## Brief Study Of Nonlinear Parameters

Modelling the constitutive behaviour of concrete is the most important part of nonlinear seismic response study of the dams. Analytical models for two-dimensional fracture propagation studies are well developed now, but the constitutive behaviour of concrete is the most important part of nonlinear seismic response study of the dams. Analytical models for two-dimensional fracture propagation studies are well developed now, but the three-dimensional application of the fracture models is still in...

## Ai rai

The above equation is a representative of a biaxial failure envelope. The principal stress a1, and the principal strain, 1, at the instance of softening initiation, are designated by a0, and 0, respectively Figure 5.10-a . Under dynamic loads, the pre-peak non-linearity decreases with increasing values for both at and t Figure 5.10-b . The strain-rate effect on the material parameter U0 is considered through a dynamic magnification factor, DMFe, as follows

## Appurtenant Features Of Dams

Appurtenances are structures and equipment on a project site, other than the dam itself. They include, but are not limited to, such facilities as intake towers, powerhouse structures, tunnels, canals, penstocks, low-level outlets, surge tanks and towers, gate hoist mechanisms and their supporting struc tures, and all critical water control and release facilities. Also included are mechanical and electrical control and standby power supply equipment located in the powerhouse or in the remote...

## At y

The hydrodynamic pressure for a harmonic ground excitation of ug e would be p(x,y,t) V -, exp -x X2m - cos Xmy elwt It can be seen that as x to the pressure does not approach zero if Am < c which was the fourth boundary condition imposed in Westergaard solution. To compare the above solution with Westergaard's solution, we can consider the real part of the ground motion as well as coefficient ag (ug ag cos( )).The Solution in this case consist of the real part of the pressure solution...

## Boundarylayer Theory

For fluids with small viscosity such as air and water with a high degree of accuracy, we can consider frictionless flow over entire fluid except for their regions around the contact areas. Here, because of high velocity gradients, we could not properly neglect frictions (Newton's viscosity law) so we consider these regions apart from the main flow, terming the boundary layers. In 1904 Ludwing Prandti introduced the concept of boundary layer and showed how Navier-Stokes equations could be...

## Compressible Fluid

Despite the existence of large pressure, fluids undergo very little change in density. Water needs a pressure change of 20,000kPa to have 1 percent change in its density. Fluids of constant density are termed incompressible fluids and assumed during computation the density is constant. A flowing fluid is said to be compressible when appreciable density changes are brought about the motion. The variation of density is usually accompanied by temperature changes as well as heat transfer. The...

## Constitutive relationships during softening

After the softening initiation, a smeared band of microcracks is assumed to appear in the direction perpendicular to the principal tensile strain. The material reference axis system, referred as the local axis system, is aligned with the principal strain directions at that instant directions n-p in Figure 5.10-c. The constitutive matrix, relating the local stresses to local strains is defined as where the parameter n (0 < n 1) is the ratio between the softening Young's modulus En, Figure...

## Contents

1.1 TYPES OF 1.1.1 Embankment 1.1.2 Concrete Arch and Dome Dams 13 1.1.3 Concrete Gravity and Gravity-Arch Dams 14 1.1.4 Concrete Slab and Buttress Dams 16 1.2 APPURTENANT FEATURES OF DAMS 16 1.3 SAFETY OF DAMS AND RESERVOIRS 18 1.4 HOW DAMS ARE 1.5 FAMOUS DAMS OF THE WORLD 21 1.6 POWER GENERATOR, FLOOD CONTROL AND IRRIGATION DAMS 1.6.1 Power Generator 1.6.2 Flood Control 1.6.3 Irrigation 1.7 INSTRUMENTATIONS AND SURVEILLANCE OF DAMS . 28 1.8 ECOLOGICAL ENVIRONMENTAL CONSIDERATION OF DAM...

## Coupling Matrix Of The Damreservoir

The coupling matrix relates the pressure of the reservoir and the forces on the dam-reservoir interface as following where f is the force vector acting on the structure due to the hydrodynamic pressure. Figure shows a line element on the interaction boundary of the damreservoir. The work done by the hydrodynamic pressure on the interaction surface of the structure must be equal to the work of the equivalent nodal forces on the interface boundary of an element. Thus, for unit thickness elements...

