conventional and fem analysis of a dam

8/10/2019 conventional and fem analysis of a dam

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IIIContents

CERTIFICATE...I

ACKNOWLEDGEMENT .II

CONTENTS .........................................................................................................................................III

LIST OF TABLES .................................................................................................................................. V

LIST OF FIGURES ............................................................................................................................... VI

ABSTRACT.....VII

CHAPTER 1

INTRODUCTION ........................................................................................................ 1-1

1.1 GENERAL.......................................................... ................................................................. ......... 1-1

1.1.1 Gravity Dam ................................................................... ..................................................... 1-1

1.1.2 Arch Dam ............................................................................................................................ 1-3

1.1.3 Buttress dams ..................................................................................................................... 1-3

1.2 CASE STUDY OF INDIRA SAGAR POLAVARAM DAM.............................................................................. 1-5

1.3 OBJECTIVES OF THE PRESENT WORK................................................................. ............................... 1-7

CHAPTER 2 LITERATURE REVIEW ................................................................................................ 2-1

2.1 GRAVITY DAMS........................................................................................................................... 2-2

2.1.1 Forces to be resisted by the Dam ............................................................................... ......... 2-2

2.1.2 Load Combinations ............................................................................................................. 2-9

2.1.3 Requirements for Stability................................................................................................... 2-9

2.1.4 Stress Analysis ................................................................ ................................................... 2-11

2.2 TWO DIMENSIONAL ANALYTICAL METHOD.................................. ................................................... 2-13

2.2.1 Assumptions ......................................................... ............................................................. 2-13

2.2.2 Analysis Procedure ............................................................................................................ 2-13

2.2.3 Stresses in the Dam ........................................................................................................... 2-14

2.3 FINITE ELEMENT METHOD (FEM) ................................................................................................. 2-15

2.3.1 Two-Dimensional Stress Distributions ............................................................. .................. 2-16

2.3.2 Shape Functions ......................................................................................................... ....... 2-16

2.3.3 Strain Displacement Relations ....................................................................................... 2-17

2.3.4 Convergence and Compatibility Requirements of Elements ............................................. 2-18

2.3.5 Stiffness equation of the Element ..................................................................................... 2-19

CONTENTS


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IVContents

CHAPTER 3 METHODOLOGY ....................................................................................................... 3-1

3.1 STRESS ANALYSIS USING FINITE ELEMENT METHOD............................................................................ 3-1

3.1.1 Finite Element Analysis Program ........................................................................................ 3-1

3.1.2 Analysis Cases ..................................................................................................................... 3-63.1.3 FEM Model ........................................................... ............................................................... 3-8

3.1.4 Validation of the modelling procedure ............................................................................. 3-17

CHAPTER 4 RESULTS & DISCUSSIONS .......................................................................................... 4-1

4.1 RESULTS................................................................................................................. .................... 4-1

4.2 TWO DIMENSIONAL ANALYTICAL METHOD.................................. ..................................................... 4-1

4.3 FEM ..................................................................................... .................................................... 4-2

4.4 STRESS ANALYSIS OF POLAVARAM DAM................................................. .......................................... 4-2

4.4.1 Load combination A (Construction Condition) .......................... .......................................... 4-2

4.4.2 Load Combination B (Full Reservoir Condition) .......................................................... ......... 4-4

4.4.3 Load Combination C (Reservoir Maximum Water Level Condition) ................................... . 4-6

4.4.4 Load Combination D (Combination A with Earthquake) ..................................................... 4-7

4.4.5 LOAD COMBINATION E (Combination B with Earthquake) ................................................. 4-9

4.4.6 Load Combination F (Combination C with extreme Uplift) ............................................... 4-10

4.4.7 Load Combination G (Combination E with extreme Uplift) ............................................... 4-12

CHAPTER 5 SUMMARY AND CONCLUSIONS ................................................................................ 5-1

5.1 CONCLUSIONS FROM THE RESULTS OBTAINED FROM TWO DIMENSIONAL ANALYTICAL METHOD.................. 5-1

5.2 CONCLUSIONS FROM THE RESULTS OBTAINED FROM FEM ........................................................... ......... 5-3

LIST OF REFERENCES ........................................................................................................................ 6-1


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VList of Tables

Table 1-1 Polavaram Dam details ......................................................... ............................................................... 1-5

Table 1-2 Compressive Strength of the Dam Material .......................................................... ............................... 1-6

Table 2-1 Values of Permissible Tensile Stresses in Concrete ............................................................................ 2-10

Table 3-1 An Example of SAP 2000 Input Table Showing Area loads (pore) ...................................................... 3-16

Table 3-2 Comparison of Vertical Normal stress of Saini and present study ..................................................... 3-18

Table 4-1 Stresses Obtained for Load Combination A from FEM ......................................................................... 4-4

Table 4-2 Stresses Obtained for Load Combination B from FEM ......................................................................... 4-5

Table 4-3 Stresses Obtained for Load Combination C from FEM ......................................................................... 4-7

Table 4-4 Stresses Obtained for Load Combination D from FEM ................................................................ ......... 4-8

Table 4-5 Stresses Obtained for Load Combination E from FEM ............................................................... ....... 4-10

Table 4-6 Stresses Obtained for Load Combination F from FEM ..................................................... .................. 4-11

Table 4-7 Stresses Obtained for Load Combination G from FEM ................................................................ ....... 4-13

Table 4-8 Stresses Obtained for All the Load Combination from Two Dimensional Analytical Method ............ 4-14

Table 4-9 Maximum Stresses for Load Combinations with Two Dimensional Analytical Method ..................... 4-15

LIST OF TABLES


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VIList of Figures

Figure 1-1 A General view of Gravity Dam .......................................................................................................... 1-3

Figure 1-2 A General view of Arch Dam .......................................................... .................................................... 1-4

Figure 1-3 A General view of Buttress Dam ..................................................................................... .................... 1-4

Figure 1-4 Cross Section of Polavaram Dam ............................................................... ......................................... 1-6

Figure 2-1 Maximum Value of Pressure Coefficient .............................................................. ............................... 2-6

Figure 2-2 Representation of Various Forces acting on Dam .............................................................................. 2-8

Figure 3-1 Flow Chart showing Modelling & Analysis in SAP 2000 ............................................................. ......... 3-2

Figure 3-2 Quadrilateral Element ........................................................................................................................ 3-4

Figure 3-3 Triangular Element ........................................ ................................................................. .................... 3-4

Figure 3-4 Finite Element Model of the Dam .............................................................................................. ......... 3-9

Figure 3-5 Finite Element Model of Foundation ................................................................................................ 3-10

Figure 3-6 Finite Element Model of Dam .............................................................................. ............................. 3-11

Figure 3-7 Define Material Command Box ........................................................................................................ 3-12

Figure 3-8 Define Area Section Command Box ..................................................................... ............................. 3-13

Figure 3-9 Joint Restraint Command Box........................................................................................................... 3-14

Figure 3-10 Joint Pattern Command Box ........................................................................................................... 3-14

Figure 3-11 Defined Load Patterns .................................................................................................................... 3-15

Figure 3-12 Defined Load Cases ............................................................ ............................................................. 3-15

Figure 3-13 Run Analysis Command Box........................................................................................................... 3-15

Figure 3-14 FEM Discretization on Saini dam .................................................................................................... 3-17

LIST OF FIGURES
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1-1Introduction

1.1 General

A dam is an obstruction placed across a stream or depression and so devised as

to hold back or obstruct the flow of the water. It may be built in such way that water

will flow over it, in which case it is termed an overflow or submerged dam or weir; or

it may be of such height that it cannot be overtopped and it is termed as non-over flow

dam. Dams are classified according to use into Storage Dams, Diversion Dams and

Detention Dams. The primary purpose of a dam is to provide for the safe retention

and storage of water. Structurally, a dam must be stable against overturning and

sliding, either or within the foundations. The rock or soil on which it stands must be

competent to withstand the superimposed loads without crushing or undue yielding.

The reservoir basin created must be watertight and seepage through the foundation of

the dam should be minimal.

