overflow section

overflow section

A case study of Longtan dam

Albin Wessling, Simon Jonsson

Engineering Physics and Electrical Engineering, master's level 2018

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

the loads acting on it was identified after an iterative flood routing calculation. The section

was then evaluated, both theoretically and numerically using FEM with a linear-elastic isotropic

constitutive model, with regards to the stability against compressive and tensile failure. Also, a

numerical analysis of the seepage through and around the structure was conducted. The results

show that, for a given design of the section, the structure can handle the compressive stresses that

arises. However, the numerical analysis indicates a possible tensile failure at the dam heel, which is

inconsistent with the theoretical stability analysis. The seepage analysis show that the maximum

seepage occurs at the dam heel, the dam toe, and at grout curtains beneath the structure. A

simple convergence analysis was done which showed stress singularities at the dam heel and dam

toe. These were discussed and connected to the St Venant’s principle, and these singularities shows

the risk of blindly trusting numerical results.

us to China and Hohai University. We would also like to thank Gunnar Hellström, associate

professor at Luleå University of Technology, for his support before and during the project, as

well as for inspiring us about the field of hydropower. Mr Tao Wo and Mengjiao is also deserving

recognition for helping us with everything from feedback and support on the project to the practical

arrangements that enabled us to stay in China. Also, great gratitude is expressed to James Yang,

adj. professor at the Royal Institute of Technology, for giving us the opportunity to do this master

thesis. The authors would finally like to thank Energiforsk AB who has supported this China

project in its dam safety program.

1.2 Spillways . . . . 1

1.3 Longtan dam . . . . 2

1.4 Problem formulation . . . . 2

1.5 Purpose and limitations . . . . 3

1.6 Structure of thesis . . . . 3

2 Theory 4 2.1 Flood routing . . . . 4

2.2 Design process of overflow section . . . . 4

2.2.1 Practical profile . . . . 4

2.2.2 Overflow weir curve . . . . 6

2.2.3 Diversion walls and gate house . . . . 8

2.2.4 Trajectory distance and scour hole depth . . . . 8

2.3 Loads . . . . 9

2.3.1 Self-weight . . . . 9

2.3.2 Hydrostatic pressure . . . . 10

2.3.3 Uplift . . . . 12

2.3.4 Hydrodynamic pressure . . . . 13

2.3.5 Silt pressure . . . . 14

2.3.6 Earthquake forces . . . . 15

2.4 Stability . . . . 17

2.4.1 Reliability of hydraulic structures . . . . 17

2.4.2 State functions and the limiting state . . . . 17

2.4.3 Probability of failure . . . . 18

2.4.4 Partial safety factors . . . . 18

2.4.5 Sliding at the dam base . . . . 20

2.4.6 Compressive stress at the dam toe . . . . 21

2.4.7 Tensile stress at the dam heel . . . . 22

2.5 Stress-strain constitutive modelling . . . . 22

2.5.1 Governing equations . . . . 24

2.6 Seepage . . . . 25

2.6.1 Governing equations . . . . 25

2.7 Finite element method . . . . 26

2.7.1 Energy methods . . . . 26

2.7.2 Method of weighted residuals . . . . 26

2.7.3 Discretisized equations: Stress and deformation . . . . 26

2.7.4 Discretisized equations: Seepage . . . . 27

2.7.5 Numerical integration . . . . 27

2.7.6 Meshing . . . . 27

2.7.7 Element types . . . . 27

2.7.8 Element nodes . . . . 28

3 Method 29 3.1 Flood routing procedure . . . . 29

3.2 Implementation of finite element method . . . . 30

3.2.1 Software . . . . 30

3.2.2 Mesh . . . . 30

3.2.3 Materials . . . . 32

3.2.4 Boundary conditions . . . . 33

4.4 Numerical results . . . . 38

4.4.1 Stress and deformation . . . . 38

4.4.2 Seepage . . . . 43

4.4.3 Verification . . . . 53

5 Discussion 55 5.1 Overflow design . . . . 55

5.2 Loads . . . . 55

5.3 Stability . . . . 56

5.4 Seepage . . . . 56

5.5 Finite element method . . . . 56

6 Conclusions 57

7 Future work 57

Appendices 60

A Final layout of the dam 60

B Hydrological data 64

3 Values of parameters . . . . 29

4 Material table showing the material properties in the spillway structure (R 1 , R 2 , and C) and the soil (B 1 and B 2 ) . . . . 33