## Damage evolution for concrete subjected to tensile strain

An element of volume of the material which can be representative of the global behaviour is now considered. This volume will be characterized by its length which provides a measure of the region over which the damage is smeared so that the global response is reproduced by this volume. This length, lch, is called characteristic length and it should be measured in the direction normal to a potential crack plane ( Figure 5.8-B). To de fine the damage evolution of concrete, The principal strains...

## Dam Reservoir Boundary Condition

At the surface of fluid-structure, it is clear that there must be no flow across the interface. This is based on the fact that face of the concrete dams are impermeable. This results into the condition that at the normal to the boundary there is no relative velocity or another word we can write it mathematically in which n is the unit normal vector to the boundary at the dam-reservoir interface and vn and vn are the velocity of the structure (dam) and fluid along the n respectively (figure...

## Dams Designed for Water Supply

It is impossible for a civilization to survive without a constant fresh water supply, since it is a basic need of all humans. In many parts of the world, the freshwater supply fluctuates from enormous amounts during the rainy season to long droughts during the dry season. for this reason dams were invented to store large amounts of water from the end of the rainy season until the end of the dry season, to ensure a constant supply of water. The oldest dam in the world is the Jawa dam found 100...

## Development of the Modern Dams

Unlike past periods in history, the modern dam is designed using specific principles. The largest change in the design of a modern dam compared to the earlier dams, has been the formation of engineering schools. This changed the individualistic craftsmanship of engineering into a profession based on scientific principles. Engineers were subdivided into specific disciplines, increasing the advancements of both dam principles and construction methods. The modern embankment dam has improved...

## Direct Integration Of The Equation Of Motion

In direct integration the equation of motion are integrated using a numerical step-by-step procedure. The term direct means that no transformation of the equation of motion into the different form is carried out.The direct integration of the equations of motion provides the response of the system at discrete intervals of time which are usually spaced. In this procedure three basic parameters of displacement, velocitiy aand acceleration are computed. The integration algorithms are based on...

## Discrete crack propagation model DCPM variable mesh

In the DCPM, global approach, a crack is represented as a discrete gap along the inter-element boundary. The growth of the crack is determined by strength or fracture mechanics based constitutive models. The progressive physical discontinuity in the system is reflected instantaneously in the finite element model by modifying the mesh during the analysis. It is generally argued that the nonlinear response of concrete dams is dominated by a few discrete long cracks. From this consideration, the...

## Ecologicalenvironmental Consideration Of Dam Operation

The following observations are generalized from the literature at large. Whether any particular concern applies to any particular dam is a function of a suite of factors that might include the dam's type (e.g. concrete or earth), its purpose (e.g. hydrologic regulation, hydroelectric generation, tailings impoundment), its size, location and operating protocol (e.g. timing of drawdowns, epilim-netic versus hypolimnetic draw, peaking versus base load generation, etc.). Finally, The observations...

## Gf

For typical dam concrete properties of E 30 000 MPa, Gf 0.2 N mm, and vt 2.0MPa, the limiting value is l0 < 3.0m. This limit on maximum dimension, which is much higher than the material characteristic dimension, wc, of Bazant and Oh (1983), also has often been considered stringent for large scale finite element analysis at reasonable cost. Figure 5.5 Nonlinear fracture mechanics in smeared crack propagation model To circumvent this limit on the size of finite elements, and at the same time...

## Ek Ed Er Ep Eq Eh

In equations and 4.42, r is the vector of the nonlinear restoring force. The absolute kinetic energy is EK, the viscous damping energy is ED, the nonlinear restoring work is ER, the work of preseismic applied force is EP, the absolute seismic input energy is EQ and the work done by the hydrodynamic pressure is EH. The relative displacement is U while Ut is the total (absolute) displacement vector Ut U + Ug . Ug is the ground displacement vector. The restoring energy, ER contributes to the...

## Ep Eq Eh Ek Ed Er inn Errr Eq Eh X

When the dam-reservoir interaction effects are represented by added masses, the hydrodynamic energy EH is excluded from the energy equation, EH 0. However, energy is added to the seismic input energy EQ and kinetic energy EK through the mass added to the structural system. In the analysis, the results of the fracture response are presented for the time before the five percent energy balance error is reached. The error in the energy balance represents an excessive amount of damage when numerical...