Almost each dam that has been constructed all over the world is unique. This

is so because a particular type is chosen because of the considerations of many

factors. In fact, dam engineering brings together a range of disciplines, like structural,

hydraulics and hydrology, geotechnical, environmental etc. A broad classification on

the basis of Structural action is as follows:

Gravity Dam

Arch Dam

Buttress Dam

1.1.1Gravity Dam

Gravity dams are solid concrete structures that maintain their stability against

design loads from the geometric shape, mass and strength of the concrete. It is

primarily the weight of a gravity dam which prevents it from being overturned when

subjected to the thrust of impounded water. This type of structure is durable, and

requires very little maintenance. Gravity dams typically consist of a non- overflow

CHAPTER 1 INTRODUCTION


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1-2Introduction

section and an overflow section or spillway. A drainage gallery is generally provided

in order to relieve the uplift pressure exerted by the seeping water through the body

and foundation of the dam.

Passages constructed either within a dam or in the periphery of the reservoir to

safely pass this excess of the river during flood flows are called Spillways. Ordinarily,

the excess water is drawn from the top of the reservoir created by the dam and

conveyed through an artificially created waterway back to the river. In some cases,

the water may be diverted to an adjacent river valley. In addition to providing

sufficient capacity, the spillway must be hydraulically adequate and structurally safe

and must be located in such a way that the out-falling discharges back into the river

do not erode or undermine the downstream toe of the dam. The surface of the spillway

should also be such that it is able to withstand erosion or scouring due to the very high

velocities generated during the passage of a flood through the spillway. The flood

water discharging through the spillway has to flow down from a higher elevation at

the reservoir surface level to a lower elevation at the natural river level on the

downstream through a passage, which is also considered as a part of the spillway. At

the bottom of the channel, where the water rushes out to meet the natural river, is

usually provided with an energy dissipation device that kills most of the energy of the

flowing water. These devices, commonly called as energy dissipaters, are required to

prevent the river surface from getting dangerously scoured by the impact of the out

falling water.

The purposes of dam construction may include navigation, flood damage

reduction, hydroelectric power generation, fish and wildlife enhancement, water

quality, water supply, and recreation. A general view of gravity dam has been shown

inFigure 1-1.

Examples of Gravity dam: Grand Dixence dam,Nagarjuna Sagar (India)


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1-3Introduction

Figure 1-1 A General view of Gravity Dam

1.1.2Arch Dam

An arch dam is curved in plan, with its convexity towards the upstream side. It

transfers the water pressure and other forces mainly to the abutments by arch action.

An arch dam is quite suitable for narrow canyons with strong flanks which are

capable of resisting the thrust produced by the arch action. The section of an arch dam

is approximately triangular like a gravity dam but the section is comparatively

thinner. The arch dam may have a single curvature or double curvature in the vertical

plane. Generally, the arch dams of double curvature are more economical and are

used in practice. A General View of Arch Dam has been shown in Figure 1-2 .

Examples of Arch dam: Hoover Dam (USA) and Idukki Dam (India).

1.1.3 Buttress dams

Buttress dams are of three types: (i) Deck type, (ii) Multiple-arch type, and

(iii) Massive-head type. A deck type buttress dam consists of a sloping deck

supported by buttresses. Buttresses are triangular concrete walls which transmit the

water pressure from the deck slab to the foundation. Buttresses are compression

members. Buttresses are typically spaced across the dam site every 6 to 30 metre,

depending upon the size and design of the dam. Buttress dams are sometimes called

hollow dams because the buttresses do not form a solid wall stretching across a river

valley. The deck is usually a reinforced concrete slab supported between the

buttresses, which are usually equally spaced. The buttress dams require less concrete


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1-4Introduction

than gravity dams. But they are not necessarily cheaper than the gravity dams because

of extra cost of form work, reinforcement and more skilled labour. The foundation

requirements of a buttress are usually less stringent than those in a gravity dam. A

General View of Buttress Dam has been shown in Figure 1-3.Examples of Buttress type: Bartlett dam (USA) and The Daniel-Johnson Dam

(Canada).

Figure 1-2 A General view of Arch Dam

Figure 1-3 A General view of Buttress Dam


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1-5Introduction

1.2 Case Study of Indira Sagar Polavaram Dam

The Indira Sagar Polavaram Dam is under construction across river Godavari

(second largest river in India), near Polavaram village about 34 km upstream of

Kovvur, Rajahmundry and 42 km upstreamof Sir Arthur Cotton Barrage in Andhra

Pradesh. The Dam is a multipurpose project (Irrigation & Hydroelectric power) and

develops power of 150,000 kW. The Dam is Located at a Longitude of 81 degrees 41

minutes and latitude of 17 degrees 13minutes.

The Reservoir created by the dam has area of 552.63 m2at FRL +45.72 m and

Storage capacity of 192 T.M.C. The Dam has Spillways on the right flank saddle with

crest level at +25.72 m. It has 50 Radial gates each of 15.24 x 12.80 m. The Section of

the Dam is shown in Figure 1-4 as currently proposed by the consultants and is a

preliminary design.

The following is the data corresponding to proposed dam at Polavaram :

Table 1-1 Polavaram Dam details

Total height of the dam 44.00 m

Height of the dam up to Maximum water level 36.02 m

Height of the dam up to Full Reservoir level 35.72 m

Axis of the dam 9.675 m

Base width of the dam 37.51 m

Slope change point distance on upstream side

from the base of the dam

35.72 m

Slope change point distance on downstream side

from the base of the dam

27.90 m

Slope in upstream side 1 in 0.1

Slope in downstream side 1 in 0.85

Elasticity of concrete (EC) 2.2 10 N/m

Elasticity of foundation rock (ER) (Assumed) 6.15 1010N/m2

Poisson ratio (C) of concrete 0.15Specific weight of concrete 2.4 10 kg/m

Specific weight of foundation rock 2.4 10 kg/m


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1-6Introduction

Figure 1-4Cross Section of

Polavaram Dam

The classification of concrete in the dam as proposed by the consultants and

the compressive strength of the concrete are shown below

Table 1-2 Compressive Strength of the Dam Material

Sl.No. Location

Compressive

Strength

(N/mm

2

)

1Concrete in non-overflow section except 2000 mm exterior

thickness on the U/S and filling up crevices in foundation.12.5

2 Concrete in foundation for filling up crevices etc. 12.5

3Concrete in exterior 2000 mm on U/S face of spillway and

non-overflow.16.5

4Concrete all around galleries, elevator shaft and other

openings.

20


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1-7Introduction

1.3 Objectives of the Present Work

Out of the methods available for the stress analysis of the Gravity Dams, The

Two Dimensional Gravity Analysis and The Finite Element Method are selected

for stresses analysis of the Indira Sagar Polavaram Dam. The Objective of the present

study is to carry out the stress analysis of the Indira Sagar Polavaram Dam, to be

constructed are as follows:

1. Carrying out the stress analysis of Indira Sagar Polavaram dam by using

conventional method.

2. Carrying out the stress analysis of Indira Sagar Polavaram dam by using finiteelement method.

3. Comparing the stress analysis results at the base of the dam for both the methods.

Stress Analysis

Gravity Method Finite Element MethodComparsion Of

Results


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2-1Literature Review

Zienkiewicz and Cheung (1967) dealt with the non- homogeneity of

foundation rock with varying elasticitysand with non-uniform thickness of buttress

dam. It was concluded that the finite element method of analysis is a more exact

approach with a little or no additional cost.

Dr S.S Saini and R.P Singh (1984) analysed a 200m high concrete gravity dam

located in a symmetrical valley considering the three dimensional behaviour using the

finite element technique. The effect of variation of valley width on the stresses and

displacements in the dam has been investigated and the results are compared with

those obtained from a two dimensional analysis. It is concluded that three dimensional

behaviour of the dam is exhibited for valleys with valley width to height ratio of less

than 6. For wider valleys, a two dimensional analysis is adequate for the design. This

fact can be usefully employed for an economical design of the dams located in narrow

valleys, considering the three dimensional behaviour.