5 The resulting values of the design parameters used in the final CAD design of the overflow spillway. . . . 35

6 Values of parameters for normal water level. . . . 36

7 Values of parameters for check water level. . . . 37

8 Stability results for anti-sliding. . . . 37

9 Stability results for the compressive stress at the dam toe. . . . 37

10 Stability results for the tensile stress at the dam heel. . . . 38

11 The ratio between the resistant forces R and the load action effects S. . . . 38

List of Figures 1 The overflow spillway design showing important dimensions. . . . 5

2 The overflow spillway highlighting the different splines constructing the overflow weir curve in colour. . . . 6

3 Detailed view of the 3 arcs constructing the first spline of the overflow design. . . . 7

4 Magnified view of the third spline making up the flip bucket. R is the anti-arc radius and θ is the bucket angle. . . . 8

5 Figure showing the trajectory distance L and the scour hole depth t k after the flip bucket (reproduced from Chen [2]). . . . 8

6 Conceptual figure describing the gravitational forces acting on the dam structure. The moment arms relative to the dam centre are also shown. . . . 10

7 Conceptual figure describing the hydrostatic pressure acting on the dam structure. The arms of the moments can also be seen here. . . . 11

8 Conceptual figure depicting the uplift pressure acting on the overflow spillway sec- tion. . . . . 12

9 Conceptual figure depicting the hydrodynamic pressures acting on the overflow sec- tion. Note that the spillway is divided into several sections between point A and E . . . . 13

10 The overflow spillway design with the horizontal and vertical silt pressures acting on the structure. . . . 14

11 The overflow spillway design with resulting earthquake forces and moment arms. . 15

12 Working state of a design model. The limit state function Z = 0 divides the model domain into three sets (reproduced from Chen [2]). . . . 17

13 Conceptual figure of the overflow spillway showing the summation of horizontal forces F H and vertical forces F V , along with the resulting moment ~ M with respect the centroid of the base. . . . 21

14 Conceptual stress-strain curve for a brittle material. . . . 22

15 Conceptual stress-strain curve for a ductile material. . . . 22

16 Conceptual figure showing two types of mesh elements: a triangular element and a quadrilateral element. The black dots are main nodes and the white dots represents secondary nodes. . . . 31

17 Close-up of the mesh at the dam heel for the stress and deformation analysis using secondary nodes. . . . 31

18 Close-up of the mesh at the weir crest for the seepage analysis using no secondary nodes. . . . 31 19 Conceptual figure of the overflow spillway showing the material distribution for the

overflow section and the soil below it in bold face and the boundary points in regular. 32

section for the normal water level. . . . 39

23 Contour plot showing the horizontal stresses on the overflow section for the normal water level. . . . 39

24 Contour plot showing the vertical stresses on the overflow section for the normal water level. . . . 40

25 Contour plot showing the maximum total stresses on the overflow section for the normal water level. . . . 40

26 Contour plot showing the minimum total stresses on the overflow section for the normal water level. . . . 41

27 Contour plot showing the total displacements of the overflow section for the normal water level. . . . 41

28 Contour plot showing the horizontal displacements for the overflow section for the normal water level. . . . 42

29 Contour plot showing the vertical displacements for the overflow section for the normal water level. . . . 42

30 Contour plot showing the combined horizontal and vertical stresses on the overflow section for the check water level. . . . 43

