conventional and fem analysis of a dam
TRANSCRIPT
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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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2-2Literature Review
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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2-3Literature Review
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
down- stream 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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2-4Literature Review
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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2-5Literature Review
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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2-6Literature Review
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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2-7Literature Review
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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2-8Literature Review
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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2-9Literature Review
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
357 uplift mwl 1 Pore MWL
358 uplift 1 pore2
358 uplift mwl 1 Pore MWL1
358 uplift mwl 1 Pore MWL
359 uplift 1 pore2
359 uplift mwl 1 Pore MWL1
359 uplift mwl 1 Pore MWL
360 uplift 1 pore2
360 uplift mwl 1 Pore MWL1
360 uplift mwl 1 Pore MWL
361 uplift 1 pore2
361 uplift mwl 1 Pore MWL1
361 uplift mwl 1 Pore MWL
362 uplift 1 pore2
362 uplift mwl 1 Pore MWL1362 uplift mwl 1 Pore MWL
363 uplift 1 pore2
363 uplift mwl 1 Pore MWL1
363 uplift mwl 1 Pore MWL
364 uplift 1 pore2
364 uplift mwl 1 Pore MWL1
364 uplift mwl 1 Pore MWL
365 uplift 1 pore2
365 uplift mwl 1 Pore MWL1
365 uplift mwl 1 Pore MWL
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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4-2Results & Discussions
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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4-3Results & Discussions
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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4-4Results & Discussions
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.
The total vertical force and the total horizontal force and the moments due to
the loadings are shown below:
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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4-5Results & Discussions
The maximum principal stress at the base is - 527.43 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 = 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)
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:
Table 4-2 Stresses Obtained for Load Combination B from FEM
Distance from
Heel to Toe
(m)
Stresses (kN/m2)
Vertical Principal Shear
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
The stresses obtained are found to be within the permissible limits and are
compressive in nature.
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4-6Results & Discussions
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.
The total vertical force and the total horizontal force and the moments due to
the loadings are shown below:
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
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 = 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)
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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4-7Results & Discussions
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
The stresses obtained are found to be within the permissible limits and are
compressive in nature.
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.
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. 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.
The total vertical force and the total horizontal force and the moments due to
the loadings are shown below:
Total vertical force = V1+V9= 19154.30 kN
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4-8Results & Discussions
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
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 = 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)
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:
Table 4-4 Stresses Obtained for Load Combination D from FEM
Distance from
Heel to Toe
(m)
Stresses (kN/m2)
Vertical Principal Shear
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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4-9Results & Discussions
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.
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. For the reservoir full
condition, the worst case occurs when the earthquake acceleration act towards theupstream direction.
The total vertical force and the total horizontal force and the moments due to
the loadings are shown below:
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
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 = 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)
and principal stresses can be calculated using the equation (2-15).
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4-10Results & Discussions
The stresses as obtained by the FEM for the above load combination are
shown below:
Table 4-5 Stresses Obtained for Load Combination E from FEM
Distance from
Heel to Toe
(m)
Stresses (kN/m2)
Vertical Principal Shear
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