## F Kj Kj Kjjj KC

Several functional forms of equation 5.1 have been proposed in the literature. In LEFM models, a sudden release of stress on the surface is assumed with the extension of the crack. Most investigators adopt the discrete crack propagation finite element model (DCPM) with the LEFM constitutive models. Techniques to apply the LEFM criteria in smeared crack propagation finite element model have also been reported in the literature. Pekau et al. (1991) have proposed a numerical procedure to apply the...

## Fracture mechanics criteria

Fracture mechanics is the theory dealing with propagation of cracks, based on the concept of energy dissipation by the structure undergoing fracturing process. It has been recognized only recently that the failure mechanism in concrete structures is different from the usual strength based concept, due to the progressive growth of a fracture process. The fracture mechanics of concrete has drawn significant attention of the research community over the last decade as a consequence, the literature...

## Famous Dams Of The World

Rogun and Nurek, in Tajikistan, are among the world's highest dams. They rise to a height of more than 1,000 feet (300 meters) each. Some of the world's other major dams are Grande Dixence and Mauvoisin, in Switzerland Chicoasen, in Mexico Inguri, in Georgia Sayano-Shushensk, in Russia Mica, in Canada and Guavio, in Colombia. Three of the foreign dams that are constructed of the largest volume of materials are Tarbela, in Pakistan Lower Usuma, in Nigeria and Guri, in Venezuela. Dams that form...

## Fd [CU

Equating of the external forces and internal forces will results in M Ut + C U + K U f (t) which f (t) is the external forces act on the structures and can be written as If there is a base displacement like ground acceleration along y direction as shown in figure 3.3, the equation of motion for a multi-degree-of-freedom system can be written as M U t + C U + K U f (t) Ugy + U . The vector of ground acceleration can be Where ugyis the ground acceleration at the base along y direction. Thus, the...

## Finite Element Modelling Of The Reservoir

As it was discussed in previous Chapter, the hydrodynamic pressure distribution in the reservoir is governed by the pressure wave equation. Assuming that water is linearly compressible and neglecting its viscosity, the small amplitude irrotational motion of water is governed by the two-dimensional wave equation where p(x,y,t) is the hydrodynamic pressure in excess of hydrostatic pressure, C is the velocity of pressure wave in water andxand y are the coordinate axes. The hydrodynamic pressure in...

## Flood Control Dams

One way to avoid floods is to take the obvious precaution of living where there is no danger of high waters. It has always been convenient and often necessary to build homes and factories on the floodplains along rivers and streams and on the seacoasts. American pioneer settlers depended upon the streams for drinking water, transportation, and power to run their mills and factories. Floodplains, deep with the silt laid down by overflowing rivers, are fertile farmlands. The earliest towns and...

## Free Surface Boundary Condition

The geometry of free surface boundary is not known a priori. This shape is part of the solution, which means we have a very difficult boundary condition to cope with. In the case of surface wave of negligible surface tension, we call them gravity wave. The free surface of a wave can be described as where p is the displacement of the free surface above the horizontal plane, say z 0,. If the surface varies with time, as would the water surface, then the total derivative of the surface with...

## General Form Of Reservoirs Equation Of Motion

Fluids are composed of molecules in constant motion and collisions. A fluid has no structure. They are not distinguished by their microscopic (molecular) structure. To take account of each molecule in a flow, may be difficult for engineering purposes. Instead we are interested in average measuring of the molecule manifestations such as density, pressure and temperature which are their macroscopic structure. The continuum concept offers a great deal of simplification in analysis. A fluid point...

## How Dams Are Built

The methods of building dams can be envisioned by following the construction of Hoover Dam, built between 1930 and 1935. The engineers constructed a concrete arch-gravity dam at an approved cost of 174,000,000. It is as tall as a 60-story skyscraper. Its crest is 45 feet (14 meters) thick and its base, 660 feet (201 meters). It stores the entire flow of the Colorado River for two years. Much preliminary work had to be done. The engineers made geologic and topographic surveys to select the site....

## Instrumentation

Instruments are used to characterize dam site condition as well as dam structure conditions. Instruments can be used for verifying design verification, safety assessment, performance assessment. Design Verification Instruments are used to verify design assumptions and to check that performance is as predicted. Instrument data from the initial phase of a project may reveal the need (or the opportunity) to modify the design in later phases. Safety Instruments can provide early warning of...