31 Contour plot showing the horizontal stresses on the overflow section for the check water level. . . . 44

32 Contour plot showing the vertical stresses on the overflow section for the check water level. . . . 44

33 Contour plot showing the maximum total stresses on the overflow section for the check water level. . . . 45

34 Contour plot showing the minimum total stresses on the overflow section for the check water level. . . . 45

35 Contour plot showing the horizontal displacements of the overflow section for the check water level. . . . 46

36 Contour plot showing the vertical displacements of the overflow section for the check water level. . . . 46

37 Contour plot showing the total displacements of the overflow section for the check water level. . . . 47

38 Close-up of the deformed mesh in the crest area of the overflow section for the check water level. . . . 47

39 Close-up of the deformed mesh in the bucket area of the overflow section for the check water level. . . . 47

40 Close-up of the deformed mesh in the heel area of the overflow section for the check water level. . . . 48

41 Close-up of the deformed mesh in the toe area of the overflow section for the check water level. . . . 48

42 Contour plot showing the total water head for the normal water level. . . . 48

43 Contour plot showing the pressure head for the normal water level. . . . 49

44 Contour plot showing the pore water pressure for the normal water level. . . . 49

45 Contour plot showing the water flux for the normal water level. . . . 50

46 Close-up picture of the area beneath the overflow section complete with flux vectors for the normal water level. . . . 50

47 Contour plot showing the total water head for the check water level. . . . 51

48 Contour plot showing the pressure head for the check water level. . . . 51

49 Contour plot showing the pore water pressure for the check water level. . . . 52

50 Contour plot showing the water flux for the check water level. . . . 52

51 Close-up picture of the area beneath the overflow section complete with flux vectors

for the check water level. . . . 53

54 Topology map of the dam site before the construction of the Longtan dam. The straight lines over the river represents the four different section of the dam to be built. 60 55 The Longtan dam with the re-designed spillways included as viewed from downstream. 62 56 The Longtan dam viewed from the above. The ship lift can be seen on the left, the

seven overflow spillways in the middle, and the seven penstocks for hydropower on

the right. . . . 63

Hydraulic projects are used to alter a natural water body (for example a river or a lake) to obtain desired characteristics. They often have multiple purposes and are therefore made of a collection of single-purpose structures, such as retaining structures, conveying structures, and special-purpose structures. Retaining structures are used to block off the stream of water, usually in the form of a dam to create a water storage reservoir. Conveying structures are artificial channels made of either natural soil and rock or manufactured materials. Special-purpose structures are usually, but not limited to, structures for hydroelectric power generation, structures for inland transportation, and fish passages.

Historically, hydraulic projects for flood control and irrigation have always been important. Water is vital for the survival of mankind, and people have been settling down close to rivers and seas to satisfy their consumption and transportation needs since very early on. The great ancient civilizations in Egypt, Mesopotamia, Europe, and China all used certain hydraulic techniques for irrigation of their lands [27]. Ancient Egypt was dependent on the river Nile for the irrigation of the crops, first limited to the annual natural flooding of the river for farming, but later on reached the point of basin irrigation to gain more control over the process. Mesopotamian engineers used the two-river system involving the higher elevation Euphrates river and the lower elevation Tigris river for irrigation and could thus plant more crops. They used dams to a greater extent to barricade rivers and control the flow of water, but the perennial irrigation lead to huge maintenance problems since silt eventually blocked the canals causing flooding, and over time soil salination became too great which led to the demise of the canal system. Roman engineers brought much more structured engineering into the construction of dams and on a larger scale, introducing reservoirs for a more permanent water supply as well as new dam inventions. Much later, the industrial revolution brought the materials necessary to scale up these dam constructions, and in the twentieth century, truly large dams of different types were constructed making it possible to alter even the largest rivers and to extract electricity from them with the help of turbines, giving birth to the hydropower era.