Surya Rao (1989) dealt the analysis and design of hydraulic structures. He

solved the formulation for finding the stress condition at the faces, Equilibrium

equations at interior points, geometric analysis and algebraic determination of shears

and geometric analysis and algebraic determination of horizontal stresses. He worked

out the numerical example in detail for finding out the horizontal normal stress,

vertical normal stress and shear stress at the certain distance of the dam. The

assumptions of conventional method are vertical normal stress varies linearly,

horizontal normal stress varies cubically and shear stress varies quadritically

Ramamurthy (2003) had attempted to assess the reliability of predicting the

uniaxial compressive strength and the corresponding modulus of rock by some of the

approaches in vogue. The elasticity of rock is estimate from the influence of theuniaxial compressive strength of the intact rock and the modulus of rock mass has

CHAPTER 2 LITERATURE REVIEW


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been calculated. The estimation of the compressive strength and the modulus of rock

mass,effectively vary from 0.1 to 10.

Bollaert and Mason(2006) has explained about scour prediction of srisailam

dam.They have analyzed the rock properties of srisailam dam.The srisailam has

designed two different rocks i.e. quartzite and shale.They have given the uniaxial

compressive strength of srisailam foundation rock.

2.1 Gravity Dams

The first dams were believed to have been built in ancient Mesopotamia

around 5000 BC, these dams were used to control the water level, for Mesopotamia's

weather affected by theTigris andEuphrates rivers, and could be quite unpredictable.

Thegravity dams were the first type of dam ever to be constructed, and were made

from stone bricks orconcretebricks.

2.1.1Forces to be resisted by the Dam

The force to be resisted by a gravity dam as recommended by IS 6512-1984

falls into two categories as given below:

a) Forces, such as weight of the dam and water pressure, which are directly

calculable from the unit weights of the materials and properties of fluid pressures.

b) Forces, such as uplift, earthquake loads, silt pressure and ice pressure,

which can only be assumed on the basis of experience and available data.

In general the following forces act on the dam:

Dead load

Reservoir and Tail water loads

Uplift pressure

Earthquake forces

Earth and silt pressures

Ice pressure

Wind pressure

Wave pressure Thermal loads
http://en.wikipedia.org/wiki/Tigrishttp://en.wikipedia.org/wiki/Euphrateshttp://www.ritchiewiki.com/wiki/index.php/Concretehttp://www.ritchiewiki.com/wiki/index.php/Concretehttp://en.wikipedia.org/wiki/Euphrateshttp://en.wikipedia.org/wiki/Tigris


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Dead Load2.1.1.1

The dead load to be considered comprises the weight of the concrete or

masonry or both plus the weight of such appurtenances as piers, gates and bridges.

The unit weight of concrete and masonry varies considerably depending upon the

various materials that go to make them. It is essential to make certain that the assumed

unit weight for concrete/masonry or both can be obtained with the available

aggregates/ stones. The total weight of dam acts at the centre of gravity of the dam

section. The unit weight of concrete is taken as 24 kN/m3.

Reservoir and Tail water loads2.1.1.2

Water pressure is the most major external force acting on gravity dams. The

horizontal water pressure exerted by the weight of water stored on the upstream and

downstream sides of the dam can be estimated by the linear distribution of the static

water pressure acting normal to the face of the dam.

Uplift pressure2.1.1.3

Uplift forces occur as internal pressures in pores, cracks and seams within the

body of the dam, at the contact between the dam and its foundation and within the

foundation. Uplift pressure distribution in the body of the dam shall be assumed to

have an intensity which at the line of the formed drains exceeds the tail water head by

one-third the differential between reservoir level and tail water level. The pressure

gradient shall then be extended linearly to heads corresponding to reservoir level and

tail water level. The uplift shall be assumed to act over 100 percent of the area. For

the loading combinations F and G to be discussed in section2.1.2,the uplift shall be

taken as varying linearly from the appropriate reservoir water pressure at the upstream

face to the appropriate tail water pressure at the downstream face. The uplift is

assumed to act over 100 percent of the area.

Earthquake forces2.1.1.4

Earthquake produces waves, which are capable of shaking the earth upon

which the gravity dams rest, in every possible direction. The effect of an earthquake

is, therefore, equivalent to imparting acceleration to the foundations of the dams in thedirection in which the wave is traveling at the moment. Generally, an earthquake


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induces horizontal acceleration (h) and vertical acceleration (v). Earthquake

loadings should be checked for horizontal as well as vertical earthquake accelerations.

While earthquake acceleration might take place in any direction, the analysis should

be performed for the most unfavourable direction. For dams upto 100m height the

seismic coefficient method of analysis is recommended by IS 1893-1984 to be used in

determining the resultant location and sliding stability of dams, horizontal seismic

coefficient shall be taken as 1.5 times seismic coefficient, at the top of the dam

reducing linearly to zero at the base. Vertical seismic coefficient shall be taken as 0.75

times the value at the top of the dam reducing linearly to zero at the base. In Seismic

Coefficient Method the design value of horizontal seismic coefficient shall be

computed as given by the following expression

h = Io ( 2-1 )

where is a coefficient depending on soil-foundation system, I is a factor

depending on the importance of the structure, o is basic horizontal seismic

coefficient according to zone factor of IS 1893-2000.

2.1.1.4.a

Effect of Vertical Acceleration (v)

Vertical acceleration may either act downward or upward. When it acts in the

upward direction, then the foundation of the dam will be lifted upward and becomes

closer to the body of the dam, and thus the effective weight of the dam will increase

and hence, the stress developed will increase. The net effective weight of the dam is

given by

W -

kvg = W(1kv) ( 2-2 )

where, Wis the total weight of the dam, kvis the fraction of gravity adopted for

vertical acceleration

When the vertical acceleration acts downward, the foundation shall try to

move downward away from the dam body; thus, reducing the effective weight and the

stability of the dam, and hence is the worst case for design. In other words, vertical


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acceleration reduces the unit weight of the dam material and that of water to (1 kv)

times their original unit weights.

2.1.1.4.b Effects of Horizontal Acceleration (h)

The horizontal acceleration causes

Hydrodynamic pressure2.1.1.4.b.i

Due to horizontal acceleration of the foundation and dam there is an

instantaneous hydrodynamic pressure exerted against the dam in addition to

hydrostatic forces. The direction of hydrodynamic force is opposite to the direction of

earthquake acceleration. Based on the assumption that water is incompressible, the

hydrodynamic pressure at depth y below the reservoir surface shall be determined as

follows: p= Cshwh ( 2-3 )

wherepis hydrodynamic pressure in kg/m2at depth y, Csis coefficient which

varies with shape and depth, h is design horizontal seismic coefficient, w is unit

weight of water in kg/m3, andhis the depth of reservoir in m.

The approximate values of Cs, for dams with vertical or constant upstream

slopes may be obtained as follows

s m ( ) ( ) ( 2-4 )

Where Cmis maximum value of Csobtained from

Figure 2-1 as given in IS 1893-1984,yis the depth below the surface and h is

the depth of the reservoir.

The approximate values of total horizontal shear and moment about the centre

of gravity of a section due to hydrodynamic pressure are given by the relations:

Vh = 0.726py ( 2-5 )

Mh= 0.299py2 ( 2-6 )


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Figure 2-1 Maximum Value of Pressure Coefficient

Horizontal Inertia Force2.1.1.4.b.ii

In addition to exerting the hydrodynamic pressure, the horizontal acceleration

produces an inertia force into the body of the dam. This force is generated to keep the

body and the foundation of the dam together as one piece. The direction of the

produced force will be opposite to the acceleration imparted by the earthquake. The

Horizontal Inertia force is given by kh. For the reservoir full condition, the worst

case occurs when the earthquake acceleration act towards the upstream direction and

for reservoir empty condition, the worst case occurs when the earthquake acceleration

act towards the downstream direction.


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Earth and silt pressures2.1.1.5

Gravity dams are subjected to earth pressures on the downstream and

upstream faces where the foundation trench is to be backfilled. Except in the abutment

sections in specific cases and in the junctions of the dam with an earth or rock

embankment, earth pressures have usually a minor effect on the stability of the

structure and may be ignored.

