure. Moments are taken about 0,0 which does not

necessarily have to be at the toe of the dam. The line of action of the resultant is then determined as

shown in Figure (23). The intersection of the resultant line of action and the sloping failure plane is

the point of action of the resultant on the structure. A crack is assumed to develop between the

base and foundation if the stress normal to the base is tensile. Since the gravity analysis technique

assumes a linear effective stress distribution along the dam base, the length of this crack is uniquely

determined by the location of the resultant and the assumption of a linear effective stress

distribution.

Dynamic loading is equally capable of causing base cracking, however, cracked base

analyses are not typically performed for dynamic loadings because of the computational difficulty

involved. The conventional gravity analysis procedure is not appropriate for dynamic loading

because it ignores the dynamic response of the structural system. Standard dynamic finite element

techniques are not appropriate because they are based on an assumption of material linearity and

structural continuity. What is typically assumed is that during the earthquake, extensive base

cracking does occur. Stability under post earthquake conditions, which include whatever damage

results from the earthquake, must be verified.

7. DESIGN & SAFETY CRITERIA

Specific stability criteria for a particular loading combination are dependent upon the degree

of understanding of the foundation structure interaction and site geology, and to some extent, on themethod of analysis. Assumptions used in the analysis should be based upon construction records

and the performance of the structures under historical loading conditions. In the absence of available

design data and records, site investigations may be required to verify assumptions. Safety factors are

intended to reflect the degree of uncertainty associated with the analysis. Uncertainty resides in the

knowledge of the loading conditions and the material parameters that define the dam and the

foundation. Uncertainty can also be introduced by simplifying assumptions made in analyses. When

sources of uncertainty are removed, safety factors can be lowered.

The basic requirement for stability of a gravity dam subjected to static loads is that force and

moment equilibrium be maintained without exceeding the limits of concrete, foundation or concrete

/foundation interface strength. This requires that the allowable unit stresses established for theconcrete and foundation materials not be

exceeded. The allowable stresses should be

determined by dividing the ultimate strengths of

the materials by the appropriate safety factors in

Table 2. In most cases, the stresses in the body

of a gravity dam are quite low, however if

situations arise in which stress is a concern, the

following guidance in Table 1 is applicable.

Table1:

Figure (23)


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Load

Condition

Shear Stress on Pre-cracked

Failure Plane1

Principal Axis Tension Within

Intact Concrete2,4

Worst Static 1.4 σn 1.7(F'c)2/3

Max. Dynamic N.A.

3 N.A.

3

The tensile strength of the rock-concrete interface should be assumed to be 0. Rock foundations may

consist of adversely-oriented joints or fractures such that even if the interface could resist tension,

the rock formation immediately below may not be able to develop any tensile capacity. Therefore,

since stability would not be enhanced by an interface with tensile strength when a joint, seam or

fracture in the rock only a few inches or feet below the interface has zero tensile strength, no tension

will be allowed at the interface.

7.1. Sliding Stability Safety FactorsRecommended factors of safety are listed in Table 2

Table 2 Recommended Minimum Factors of Safety 1/

Dams having a high or significant hazard potential.

Loading Condition 2/ Factor of Safety 3/

Usual 3.0

Unusual 2.0

Post Earthquake 4/ 1.3

Dams having a low hazard potential.

Loading Condition Factor of Safety

Usual 2.0

Unusual 1.25

Post Earthquake Greater than 1.0

Notes:

1. Safety factors apply to the calculation of stress and the Shear Friction Factor of Safety within

the structure, at the rock/concrete interface and in the foundation.

2. Loading conditions as

zeidan dam chapter 2014

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