## Instruments

In the following section, Some instruments used in monitoring of dams are briefly described. Piezometers measure pore-water pressure and ground water levels. Piezometer measurements help engineers to Monitor water levels, Predict slope stability, Design and build for lateral earth pressures, Design and build for uplift pressures and buoyancy, Monitor seepage and verify models of flow. Inclinometers may be installed to check that actual movements of a structure correlate to those predicted...

## Introduction

The cracking behaviour of concrete dams has been the subject of extensive research during the past decade because few dams suffered severe cracking during earthquakes. Rescher (1990) indicated that most concrete gravity dams will experience cracking even under operational loading conditions and moderate earthquake ground motions. Therefore, the assumption of linear behavior may not be appropriate in the analysis of the seismic response of concrete gravity dams. Concrete dams are distinguished...

## Irrigation Dams

Irrigation is the artificial supply of water to agricultural land. It is practiced by more than half the farmers in the world because they need more water for their crops than is available from rainfall. Irrigation projects must also allow for removal of excess water. Generally the need for irrigation water is highest in the dry season when river flows are lowest. To ensure continuous supply, water must be stored on a seasonal and sometimes annual basis. By erecting a dam, an artificial lake,...

## Linear elastic fracture mechanics parameters

The principal parameter applied in the linear elastic fracture mechanics crack propagation models is the fracture toughness, K1c, of concrete. No definite relationship is readily available in the literature of dam concrete to determine the fracture toughness from standard material parameters such as strength, elastic modulus, and aggregate size. And only a handful of experimentally a'a 1.25(1.3 x 0.324) 2 3 0.526 2 3 determined results has been reported so far. Saouma et al. (1989) found a Kic...

## Linear Momentum Equation

If we take linear momentum as parameters N which is used as a general term (vector and scalar) in the Reynold transport equation, we have N P mv. The term n in this case would become momentum per unit mass, or simply v, then, P JJJ v(pdV). Then the Reynold transport equation can be written as followings The equation of linear momentum is a vector equation and can be divided into its three components. In above equation, F if the resultant of the all external forces acting on the system and v the...

## M i d[CoM

Since Y is a quadratic function of strain, it is positive and, thus, d is always increasing as shown in equation 5.9. This is a characterization of the irreversibility of damage. Different hypothesis have been proposed for isotropic damage model. The basic assumption in behaviour of the damaged and equivalent undamaged element will results in different model. Among several models, special attention will be given to the two models. They are mainly based on two concepts of the modelling. In the...

## Mn Mm

Figure 3.4 An example of MDF system with two degrees of freedom at each mass For a system of MDF system when subjected to ground acceleration in two directions and there are two degree of freedoms for each mass,as shown in figure 3.4,we can write the equation of motion based on the both degrees of freedom at each mass.For this system, we can write the displacement vector as The same can be written for U and U . The equation of motion can be written as following m u + c U + K u f (t) - m u9 When...

## Modified Staggered Pressure Method

Most of the available nonlinear solutions assume a diagonal mass matrix for the purpose of analysis. The staggered displacement method is the most suitable coupled field problem solution procedure for the case of nonlinear analysis. In the case of the staggered pressure method some difficulties may arise due to added mass effect in equation 4.27 which changes the mass matrix from diagonal to a full matrix. For this reason the staggered pressure method was modified to apply to nonlinear...

## Nonlinear fracture mechanics parameters

Fracture energy, Gf, is the key parameter that is combined with elastic modulus, E, and tensile strength, at, to define the entire constitutive behaviour of concrete in the nonlinear fracture mechanics models. Usually, the tensile strength, at, beyond which a strain softening process is assumed to take place, is determined from uniaxial or split cylinder tests and the fracture energy, Gf, from wedge splitting tests (Briihwiler and Wittmann 1990). Values for at and E can be selected according to...

## Nonlinear Modelling Of Concrete Dams Using Damage Mechanics

Micro cracking in concrete is believed to occur at relatively low levels of loading. Therefore, cracking progresses in a heterogeneous medium because of an increase in micro cracking, and because of the linking of various micro cracked zones. Experiments performed on cement paste as well as concrete show that micro cracking has an arbitrary orientation. When the load is increased, macroscopic cracks develop and the crack orientations follow the principal stress directions in the material....