Ice pressure2.1.1.6

Ice expands and contracts with changes in temperature. In a reservoir

completely frozen over, a drop in the air temperature or in the level of the reservoir

water may cause the opening up of cracks which subsequently fill with water and

freezes into solid. When the next rise in temperature occurs, the ice expands and, if

restrained, it exerts pressure on the dam. In some cases the ice exerts pressure on the

dam when the water level rises. The problem of ice pressure in the design of dam is

not encountered in India except, in a few localities.

Wind pressure2.1.1.7

Wind pressure does exist but is seldom a significant factor in the design of a

dam, but the Wind causes waves in the reservoir which exert pressure on dam. Wind

loads may, therefore, be ignored.

Wave pressure2.1.1.8

In addition to the static water loads the upper portions of dams are subject to

the impact of waves. Wave pressure against massive dams of appreciable height is

usually of little consequence. The force and dimensions of waves depend mainly on

the extent and configuration of the water surface, the velocity of wind and the depth

of reservoir water. The height of wave is generally more important in the

determination of the free board requirements of dams to prevent overtopping by wave

splash. An empirical method based upon research studies on specific cases has been

recommended by T. Saville for computation of wave height. It takes into account the

effect of the shape of reservoir and also wind velocity over water surface rather than

on land by applying necessary correction. It gives the value of different wave heights

and the percentage of waves exceeding these heights so that design wave height for

required exceedance can be selected.


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Thermal loads2.1.1.9

Measures for temperature control of concrete in solid gravity dams are

adopted during construction. Yet it is noticed that stresses in the dam are affected due

to temperature variation in the dam on the basis of data recorded from the

thermometers embedded in the body of the dam. The cyclic variation of air

temperature and the solar radiation on the downstream side and the reservoir

temperature on the upstream side also affect the stresses in the dam. Even the

deflection of the dam is maximum in the morning and it goes on reducing to a

minimum value in the evening. The magnitude of deflection is also affected

depending on whether the spillway is running or not. It is generally less when

spillway is working than when it is not working. While considering the thermal load,

temperature gradients are assumed depending on location, orientation, surrounding

topography, etc.

Figure 2-2 Representation of Various Forces acting on Dam


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2.1.2Load Combinations

Gravity dam design should be based on the most adverse load combination.

The IS code 6512-1984 recommends that a gravity dam should be designed for the

most adverse load condition of the seven given type using the safety factors

prescribed. The seven types of load combinations are:

Load Combination A (Construction Condition) - Dam completed but

no water in reservoir and no tailwater.

Load Combination B (Normal Operating Condition ) - Full reservoir

elevation normal dry weather tailwater, normal uplift; ice and silt ( if

applicable )

Load Combination C ( Flood Discharge Condition) - Reservoir at

maximum flood pool elevation, all gates open, tailwater at flood

elevation, normal uplift, and silt ( if applicable ).

Load Combination D - Combination A, with earthquake.

Load Combination E - Combination B, with earthquake but no ice.

Load Combination F - Combination C, but with extreme uplift (drains

inoperative).

Load Combination G - Combination E, but with extreme uplift (drains

inoperative).

2.1.3 Requirements for Stability

The design of gravity dams shall satisfy the following requirements of

stability as per IS 6512-1984:

The dam shall be safe against sliding on any plane or combination of

planes within the dam, at the foundation or within the foundation;

The dam shall be safe against overturning at any plane within the dam,

at the base, or at any plane below the base; and

The safe unit stresses in the concrete or masonry of the dam or in the

foundation material shall not be exceeded.


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Resistance against Overturning2.1.3.1

Before a gravity dam overturns bodily, other types of failures may occur, such

as cracking of the upstream material due to tension, increase in uplift, crushing of toe

material and sliding. A gravity dam is, therefore, considered safe against overturning

if the criteria of no tension on the upstream face, the resistance against sliding as well

as the quality and strength of concrete/masonry of the dam and its foundation is

satisfied assuming the dam and foundation as a continuous body.

Compression or Crushing2.1.3.2

A dam may fail by the failure of its materials, i.e., the compressive stresses

produced may exceed the allowable stresses, and the dam material may get crushed.

The vertical direct stress distribution at the base is given by

yy = ( 2-7 )where, e is the eccentricity of the resultant force from the centre of the base, the

maximum value of which can be permitted on either side of the centre of the base is

equal to B/6; Vis the total vertical force andBis the base width of the dam.

Tension2.1.3.3

No tensile stress shall be permitted at the upstream face of the dam for load

combination B (refer Section2.1.2). Masonry and concrete gravity dams are usually

designed in such a way that no tension is developed anywhere, because these

materials cannot withstand sustained tensile stresses. However tension, may be

permitted in load combinations other than load combination B and their permissible

values shall not exceed the values as specified in Is Code 6512-1984 and the

permissible stresses are:

Table 2-1 Values of Permissible Tensile Stresses in Concrete

Load Combination Permissible Tensile Stress

C 0.01fc

E 0.02fc

F 0.02fc

G 0.04fc


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Sliding2.1.3.4

Many of the loads on the dam are horizontal or have horizontal components

which are resisted by frictional or shearing forces along horizontal or nearly

horizontal planes in the body of the dam and the foundation. The stability of a dam

against sliding is evaluated by comparing the minimum total available resistance

along the critical path of sliding (that is, along that plane or combination of planes

which mobilizes the least resistance to sliding) to the total magnitude of the forces

tending to induce sliding. Sliding or shear failure will occur when the net horizontal

force above any plane in the dam or at the base of the dam exceeds the frictional

resistance developed at that level. The factor of safety against sliding (F.S.S.) is given

by

F.S.S = ( 2-8 )

where V is the shear resistance in which V is the total vertical forces, is

the coefficient of friction between the dam base and foundation, which varies from

0.65 - 0.75, and H is the total external horizontal forces.

2.1.4Stress Analysis

The stability of a gravity dam can be approximately and easily analysed by

Two Dimensional Analytical Method and three dimensional methods such as slab

analogy method, trial load twist method, Finite Element Method (FEM), or by

experimental studies on models.

Two Dimensional Analytical Method2.1.4.1

The Gravity Method of Stability Analysis is used a great deal for preliminary

studies of gravity dams, depending on the phase of design and the information

required. The gravity method is also used for final designs of straight gravity dams in

which the transverse contraction joints are neither keyed nor grouted. This method is

recommended by IS 6512-1984 for the stability analysis of the solid gravity dams.

The assumptions of conventional method are vertical normal stress varies linearly,

horizontal normal stress varies cubically and shear stress varies quadritically.


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Trial Load methods of analysis2.1.4.2

For dams in which the transverse joints are keyed and grouted, the Trial Load

twist analysis should be used. A gravity dam may be considered to be made up of a

series of vertical cantilever elements from abutment to abutment. If the cross-canyon

profile is narrow with steep sloping walls, each cantilever from the centre of the dam

towards the abutments will be shorter than the preceding one. Consequently, each

cantilever will be deflected less by the water load than the preceding one and more

than the succeeding one. If the transverse contraction joints in the dam are keyed, and

regardless of whether grouted or ungrouted, the movements of each cantilever will be

restrained by the adjacent ones. The longer cantilever will tend to pull the adjacent

shorter cantilever forward and the shorter cantilever will tend to hold it back. The

interaction between adjacent cantilever elements causes torsional moments, or twists,

which materially affect the manner in which the water load is distributed between the

cantilever elements in the dam. This changes the stress distribution from that found by

the ordinary gravity analysis in which the effects of twist, as well as deformation of

the foundation rock, are neglected. All straight gravity dams having keyed transverse

contraction joints should therefore be treated as three-dimensional structures and

designed on that basis.

The Finite element method2.1.4.3

The Finite element method (FEM) is a numerical method for determining

responses (deformation, strain, stress, etc.) of a body under external loads. The finite-

element method (FEM) uses a concept of piecewise approximation. In theory, the

elements can be of different shapes and sizes. Until developing FEM it was almost

difficult to calculate point to point responses (approximately) of a body of any

geometric shape and any complex type of loading conditions. In this method, the

entire dam body is divided by using equivalent system of small triangular element for

obtaining responses within and boundary (node) of the element. This method

determines first the global deformations at the nodes of the element then determines

successively other responses such as strains, stresses, etc. Nowadays, this method is

available as a commercial FE programming or software (SAP 2000, ANSYS, etc.) for

solving large problems. Most of the FE programming used for either general purpose

or special purpose follows the same basic procedure of finite element method.