1.1 Dam types

Dams can generally be divided into three structure types. Embankment dams are probably the earliest type of dams and consist of compacted earth to block up a river or a lake. They can either be simple earth-fill dams, made of a single material or multiple materials (usually a different core material), or rock-fill dams, with larger particle granular earth with an impervious zone, often made of concrete, masonry, moraine, or clay. Embankment dams are often very cost effective since the material is often close-by or even made available from the excavation of the dam site. They can also be made very aesthetically pleasing since the earth foundation allows it to blend in well with the surrounding environment, in contrast to gravity dams which make quite a significant mark on the surroundings. Arch dams are elegant concrete dams which gain their stability from an upstream curvature which compresses the concrete material (which has very weak tensile strength, but much stronger compressive strength) and the surrounding abutments. In recent times these have become truly massive in size, and the most famous one is probably the Hoover dam located in the Colorado River in the United States. As the name suggests, gravity dams use their sole weight to counteract the overturning moment created by the water in the upstream reservoir, which makes it rather dependent on a strong and impervious foundation. The basic gravity dams usually consists of triangular non-overflow sections used for retaining the water and a spillway section for releasing excess water.

1.2 Spillways

The spillway is often the most important structure in a larger dam project. It is responsible for

safely discharging excess flooding to avoid overtopping of the dam, which would lead to catastrophic

scenarios for inhabitants and environment downstream, as well as the dam itself. Being able to

discharge enough water is thus of critical importance when designing a new spillway. Different

of a chute which smoothly leads the water from the higher potential reservoir to the downstream water body. These can either be pure overflow spillways which automatically starts releasing water when the water level become too high for the reservoir, or be accompanied by a gate house with a sluice gate making it possible to control the release of water. As the water is released, the potential energy transforms into kinetic energy which can put a lot of strain on the materials in the dam or on the river surroundings downstream. For this reason, the spillway is often accompanied by an energy dissipator either in the form of stilling basin or as a bucket lip, airing the water to dissipate energy.

1.3 Longtan dam

In this thesis, the design and simulation of an overflow spillway with an bucket lip dissipator di- mensioned for the Longtan dam will be carried out. The Longtan dam is with its 216.5 metres the highest roller-compacted concrete (RCC) gravity dam in China [23] and is located in the Hongshui River close to Tian’e County of the Guangxi Zhuang Autonomous Region in the People’s Republic of China. The construction of the dam started in 2001 with the main purpose of being a hy- dropower generation facility, but also for flood control and navigation purposes [22]. The river was successfully dammed in 2003 according to schedule and the first generator of the plant passed trial operation in 2007 [1, 4, 5]. The last of the nine generators became operational in 2009, creating an estimated total annual generation of electricity of 18.7 TWh.

Thanks to the submerging of a large number of shoals upstream after construction and an im- pressive ship lift capable of lifting vessels up to 500 metric tonnes, the Longtan dam opens up inland China to the South China Sea via the Pearl River. This connects the resource rich western parts of China to the more developed eastern parts, but also benefits the inland provinces since transportation of minerals and coal between them greatly improves.

Apart from the power generation aspect of the dam, flood control was also of main interest since the Hongshui River have been prone to flooding in the 1980’s and 1990’s, causing heavy economic damages and loss of lives [29]. Longtan dam has a normal water level of 377.1 m and a storage capacity of 27.3 billion cubic metres and seven overflow spillways capable of discharging over 27 000 cubic metres per second, which allows the structure to prevent flooding downstream, potentially saving many lives in the case of extreme flooding scenarios. It is the design process and stability regarding overturning, seepage, sliding, and structural integrity against cracking and crushing of one of these overflow spillway sections that is of interest in this thesis.

1.4 Problem formulation

For an overflow spillway, finding the design head water level is the first step taken, and to find this, some engineering hydrology is needed, which is very dependent on both spatial and temporal parameters. Much data is needed when applying engineering hydrology to dam projects to ensure safe operation, such as water levels, flow discharge, precipitation, run-off data, silt, and evaporation.

Data of main interest when initially designing a new overflow spillway is however the design run-

off, which yield information about the historical hydrological events, the design flood, which is in

essence the flood the dam is designed to handle, and the reservoir design flood, ensuring that the

reservoir can store or partially store the flow of water. The spillway should be able to pass the

probable maximum flood (PMF) according to regulations, and proper flood routing calculations

will help set the proper elevation and design of the overflow weir. When this is done, identification

and calculation work can be initiated to find the loads and moments acting on the spillway, which

is essential when doing a stability analysis as well as a numerical finite element analysis of structure

to check if its integrity remains satisfactory. If not, as in most engineering, modifications are made

and the analysis is repeated until the spillway meets expectations. This thesis work is however

based on a dam construction already finished, and the sole purpose of it is therefore to gain insight

in how the design process takes place based on real world data.