## P Vp Vv B

The operator V gx + dy2 + is called the Laplacian operator. The above equation is called Navier-Stokes equation which first developed by Navier in France. For inviscid fluids, the equations 2.17, 2.18 and 2.19 will result in This is Euler equation which is widely used for describing flow systems in which viscous effects are relatively unimportant.

## Power Generator Dams

Most electric power is generated in large plants that use coal, gas, oil, or nuclear energy. Electric energy may also be obtained from waterpower. The roar of a waterfall suggests the power of water. Rampaging floodwaters can uproot strong trees and twist railroad tracks. When the power of water is harnessed, however, it can do useful work for man. Since ancient times man has put to work the energy in the flow of water. He first made water work for him with the waterwheel a wheel with paddles...

## PVv v p Vp

For small amplitude motion the effect of convective term of the left hand side of the above equation is negligible, thus we can write For a linearly compressible fluid, we define the following p K (.xx + .yy + .zz) (2.25) Derivating of whole equation with respect to time The .xx can be written as following 6xx (6xx) 8x) 6x dt) XX I V XX J All A I O I I A In which ux is the displacement in the x-direction. Thus we can write If we use equation 2.24 to take the divergence of the whole equation we...

## Reservoir Boundary Conditions

The formulation of the boundary conditions associated with the reservoir boundaries is simply the expression in mathematical terms of the physical situations. There are an infinite number of solutions to the differential equation. The task is to find the solution that is relevant to the boundary condition. It should be noted that in addition to the spatial (or geometric) boundary conditions, there are temporal boundary conditions which specify the state of the variable of the interest at some...

## Reservoir Farend Truncated Boundary Condition

The truncated boundary for the finite element and boundary element modeling of the infinite reservoir was worked by so many researchers. Sommerfeld boundary condition is the most common one that is based on the assumption that at long distance from the dam face, water wave can be considered as plane wave. A plane wave can be represented by an equation of the form in which n is the normal at the truncation boundary. This represents well-known Sommerfeld radiation condition. It introduces a...

## Reservoir Foundation Boundary Condition

If there is no absorption or penetration of water into the reservoir bottom, the same dam-reservoir boundary condition that was obtained previously, can be used for reservoir-foundation boundary. In case of reservoir with sediment at the bottom, we can account it in a very simplified manner. The boundary condition at the reservoir bottom relates the hydrodynamic pressure to the sum of the normal acceleration and acceleration due to interaction between impounded water and the reservoir bottom...

## Reservoirs Equation Of Motion

Based on the information presented in the previous sections, the Euler equation is valid for the case of dam's reservoir. In the Euler equation, if we divide the pressure into static and dynamic pressure we can write for static pressure Vps + B 0, therefore for only hydrodynamic pressure we can write the following The p is the dynamic pressure in excess of static pressure. The above equation can be written as

## Safety Of Dams And Reservoirs

There are Several types of reservoirs as defined by their locations. The common type is a reservoir behind a dam in a valley. Increasing use is being made of tidal-storage reservoir for power generation along coast lines and excavated reservoirs for water storage for municipal or other use. In the latter type the excavated material commonly is employed to construct an embankment on one or all sides of the reservoir. Reservoirs and their associated dams serve many purposes including electric...

## Seismic Energy Balance

In the design of structure subjected to earthquake loading, the energy equation can be used to study the energy absorbtion of different components. In a satisfactory design, the energy supply must be larger than the energy demand. In this regard, two approaches can be considered for the energy equation. Uang and Bertero (1990) used absolute and relative energy formulations for a single degree of freedom system. They found that absolute energy formulation is simple and more straightforward....

## Shear resistance of fractured concrete

After the initiation of fracture process on a plane perpendicular to the direction of major principal tensile stress, it is not unlikely to expect the rotation of principal stress directions under varying deformation modes. Hence, shear deformation may take place on the partially formed rough fracture plane. The definition of shear stress-strain behaviour of concrete during the fracture process seems to be difficult. Bazant and Oh (1983) ignored the shear deformation on the fracture plane, and...