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2.2 Two Dimensional Analytical Method

The assumptions of conventional method are vertical normal stress varies

linearly, horizontal normal stress varies cubically and shear stress varies quadritically.

This method is recommended by IS 6512-1984 for the stability analysis of the Solid

gravity dams.

2.2.1Assumptions

The assumptions made in the Two Dimensional Analytical Method are:

The Vertical normal stress varies linearly, horizontal normal stress

varies cubically and shear stress varies quadritically

The dam is considered to be composed of a number of cantilevers,

each of which is 1 m thick and acts independent of the other

No loads are transferred to the abutments by beam action

The Foundation and the Dam behave as a single unit

The materials in the dam body and foundation are isotropic and

homogeneous

The stresses developed in the dam and foundation are within the elastic

limits and no movement of the foundation is caused due to the

transference of loads.

2.2.2Analysis Procedure

The steps involved in the analysis of gravity dam by analytical Two

Dimensional Analytical Method are:

Consider unit length of the dam;

Work out the magnitude and directions of all the vertical and

horizontal forces acting on the dam and their algebraic sum;

Determine the lever-arm of all these forces about the toe;

Determine the moments of all these about the toe and find out the

algebraic sum of all those moments;

Find out the location of the resultant force by determining its distance

from the toe;


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Find out the eccentricity of the resultant, which must be less than B/6

in order to ensure that no tension is developed anywhere in the dam;

Determine the vertical stresses at the toe and heel;

Determine the maximum normal stresses at the toe and heel;

Determine the factor of safety against overturning, sliding and

crushing.

2.2.3Stresses in the Dam

The stresses at the Toe (Downstream side) and Heel (Upstream side) of the

dam are determined as follows:

The vertical normal stress at toe and heel are determined by

(yy)d = ( 2-9 )

(yy)u = ( 2-10 )

The horizontal normal stress at toe and heel are determined by

(

xx)d =p+ (

yydp) tan

2 ( 2-11 )

(xx)u = p + (yyu - p) tan2 ( 2-12 )The Shear Stress at Toe & Heel are determined by

(xy)d = (yydp) tan ( 2-13 )(xy)u= (pyyu) tan ( 2-14 )

The Principal Stress is determined by

11=yy xx

yy-xx ) 2 xy2 ( 2-15 )

where, e is the eccentricity of the resultant force from the centre of the base,

Vis the total vertical force, andBis the base width of the dam,pis pressure due to

upstream water, p is pressure due to water on upstream side, tan is downstream

slope, tan is upstream slope.


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The variation of vertical normal stress, horizontal normal stress and shear

stress with respectxis given by

yy = a + b x ( 2-16 )

xy =c + dx+ e x2 ( 2-17 )

xx = f+g x +x

2+x

3 ( 2-18 )

where x is the distance at the base of the dam from the upstream side to

downstream side and a, b, c, d, e, f, g, & h are constants and are determined by

substituting the boundary conditions that is stresses at the toe & heel and by using the

following equations

xydx = H ( 2-19 )g + hx + ix2 = - (d + 2 ex) (dx/dy) ( 2-20 )

2.3 Finite Element Method (FEM)

The basic concept of FEM is that a body or a structure may be divided into

smaller elements of finite dimensions called finite elements. The original body or the

structure is then considered as an assemblage of these elements connected at a finite

number of joints called nodes. The properties of the elements are formulated and

combined to obtain the solution for the entire body or structure. In the displacement

formulation widely adopted in FEM analysis, simple functions known as shape

functions are chosen to approximate the variation of displacement within an element

in terms of the displacement at the nodes of the element. The strains and stresses

within an element will also be considered in terms of the nodal displacements. Then

the principle of virtual forces is used to derive the equations of equilibrium for the

element with nodal displacements as unknowns. The equations of equilibrium of the

entire structure are then obtained by combining the equilibrium equation of each

element such that the continuity of displacement is ensured at each node where the

elements are connected. The necessary boundary conditions are imposed and the

equations of equilibrium are solved for the nodal displacements.


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2.3.1Two-Dimensional Stress Distributions

In many practical situations, the geometry and loading will be such that the

problems can be reduced to one or two dimensional problems without much loss of

accuracy. The two dimensional idealizations are discussed below.

Plane Stress2.3.1.1

The Plane Stress condition is characterised by very small dimensions in one of

the normal directions. In these cases the stress components x,xz , yz are zero and itis assumed that no stress component varies along the thickness.

Plane Strain2.3.1.2

The Plane Strain condition is characterised by a long body whose geometry

and loading do not vary significantly in the longitudinal direction. In these situations,

a constant longitudinal displacement corresponding to a rigid body translation and

displacements in linear z corresponding to rigid body rotation does not result in strain.

In these cases the stress components x,xz , yz are zero. The Linear Constitutiverelation for elastic isotropic material derived from the Hooks law is given by

xy

y

x

xy

y

xE

2

2100

01

01

211 ( 2-21 )

and z= (x+y) ( 2-22 )

Some of the typical examples are the long retaining walls and tunnels

subjected to uniform pressure along their length.

2.3.2Shape Functions

The Shape Function directly expresses the displacement at any point within

the element in terms of the nodal displacements. The displacements at any point (x, y)

within the element can be expressed in the following form,


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e

n

n

n

ne

ii dN

v

u

v

u

N

N

N

N

N

NdN

v

uu

:

:0

.............

.......

0

0

0

1

1

2

2

1

1 ( 2-23 )

Where {u} is the vector of displacements at any point (x, y), [N] is the shape

function matrix, {de} is the vector of nodal displacements. The properties of the

shape functions are,

n

iiN1 1 , where n is the number of nodes used to define the element.

2.3.3Strain Displacement Relations

For small strains, the strain displacement relations and the theory of elasticity

could be used to derive the relation between the strains at any point inside an element

in terms of the nodal displacements. The Strains can be written in a matrix form as,

( 2-24 )

where [L] is the matrix of linear operators as shown in the square

brackets in the above equation. The strains within an element can be expressed as

follows by substituting equation (2-11),

ee dBdNLuL ( 2-25 )

Where [B] is the strain displacement matrix and is the product of [L] and [N]

matrices and is obtained as,

uLv

u

xy

y

x

xy

y

x

0

0


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.........

.........0

........0

11

1

1

xN

yN

y

Nx

N

NLB ( 2-26 )

The stresses {} can be computed from the strains by substituting equation

(2-25) as follows:

{} = [C]{} = [C][B]{de} ( 2-27 )

where [C] is the constitutive matrix relating the stresses and strains, {} is the

strain vector.

2.3.4Convergence and Compatibility Requirements of Elements

2.3.4.1.a Convergence Requirements of Elements

The FEM provides a numerical solution to a complex program. It may,

therefore, be expected that the solution must converge to the exact solution under

certain circumstances. In the FEM, with increase in the degrees of freedom the

numerical solution tends to the exact solution, which is called the convergence. As the

finite element subdivision of the structure is made finer, the solution may not be

always converging. To make the solution to converge the displacement solution to the

exact solution, the following conditions must be met.

The displacement function must be continuous within the element.

The displacement function must be capable of representing rigid body

displacements of the element.

The displacement function must be capable of representing constant

strain states within the element.

2.3.4.1.b Compatibility Requirements of Elements

The displacements must be compatible within two adjacent elements. That is

when the elements deform there must not be any discontinuity between elements (i.e

elements must not overlap or separate). The elements which conform to the


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compatibility requirements are called as compatible elements and the elements which

violate the compatibility requirements are known as incompatible elements. The

elements which satisfy all the three convergence requirements and compatibility

conditions are called compatible or conforming elements.