The purpose of the thesis is to make the necessary preliminary calculations and design steps to create a numerical finite element analysis of an overflow spillway section dimensioned for the Longtan dam in the People’s Republic of China. The analysis will check stress and deformation for the overflow spillway as well as seepage underneath and through the structure. The analysis is limited to a two-dimensional model of the overflow section only, leaving out the non-overflow retaining structures. Because of time limitations, many parameters are also handed "as is" by the supervisors without closer investigation. In reality, all parameters need to be studied in detail and many factors have to be considered when constructing a structure of this magnitude. Also, the basic profile has not been determined by the authors, rather, it was already available.

1.6 Structure of thesis

The following steps will be carried out to complete the design model:

1. A flood routing calculation will be performed with the help of hydrology data to find the flood control standard of the overflow design. This includes the design water head, the check water head, as well as the elevation of the weir crest.

2. Based on the above results as well as many other design parameters based in standards and theory, a two-dimensional overflow spillway design is constructed using Computer Aided Design (CAD) software.

3. The general loads and moments acting on the overflow spillway are then identified and cal- culated, which are later used in a basic stability analysis of the structure.

4. Finally, a two-dimensional finite element analysis is made on the overflow spillway using the above data for geometry and boundary conditions. GeoStudio ^® SIGMA/W and GeoStudio ^® SEEP/W are used to perform the stress-deformation analysis and seep analysis, respectively.

A simple verification of the results and the meshing of the model is also made to see whether or not the model converges numerically.

The structure of the thesis is as follows:

• Section 2 introduces the reader to the theory needed for the thesis.

• Section 3 presents the methods used when appropriate, for example to obtain the time- dependence of the average in- and outflow in the reservoir for the flood routing calculation.

Also, the methods, such as meshing, boundary conditions, geometry, and materials, used in the finite element analysis are presented.

• Section 4 presents the results of the flood routing calculation, the resulting design of the overflow section, loads and moments acting on it, and finally the results of the numerical finite element analysis.

• Section 5 analyses the results above and concludes the thesis. Recommendations on future work is also made.

• section 6 covers the conclusions drawn about the thesis and, finally, section 7 covers some suggested future work on the subject.

A set of appendices are included outside the thesis structure in the end in an effort to not get in the way of the reader. Section B lists the hydrological data used for the flood routing calculation.

Section A shows the modified final layout of Longtan dam for the interested reader.

In the section the underlying theory of this thesis is presented and reviewed. The topics are flood routing calculation, design process of the overflow section, loads, stability, stress-strain modelling, seepage, and the finite element method.

2.1 Flood routing

The flood routing calculation is made to make sure that the overflow spillway can pass the PMF, and is in essence rooted in the continuity. In the case of a large dam reservoir as in the Longtan dam, the water level can be assumed to be horizontal at all times which eliminates the dynamic effects in the momentum equation. Thus, only the continuity equation, seen in eq. (1), is necessary for the routing calculations.

dS

dt = Q in − Q out (1)

This equation states that the rate of change of storage volume S in the reservoir is equal to the difference in inflow Q in to the reservoir and outflow Q out out of the reservoir. The inflow is given by a hydrograph, which is a record of flow to reveal the run-off characteristics of the reservoir, and the storage volume is a function of the water level relative to a datum. The outflow can be calculated from the equation of flow over a weir of a spillway, seen in eq. (2), assuming of course that the entire outflow is passed over the weir and through the turbines [2].

Q = N · ς · ξ · M · B · q

2gH w 3 (2)

Here, Q is the flow over the weir, N is the number of sluice vents, ς is the discharge reduction coefficient due to the degree of submergence, ξ is the lateral contraction coefficient, M is the coefficient of discharge, B is the net width of the sluice vent, g is the gravitational constant, and H _w is the water head over the weir crest. The equation can often be simplified to eq. (3), where the coefficient k can be modified to design flood and check flood specifications.