## Solution Of The Reservoir Equation

Westergard's classic work (1933) on the hydrodynamic water pressure on dams during the earthquake started a new area for the researcher in this field. Westergaard's solution to wave equation for rigid dams during earthquakes was obtained based on the assumptions of dam with rectangular reservoir subjected to horizontal earthquake. In the solution, the reservoir extended to infinity in the upstream direction and the effect of surface wave was neglected. The system was subjected to the horizontal...

## Staggered Displacement Method

In this method, equation 4.35 can be approximated as following M u K+1 F1 i+1 + Q p P+1 C U P+1 (1 + a) K U P+1 + a K U Combining equations 4.37 and 4.35 gives M U i+i M U *+1 + At2 Q p m y At C U i+i (1+a) At2 K U i+i Taking advantage of the lumped mass which results in a diagonal mass matrix, equation 4.38 can be modified as M U i+1 M U *+1 + PAt2 Q p i+1 (4.39) Substituting equation 4.39 into equation 4.36, then ( G + pPAt2 Q T M -1 Q ) P i+1 + C p i+1 + (1 + a) K' p i+1 F2 i+1 - p Q T UK+1...

## Staggered Pressure Method

In this method, the pressure can be approximated using equation 4.8 as following G PK+1 F2W1 - C' p +1 - K'Mf+1 (4.24) Substituting equation 4.24 into equation 4.8, there is obtained G p i+1 G pK+1 - p Q t U i+1 - 7At C p i+1 - pAt2 K' p m ( G + pAt2 K + yAt C ) P i+1 G P *+1 - p Q T U i+1 (4.26) Substituting equation 4.26 into equation 4.7 with H G + PAt2 K' + YAt C , gives ( M + pP At2 Q H -1 Q T ) U i+i + C U i+i + K U i+i (4.27) Fi W + Q ( p p+i + P At2 H -1 G p *+i) Using equation 4.27,...

## Strengthbased criteria

The early investigations on cracking of concrete have mostly applied simple criteria based on the concepts of strength of material (SOM). A crack is assumed to propagate when the predicted stress or strain at the crack-tip exceeds the critical value representative of the strength of the material. The crack propagation criterion, in this approach, is identical to the new crack initiation criterion. A sudden release of stress on the fracture plane is commonly assumed upon reaching the material...

## Surveillance

External and internal examination of dams are required during their life time. Long term surveillance program can ensure dams owner of their safe operation. Long term surveillance requires instrument installation and monitoring to assess the performance of structures with respect to design parameters. Design of monitoring installations is to measure displacement, settlement, strain, stress, piezometric pressures, seepage, uplift, flow velocity and water levels, and alarm systems as required for...

## T n m m

The term, (2m4-hi)c is the mthperiod of the reservoir, Tm,. Therefore, the westergaard's solution is valid if the period of loading, T, is greater than the first period of the reservoir, T ,. If T Tm, then p to and we have resonance. The Westergaard pressure equation at the upstream face of the dam can be written as followings

## The Moslem World

Arabs quickly developed their country within one century after the impact of new Islamic religious founded by prophet Mohammed. Several irrigation dams around the new power centres of Mecca and Madina, were built. The following is a summary of 7th and 8thcentury dams in central Arabia. Table 1.4 Summary of 7th and 8th century dams in central Arabia All of these dams were of the gravity type. They all have two outer walls of dry masonry and an earth or rubble core in between. The walls of some...

## Types Of Dams

Dams range in size and complexity of construction from low earth embankment constructed to impound or divert water in small streams to massive earth or concrete dams built across major rivers to store water. The type of dam that is built and its size are a complex function of a demonstrated necessity for water storage or diversion, the amount of water available, topography, geology, and kinds and amount of local materials for construction. Although large embankment dams do not posses the...

## Tzz

Figure 2.10 Displacement thickness in boundary layer The motivation for using displacement thickness is to permit the use of a displaced body in place of the actual body, such that the frictionless mass flow around the displaced body is the same as the actual mass flow around the real body. Use is made of the displacement thickness in the design of wind tunnels, air intakes for airplane jet engine. As the boundary layer thickness growth, the fluid is pushed away from the plate, which means that...

## U cos C h

The difference between the two solutions depends on the magnitude of the quantity c. Analysis of strong ground motions of the past earthquakes seems to indicate that frequencies of most of the significant harmonic components lie in the range 1 < u < 120 rad sec. Thus, the largest value of C may be taken as 6.82 x 10-3(u 1, C 4720fps). However, such low frequency harmonics contribute little to the response in the case of typical reservoir depths encountered, because natural frequencies for...