2.3.5Stiffness equation of the Element

The Potential energy of a system can be written as follows:

= v s

sss

v

dswZvYuXdVwzvyuxdU ( 2-28 )

Where u,v,w are the displacements along x, y and z direction respectively, -

zyx ,, are the components of body forces in x, y and z directions, Xs, Ys, Zs, are

surface tractions along x, y, and z direction, U is strain energy, V is Volume and s is

surface area. The strain energy density (strain energy per unit volume) can be

evaluated as,

222

zzyyxx

dV

dU

( 2-29 )

dU = dVdV T

z

y

x

zyx

2

1

2

1 ( 2-30 )

Hence by substituting equation ( 2-30) into equation (2-28), we have

v S

T

V

TTdspudVxudV

2

1 ( 2-31 )

Now, by substituting equations (2-23 & 2-25) into equation (2-31) and by

the principle of stationary potential energy, we have

s

T

V

TdspNdVxNqK ( 2-32 )

which can be written as


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[K]{q} = {Q} ( 2-33 )

Comparing equation (2-21 & 2-20) we get [K] the element stiffness matrix as,

[K] = v [B]T

[C][B]dV ( 2-34 )

and {Q} is the element load vector,

[Q] = dspNdVxNV s

TT

(2-35)


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3-1Methodology

CHAPTER 3 METHODOLOGY

3.1 Stress Analysis Using Finite Element Method

The analysis in the present study is carried out by equivalent static analysis.

3.1.1Finite Element Analysis Program

SAP 2000 is a registered trade mark of Computers & Structures, Inc.

University Avenue, Berkeley, California. SAP 2000

SAP 2000 is a full-featured program, which can be used for the simplest

problems or the most complex projects. The SAP 2000 structural analysis program

offers the features such as static and dynamic analysis; linear and nonlinear analysis;

dynamic seismic analysis and static pushover analysis; vehicle live-load analysis for

bridges; geometric nonlinearity, including P-delta and large-displacement effects;

Frame and shell structural elements, including beam-column, truss; membrane, and

plate behaviour; two-dimensional plane and axisymmetric solid elements; Three-dimensional solid elements. SAP 2000 provides manuals, which are designed to help

users quickly become productive with SAP 2000.

Structural Analysis and Design3.1.1.1

The steps required to analyse and design a structure using SAP 2000 are

represented by a flow chart as shown below:


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3-2Methodology

Figure 3-1 Flow Chart showing Modelling & Analysis in SAP 2000

MathematicalModelling

Define

Materials

Elements(Plane-Strain)

Load Patterns

Load Cases

LoadCombinations

BoundaryConditions

Discritisze

Assign

Materials to

Elements

Loadings

Analyse

Set AnalysisOptions

Run theAnalysis

ResultsStresses in the

element


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3-3Methodology

Plane Element3.1.1.2

The Plane element also called as 2D solid element is used to model plane-

stress and plane-strain behaviour in two-dimensional solids. The plane element is a

three or four noded element for modelling two-dimensional solids of uniform

thickness. It is based upon an isoparametric formulation that includes four optional in

compatible bending modes. The incompatible bending modes significantly improve

the bending behaviour of the element if the element geometry is of a rectangular form.

Structures that can be modelled with this element include:

Thin, planar structures in a state of plane stress

Long, prismatic structures in a state of plane strain

The stresses and strains are assumed not to vary in the thickness direction.

Each Plane element has its own local coordinate system for defining Material

properties and loads, and for interpreting output. Temperature-dependent, orthotropic

material properties are allowed. Each element may be loaded by gravity (in any

direction); surface pressure on the side faces; pore pressure within the element; and

loads due to temperature change.

Each plane element (and other types of area objects/elements) may have either

of the following shapes, as shown inFigure 3-2 &Figure 3-3 .

Quadrilateral, defined by the four joints j1, j2, j3, and j4 (Figure

3-2Error! Reference source not found.)

Triangular, defined by the three jointsj1,j2, andj3. (Figure 3-3 )

The quadrilateral formulation is the more accurate of the two.


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3-4Methodology

Figure 3-2 Quadrilateral Element

Figure 3-3 Triangular Element


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3-5Methodology

A 2 x 2 numerical integration scheme is used for the Plane. Stresses in the

element local coordinate system are evaluated at the integration points and

extrapolated to the joints of the element.

Local Coordinate System3.1.1.3

Each plane element has its own element local coordinate system used to define

material properties, loads and output. The axes of this local system are denoted 1, 2

and 3. The first two axes lie in the plane of the element with an orientation that user

specify; the third axis is normal.

Self-Weight Load3.1.1.4

Self-Weight Load activates the self-weight of all elements in the model. For a

Plane element, the self-weight is a force that is uniformly distributed over the plane of

the element. The magnitude of the self-weight is equal to the weight density

multiplied by the thickness. Self-Weight Load always acts downward, in the global

Z direction.

Surface Pressure Load3.1.1.5

The Surface Pressure Load is used to apply external pressure loads upon any

of the three or four side faces of the Plane element. Surface pressure always acts

normal to the face. Positive pressures are directed toward the interior of the element.

Pore Pressure Load3.1.1.6

The Pore Pressure Load is used to model the drag and buoyancy effects of a

fluid withina solid medium, such as the effect of water upon the solid skeleton of a

soil. Scalar fluid- pressure values are given at the element joints by Joint Patterns, and

interpolated over the element. The total force acting on the element is the integral of

the gradient of this pressure field over the plane of the element. This force is

apportioned to each of the joints of the element. The forces are typically directed from

regions of high pressure toward regions of low pressure.

Stresses and Strains3.1.1.7

The Plane element models the mid-plane of a structure having uniform

thickness, and whose stresses and strains do not vary in the thickness direction.


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3-6Methodology

Plane-stress is appropriate for structures that are thin compared to their planar

dimensions. The thickness normal stress (z) is assumed to be zero. The thickness

normal strain ( ) may not be zero due to Poisson effects. Transverse shear stresses

(xy, xz) and shear strains (xy, xy) are assumed to be zero. Displacements in thethickness (local 3) direction have no effect on the element.

Plane-strain is appropriate for structures that are thick compared to their planar

dimensions. The thickness normal strain ( ) is assumed to be zero. The thicknessnormal stress (

z) may not be zero due to Poisson effects. Transverse shear stresses

(xy

, xz

) and shear strains (xy

, xy

) are dependent upon displacements in the

thickness (local 3) direction.

3.1.2Analysis Cases

General3.1.2.1

An Analysis Case defines how the loads are to be applied to the structure (e.g.,

statically or dynamically), how the structure responds (e.g., linearly or nonlinearly),

and how the analysis is to be performed (e.g., modally or by direct-integration.). User

may define as many named analysis cases of any type that he wishes. For analysingthe model, user may select which cases are to be run and may also selectively delete

results for any analysis case. The results of linear analyses may be superposed, i.e.,

added together after analysis.

Each different analysis performed is called an Analysis Case. For each

Analysis Case user has to define following type of information:

Case name3.1.2.2

This name must be unique across all Analysis Cases of all types. The case

name is used to call analysis results displacements, stresses, etc., for creating

Combinations, and sometimes for use by other dependent Analysis Cases.

Analysis type3.1.2.3

This indicate the type of analysis (static, response-spectrum, time history,

etc.), as well as available options for that type (linear, nonlinear, etc.).


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3-7Methodology

Loads applied3.1.2.4

For most types of analysis, specify the Load Cases that are to be applied to the

structure. Additional data may be required, depending upon the type of analysis being

defined.

Running Analysis Cases3.1.2.5

After defining a structural model and one or more Analysis Cases, run the

Analysis Cases to get results for display, output, and design purposes. When an

analysis is run, the program converts the object-based model to finite elements, and

performs all calculations necessary to determine the response of the structure to the

loads applied in the Analysis Cases. The analysis results are saved for each case for

subsequent use.

Model Definition and Analysis Results Tabular Data3.1.2.6

Model definition data include all input components of the structural model

(properties, objects, assignments, loads, analysis cases, design settings, etc.), as well

as any options you have selected, and named result definitions you have created.