## U DMFe Uo

Where the primed quantities correspond to the dynamic constitutive parameters. The increased material resistance due to inertia and viscous effects under dynamically applied loads has been explicitly considered in the dynamic equilibrium equations. Reviewing the literature on the dynamic fracture behaviour of concrete, a 20 percent dynamic magnification of the apparent tensile strength is considered adequate for seismic analyses of concrete dams. Under dynamic loading, the material parameter U0...

## Um m

Another solution to the wave equation presented by Chopra. Chopra (1967) presented a closed form solution for hydrodynamic pressure in the case of rigid dam with the vertical up-stream face under the horizontal and vertical ground motion. For the excitation frequency less than the fundamental frequency of the reservoir, both Westergaard and Chopra's solution are the same. The greater excitation frequency causes an out-of-phase pressure compared to Westergard's solution. In this case the...

## USING aMethod For The Coupled Equations

In this method, U i+1, U i+1, P i+1 and p i+1 can be written same as Newmark- method. The governing field equations at time i + 1 can be written as follows M U t+1 + C U i+1 + (1 + a) K U i+1 F1U1 + Q p m + a K U G p i+1 + C 0 p i+1 + (1+a) K0 p i+1 F2 i+1 p Q T U i+1 +a K 0 p where a is the integration parameter which is introduced in the coupled field equation. The coupled field equations 4.35 and 4.36 can be solved using the staggered displacement solution scheme.

## Using Newmark Method For The Coupled Equations

Direct integration scheme is used to find the displacement and hydrody-namic pressure at the end of the time increment i + 1 given the displacement and hydrodynamic pressure at time i. The Newmark- method is used for discretization of both equations (implicit-implicit method). In this method U j+i, U j+i, P j+i and P i+1 can be written as follows U i+i U p+i + 7Ai U i+i (4.3) U i+i U p+i + At2 U i+i (4.4) U P+i U i + At U7 i + (0.5 - )At2 U i P i+i P P+i + Y At p m (4.5) p i+i p +i + Ai2 p m...

## Velocity Field

In a deformable system there are an infinite number of particles. In order to define the velocity of a particle we must define the position of the particles, spatial coordinates, at the specific time. Using this, the velocity of all particles can be as v(x, y, z, t) which is consisted of three components of velocity vx, vy, vz in x, y and z direction of coordinates, respectively. v is called field velocity vector. For steady flow, the value of the velocity at a position remains invariant with...

## Y d MT IV

The formulation of a damage model first requires the definition of threshold of damage. which is the conditions that initiates the propagation of damage. Secondly, the evolution of the damage with loading must also be defined, and it is a function of a measure of strains, stresses or energy. 5.6.3 ISOTROPIC DAMAGE MODEL FOR CONCRETE To establish the damage constitutive equation, it is necessary to relate the damage variable d to the other internal variables by some physical hypothesis. Here we...

## [CdG [RT [Cd[R

Where Cd G is material matrix in the global axes and R is transformation matrix defined by and 3 is angle of principal strain direction. If di d2 d, the following relation is obtained which represents the damaged isotropic model. This model differs from equation 5.12 by a square factor because of the energy equivalence. Now we try to relate the damage variable to the state of an element using uniaxial behaviour of a concrete specimen Consider the elastic brittle uniaxial behavior as shown in...

## [Dxy [TT [Dnp[T

Where T is strain transformation matrix defined as follows in terms of the inclination of the normal to it crack plane, 6 Figure 5.10-c sin2 0 cos2 0 sin 0cos0 2 sin 0cos0 2 sin 0cos0 cos2 0 sin2 0 With an increasing strain softening, the damaged Young's modulus, En, (Figure 5.10-d), and hence the parameters n and p decrease gradually, and may eventually reach zero values after the complete fracture (en > f or ef). The constitutive matrix in equation 5.26 is updated as the parameters n and p...