Model definition data are always available, whether or not analyses have been run or

design has been performed. Analysis results data include the deflections, forces,

stresses, energies, and other response quantities that can be produced in the graphical

user interface. These data are only available for analysis cases that have actually been

run. These tables can be edited, displayed, exported, imported, and printed by using

definition data tables on-screen in the graphical user interface, or export and import in

one of the following formats:

Microsoft Access database

Microsoft Excel spread sheet

Plain (ASCII) text

Whereas the analysis results tables cannot be edited or imported, but

displayed, exported, and printed on-screen in the graphical user interface, or export

and import into the above formats.


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3-8Methodology

3.1.3FEM Model

Two dimensional finite element (plane-strain) is adopted for analysis and

finite element model is prepared in SAP 2000 .Figure 3-4;

Figure 3-5;andFigure 3-6 shows the finite element mesh generated to model

dam and foundation. 174 area elements are used to represent the dam and 456 area

elements for foundation geometry. The base width of the dam at foundation level is

37.50 m and height of dam is 44 m. The size of foundation block is 137.50 m wide

and 50 m deep.


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3-9Methodology

Figure 3-4 Finite Element Model of the Dam


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3-10Methodology

Figure 3-5 Finite Element Model of Foundation


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3-11Methodology

Figure 3-6 Finite Element Model of Dam


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3-12Methodology

Modelling In SAP 20003.1.3.1

From studying data of dam, extreme outer points of the geometry are defined

which is necessary for creating dam profile by using area element. By using Edit >

Mesh Area command, geometry is discretised into finer mesh as shown inFigure 3-6.

Material Properties3.1.3.2

On the Define menu, click Materials, which displays the materials dialog box

with default materials like concrete and others. Then assign properties of concrete like

Modulus of Elasticity, Poissons Ratio etc. as mentioned earlier. Also define ROCK

properties and assign the properties for foundation.

Figure 3-7 Define Material Command Box


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3-13Methodology

Define Plane-Strain Elements3.1.3.3

When we create geometry by using area element, by default the software

considers it as a shell element. In this work plane elements are used so that to convert

shell element into plane element, use define Menu > Area Section > Modify Option,

which displays Area section dialog box. Define two types of plane elements dam,

rock each having different material i.e. concrete, and rock respectively. Also assign

the unit thickness and define problem as plane strain problem.

Figure 3-8 Define Area Section Command Box

After defining two types of plain-strain elements dam, rock assign them to

model to corresponding dam, and rock respectively by using assign>area>section

option.


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3-14Methodology

Boundary Condition for Foundation3.1.3.4

Assign fixed supports at base and at sides of foundation block by using

Assign > Restraints option.

Figure 3-9 Joint Restraint Command Box

Loads3.1.3.5

The Loads such as Horizontal Hydrostatic Loads, Uplift are to be applied by

defining the Joint Load pattern as shown in and then are to be applied as Surface

Pressure and Pore pressure respectively.

Figure 3-10 Joint

Pattern Command Box


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3-15Methodology

Load Cases3.1.3.6

The Following Load Patterns, Load Cases as shown inFigure 3-11,andFigure

3-12 and Load Combinations according to IS 6512-1984 have been defined.

Analysing in SAP 20003.1.3.7

To run analysis first specify that analysis is a two dimensional analysis from

analysis > set analysis option. Then run the analysis by using analysis > run

analysis option.

Figure 3-13 Run Analysis Command Box

Figure 3-11 Defined Load Patterns Figure 3-12 Defined Load Cases


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3-16Methodology

Input and Output Table3.1.3.8

In Sap 2000 software Input information of model in form of tables obtain from

display > show model definition table. A table obtained has been shown inTable 3-1

Similarly output information in form of tables obtain from display > analysis

result tables.

Table 3-1 An Example of SAP 2000 Input Table Showing Area loads (pore)

TABLE: Area Loads - Pore Pressure

Area Load Pattern Pressure Joint Pattern

Text Text kN/m2 Text

356 uplift 1 pore2356 uplift mwl 1 Pore MWL1

356 uplift mwl 1 Pore MWL

357 uplift 1 pore2

357 uplift mwl 1 Pore MWL1


358 uplift 1 pore2



359 uplift 1 pore2



360 uplift 1 pore2



361 uplift 1 pore2



362 uplift 1 pore2

362 uplift mwl 1 Pore MWL1362 uplift mwl 1 Pore MWL

363 uplift 1 pore2



364 uplift 1 pore2



365 uplift 1 pore2



366 uplift 1 pore2


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3-17Methodology

3.1.4 Validation of the modelling procedure

For validation purpose the Finite element analysis using SAP 2000 is

carried out for a gravity dam as carried out by Saini (7).

Figure 3-14 FEM Discretization on Saini dam

The results vertical normal stress obtained at the base of the dam are tabulated

inTable 3-2 .

The numerical data is given for validation problem are

Elasticity of concrete ( EC ) = 2.2 1010N/m2

Elasticity of foundation rock ( ER ) = 2.2 1010N/m2

Poisson ratio () of concrete = 0.15Poisson ratio () of foundation rock = 0.15Specific weight of concrete = 2.4 104N/m3

Specific weight of foundation rock = 2.4 104N/m3

Specific gravity of water = 10000 N/m3


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3-18Methodology

Comparison of Vertical Normal Stress variation by Saini and present study at

base of the dam are presented in TABLE 3-2 and the results are compared with those

obtained by saini and are found to be satisfactory.

Table 3-2 Comparison of Vertical Normal stress of Saini and present study

Analysis Elements Nodes Heel (N/m2) Toe (N/m2)

Saini 18 73 (-) 90 104 (-) 230 104

Present 18 29 (-) 86 10 (-) 202 10


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4-1Results & Discussions

4.1 Results

The results for Two Dimensional Analytical Method and FEM are presented

for all the load combinations calculated according to IS 6512-1984. The results are

tabulated for the maximum & minimum vertical normal stress, principal stress and the

shear stress.

4.2

Two Dimensional Analytical Method

The stress analyses for the load combinations A to G are carried out. The

results are tabulated as shown below in Table 4-8 containing the resisting and

opposing moments; vertical and horizontal forces; vertical, principal and shear

stresses, factor of safety against sliding and overturning.

The vertical and horizontal loadings due to the various loadings such as self-

weight, hydrostatic, uplift, earthquake forces are designated as shown below:

Vertical forces

V1is vertical force due to weight of dam V2is vertical force due to FRL on upstream slope of dam

V3is vertical force due to MWL on upstream slope of dam

V4is vertical force due to TWL on downstream slope of dam

V5is vertical force due to uplift when drains are operative for FRL

V6is vertical force due to uplift when drains are operative for MWL

V7 is vertical force due to uplift when drains are inoperative for FRL

V8is vertical force due to uplift when drains are inoperative for MWL

V9is vertical force due to self-weight of dam

CHAPTER 4 RESULTS & DISCUSSIONS


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Horizontal forces

P1is horizontal force due to water upto FRL

P2is horizontal force due to water upto MWL

P3is horizontal force due to TWL

P4is horizontal inertial force

Peis hydrodynamic pressure due to effect of horizontal earthquake

where FRL stands for full reservoir level, TWL stands for tail water level,

MWL stands for maximum water level

The forces and moments determined for different loading conditions are

shown in section4.4.The variation of vertical normal stress, horizontal normal stress,

shear stress, and principal stresses are calculated analytically.

4.3 FEM

The Stress analysis using SAP 2000 has been carried out. The analysis of

present study is done is equivalent static analysis for all load combinations specified

in IS 6512-1984. From the finite element analysis of the dam the vertical normal

stress, principal stresses and shear stress are obtained and they are tabulated in tables

given in section4.4.

4.4 Stress Analysis of Polavaram Dam

The stress analysis by two-dimensional analytic method and FEM are carried

out and are specified according to load combinations.

4.4.1 Load combination A (Construction Condition)

In the load combination A as discussed earlier in section2.1.2 , only the self-

weight of the dam is considered.

The total vertical force and the total horizontal force and the moments due to

the loadings are shown below:


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Total vertical force = V1=19871.15 kNTotal horizontal force = 0

Moment of resisting force r = 482495.88 kN-mMoment of opposing force o = 0The maximum principal stress at the base is - 1008.02 kN/m2 and it is within

the permissible limits.