## [MU ii [CUii [KU ii Fiii [Qpii

G pW + C0 p i+i + K0 p i+i F2 i+i - p Q T U i+i (4.8) The coupled field equations 4.7 and 4.8 can be solved using the staggered solution scheme. The procedure can be started by guessing P i+i in equation 4.7 to solve for U i+1 and its derivatives. Then equation 4.8 can be solved to find P i+1. This method can not guarantee the unconditional stability of the solution. Similarly, guessing U i+1 at first to calculate P i+1 from equation 4.8 and then calculating U i+1from equation 4.7 can not...

## [MUii [CU ii [K U ii [Qpii

Combining equations 4.10 and 4.12 and substituting them into equation 4.8 without the force term, gives G p i+1+ C 0 p i+1+ K0 p i+1+p Q T M -1( M +y At C + At2 K ) U i+1 0 The modally decomposed system is represented by a single degree of freedom equation. The single degree of freedom equivalent of equations 4.17 and 4.18 will be obtained by substituting the mass, damping and stiffness values m, c and k instead of M , C and K in equation 4.17 and g, c andk instead of G , C and K in equation...

## Concrete Gravity and Gravity Arch Dams

A concrete gravity dam has a cross-section such that, with a flat bottom, the dam is free-standing that is, the dam has a center of gravity low enough that the dam will not topple if unsupported at the abutments. Gravity dams require maximum amounts of concrete for their construction as compared with other types of concrete dams, and resist dislocation by the hydrostatic pressure of reservoir water by sheer weight. Concrete gravity dams have been constructed up to 285 m (950 ft)high. Properly...

## Embankment Dams

A broad spectrum of natural and fabricated materials have been used in the construction of embankment dams. Embankment dams are made by building an embankment of gravel, sand, and clay across a river. To prevent leakage, often a core, or inner wall, of concrete or other watertight materials is used. In a rolled-fill dam, earth is hauled by vehicles onto the dam and rolled tight with heavy machinery. In the hydraulic fill dam, earth is carried to the dam by water in pipes or flumes and also...

## The Equation of the Motion

The continuity equation and the linear momentum equation was developed for finite systems and their were related to control volume. The equations give an average values of quantities or only components of resultant forces. They don't give any detail information of the flow everywhere inside the control volume. Here, we are trying to find out differential equations valid at any point of fluid. Writing continuity equation in the form of differential equation, we consider an infinitesimal control...

## Reynolds Transport Equation

Consider N as properties of a substance whose measure depends on the amount of the substance presented in a system. An arbitrary flow field of v v x, y, z, t is observed from some reference xyz. We consider a system of fluid with finite mass at time t and t At figure 2.3 . The streamlines correspond to those at time t. The control volume is considered with system at time t and is fixed. The distribution of N per unit mass will be given as n, such that N HI npdV with dV representing an element...

## [MU [CU [KU f [MUg

In a MDF sytem, a system is discretized into its degrees of freedom. The number of degrees of freedom depend on the number of elements and the degree of freedom for each element. Usually in the case of concrete gravity dams isoparametric plane stress or plane strain are used. These types of elements are used for two dimensional problems. In some cases triangular elements may be used. In both plane stress or plane strain the displacement field is given by the u and v in directions of cartisian x...

## E doE Gf doGf

5.6.8 ANALYTICAL PROCEDURES IN A FINITE ELEMENT MODEL In a finite element model, the four-node isoparametric element is preferred in the implementation of the local approach of fracture-based models and has been used for the implementation of the described constitutive model. The standard local definition of damage is modified such that it refers to the status of the complete element. The average of the strain at the four Gauss points is obtained and the damage is evaluated from the...

## Concrete Arch and Dome Dams

The ultimate complexity of design and analysis of stresses is attained in arch and dome dams. These dams are thin, curved structures commonly containing reinforcing, either steel rods or prestressed steel cables. Volume requirements for aggregate for manufacture of concrete are much less than in gravity and gravity-arch dams, but the competency of bedrock in foundations and abutment to sustain or resist loads must be of a high order. Arch dams usually are built in narrow, deep gorges in...

## Strengthofmaterial parameters

The governing material parameter in SOM-based fracture propagation models is either critical stress or strain. From a rigorous study with some 12 000 published test results, Raphael 1984 proposed the following relationship between tensile and compressive strengths of concrete under static loading where f'c and at are respectively static compression and tensile strengths of concrete in MPa. In absence of experimentally determined values, the above equation can be applied as an approximation to...