The variation of vertical normal stress, horizontal normal stress, shear stresses

along the base of the dam are obtained using the equations (2-16 to 2-20) and are

shown below:

yy = 998.04 - 24.9684 x (4-1)

xy =- 99.80 + 7.856 x - 0.101 x2 (4-2)

xx = 9.98 + 0.7856 x + 0.03775 x2- 0.00091 x3 (4-3)

and principal stresses can be calculated using the equation (2-15).

The stresses as obtained by the FEM for the above load combination are

shown below:


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Table 4-1 Stresses Obtained for Load Combination A from FEM

Distance from

Heel to Toe (m)

Stresses (kN/m2)

Vertical Principal Shear

0 -934.98 -241.18 113.13

2.68 -817.56 -195.39 34.02

4.92 -744.8 -172.9 22.66

8.04 -687.46 -156.57 14.71

10.72 -631.14 -142.81 12.69

13.4 -574.85 -130.06 10.56

16.08 -519.62 -118.19 8.6

18.76 -462.63 -106.42 7.1421.43 -408.52 -95.6 6.31

24.11 -353.16 -84.91 5.29

26.79 -297.15 -74.63 4.48

29.47 -676 -125.33 16.2

32.15 -241.73 -63.57 2.31

34.83 -192.05 -52.1 -1.12

37.51 -134.71 -44.14 14.38

The stresses obtained are found to be within the permissible limits and are

compressive in nature.

4.4.2Load Combination B (Full Reservoir Condition)

In the load combination B as discussed earlier in section2.1.2 , self-weight of

the dam, water pressure and uplift force are considered as the dam is assumed to be in

full reservoir condition.



Total vertical force = V1+V2+V4+V5 = 16002.39 kNTotal horizontal force = P1+P3 = 6109.46 kNMoment of resisting force r = 505697.02 kN-m

Moment of opposing force o = 182342.19 kN-m


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The maximum principal stress at the base is - 527.43 kN/m2 and it is within the

permissible limits.



shown below:

yy = 525.68 - 5.28187 x (4-4)

xy =- 17.53 - 36.57 x + 1.152 x2 (4-5)xx = 352.163 - 3.657 x + 1.11 x

2 - 0.0289 x3 (4-6)



shown below:

Table 4-2 Stresses Obtained for Load Combination B from FEM

Distance from

Heel to Toe

(m)

Stresses (kN/m2)


0 -287.86 -146.87 -35.76

2.68 -566.01 -128.22 -16.89

4.92 -576.68 -134.06 -7.88

8.04 -567.95 -133.87 -4.71

10.72 -550.98 -131.92 -3.83

13.4 -532.61 -129.69 -3.59

16.08 -510.61 -126.59 -4.49

18.76 -483.83 -123.58 -5.42

21.43 -458.66 -119.24 -6.6124.11 -430.34 -113.29 -9.04

26.79 -398.38 -107.49 -12.58

29.47 -676 -125.33 16.2

32.15 -362.78 -102.48 -18.67

34.83 -333.72 -102.94 21.12

37.51 -262.84 -105.34 17.24




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4.4.3Load Combination C (Reservoir Maximum Water Level Condition)

In the load combination C as discussed earlier in section2.1.2,the self-weight

of the dam, water pressure and uplift force are considered as the dam is assumed to be

filled upto maximum water level.



Total vertical force = V1+V3+V4+V6 = 15981.89 kNTotal horizontal force = P2+P3= 6215.03 kN

Moment of resisting force r = 505887.92 kN-mMoment of opposing force o = 184928.66 kN-mThe maximum principal stress at the base is - 538.985 kN/m2 and it is within




shown below:

yy = 516.56 - 4.8246 x (4-7)

xy =-16.30 - 37.52 x+ 1.182 x2 (4-8)xx = 355.190 - 3.752x+ 1.1505x

2- 0.02987x3 (4-9)



shown below:


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Table 4-3 Stresses Obtained for Load Combination C from FEM

Distance from

Heel to Toe

(m)

Stresses (kN/m2)

Vertical Principal Shear0 -264.29 -148.28 -39.28

2.68 -538.39 -120.5 -14.97

4.92 -560.29 -131.33 -5.922

8.04 -561.35 -134.05 -3.87

10.72 -547.18 -132.23 -3.6

13.4 -530.9 -132.22 -3.64

16.08 -510.61 -127.85 -4.67

18.76 -485.34 -124.6 -5.68

21.43 -461.36 -120.29 -6.95

24.11 -434.05 -114.58 -9.45

26.79 -403.05 -108.95 -13.12

29.47 -676 -125.33 16.2

32.15 -368.17 -104.16 -19.39

34.83 -339.74 -104.93 -21.94

37.51 -267.97 -104.67 17.37



4.4.4Load Combination D (Combination A with Earthquake)

In the load combination D as discussed earlier in section2.1.2,along with the

load combination A, self-weight of the dam is assumed to be in empty condition and

the vertical and horizontal inertia force are considered.



stability of the dam, and hence is the worst case for design. For the reservoir empty

condition, the worst case occurs when the earthquake acceleration act towards the

downstream direction, and hence is the worst case for design.



Total vertical force = V1+V9= 19154.30 kN


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Total horizontal force A = P4= 1075.28 kNMoment of resisting force r = 482495.88 kN-m

Moment of opposing force o = 33650.50 kN-mThe maximum principal stress at the base is -901.688 kN/m2 and it is within




shown below:

yy = 892.76 - 20.3741 ( 4-10)

xy = -89.28 - 0.8905 x + 0.165 x2 ( 4-11)xx = 8.928 - 0.08905 x + 0.444 x

2 - 0.01017 x3 ( 4-12)



shown below:

Table 4-4 Stresses Obtained for Load Combination D from FEM

Distance from

Heel to Toe

(m)

Stresses (kN/m2)


0 -3771.66 -1136.73 603.78

2.68 -2930.87 -881.01 239.89

4.92 -2255.91 -660.86 231.64

8.04 -1611.34 -472.78 163.0910.72 -1144.77 -308.24 129.49

13.4 -723.7 -119.73 109.87

16.08 -359.57 109.95 101.36

18.76 -54.45 416.07 93.69

21.43 223.26 730.41 90.92

24.11 436.44 1021.59 90.72

26.79 616.56 1313.68 94.67

29.47 762.28 1601.25 108.34

32.15 866.38 1893.9 131.24

34.83 976.02 2255.24 151.76

37.51 978.32 2610.11 -27.9


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Tension is observed in FEM analysis and it is more than the maximum

permissible limits.

4.4.5LOAD COMBINATION E (Combination B with Earthquake)

In the load combination E as discussed earlier in section2.1.2,along with the

load combination B, self-weight of the dam, water pressure and uplift force are

considered and inertia forces and the hydrodynamic loadings are considered.



stability of the dam, and hence is the worst case for design. For the reservoir full

condition, the worst case occurs when the earthquake acceleration act towards theupstream direction.



Total vertical force = V1+V2+V4+V5+V9 = 15285.54 kNTotal horizontal force = P1+P3 + P4+ P5= 9243.04 kNMoment of resisting force r = 505697.02 kN-mMoment of opposing force o = 299067.96 kN-mThe maximum principal stress at the base is -1254.12 kN/m2 and it is within




shown below:

yy = 66.14 + 18.20156 x ( 4-13)

xy =28.43 - 73.94 x + 2.371 x2 ( 4-14)

xx = 347.567 - 7.394 x + 3.194 x2- 0.076 x3 ( 4-15)



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shown below:

Table 4-5 Stresses Obtained for Load Combination E from FEM

Distance from

Heel to Toe

(m)

Stresses (kN/m2)


0 3631.18 5626.13 -815.5

2.68 2010.65 3560.27 -405.19

4.92 1194.36 2402.47 -325.9

8.04 454.71 1575.54 -215.62

10.72 -43.61 1048.41 -164.87

13.4 -477.12 661 -137.65

16.08 -842.14 364.93 -126.14

18.76 -113

conventional and fem analysis of a dam

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