Q = k · B · H w 3/2 (3)

For the current spillway design, the given coefficient by the supervisor is k = 2.01 for the design flood specification and k = 2.04 for the check flood specification. Relevant hydrological data for spillway design for the Longtan dam can be seen in section B. Here is the inflow hydrograph for a period in June and July represented, both for a 500 years recurrence interval and a 10 000 years recurrence interval, as well as a table representing the storage volume versus the reservoir water level. With the help from this data and eq. (1) and eq. (3), along with given design parameters, the goal of the routing calculation is to find when the average inflow curve intersects the average outflow curve. At this moment, the maximum reservoir storage is found and the design head for the overflow spillway design is found. Two water heads are decided. The design water head is the amount of water the overflow section should be designed to handle, while the check water head is based on rare catastrophic flooding scenarios involving even larger amount of water.

2.2 Design process of overflow section

2.2.1 Practical profile

This section covers the theory of the design of the overflow section. This includes the practical profile, the overflow weir curve and its components, as well as trajectory distance and scour depth.

The basic profile, i.e. slopes of upstream and downstream faces along with the slope turning point,

of an overflow spillway may be divided into three types; vertical, inclined or partially inclined

upstream face. The choice of profile is made based upon stress analysis, e.g. if the dam foundation

has a relatively small shear strength, an inclined or partially inclined upstream face should be con-

sidered in order to counteract sliding. If, on the other hand, the foundation inhabits a relatively

large shear strength, the first principle stress will govern the profile and a vertical upstream face

y

x

1:m

1:n

H c

H d

H _t

H _w

H s

d

H l

W

Figure 1: The overflow spillway design showing important dimensions.

may be allowed.

The authors of this report have not conducted a stress analysis in order to obtain the basic profile, rather, the chosen profile along with values of the slopes was already available. The chosen profile has a partially inclined upstream face where the upstream slope is n = 0.15 and the downstream slope is m = 0.75, see fig. 1. From design requirements given by the supervisors, the elevation of the upstream slope turning point H s was subjected to the condition

1

3 H _t ≤ H _s ≤ 2

3 H _t (4)

where H t is the elevation from the bottom of the dam to the check water level H c . Also, the elevation of the turning point has to be lower than the elevation of the steel penstock H p , which, according to the supervisors, is given by

H p = H dead − Scr − d (5)

where H dead is the dead water level and S cr is the vertical distance between the dead water level and the top of the penstock, given by

S cr = cv √

D (6)

where the empirical coefficient c satisfies 0.55 ≤ c ≤ 0.73, the economical flow velocity ν satisfies 4 ≤ ν ≤ 6 m/s and D is the diameter of the penstock, which can be obtained using knowledge of the full load discharge.

Sometimes, the required spillway crest profile is wider than what is economically feasible. In order

to make the overall profile thinner, while still satisfying the required profile, the upstream face is

designed to have an offset into the reservoir. This offset does not impact the discharge coefficient

y

x Spline 1 Spline 2

Line

Spline 3

Figure 2: The overflow spillway highlighting the different splines constructing the overflow weir curve in colour.

noticeably as long as the offset height d satisfies [2]

d ≥ H _max

2 (7)

where H max is the maximum head over crest, i.e. H max = H c − H w , see fig. 1.

2.2.2 Overflow weir curve

The typical design of an open overflow spillway is as a reversed S-shaped ogee, as it closely approx- imates the contours of free flow over a sharp weir, making it very effective in guiding the flow of water over the crest. It makes the flow of water adhere to the floor face of the spillway preventing air from slipping underneath the nappe which would otherwise produce a negative pressure and thus have a negative impact on the energy dissipation and the discharge coefficient.

The overflow curve consists of three splines and a straight line. The first spline is made of three arcs with different radii, each a different fraction of the design water head. The second spline is the WES curve made according to design specifications of the State Economy and Trade Commission of the People’s Republic of China [7]. The third spline is a plain flip bucket responsible for the trajectory of water at the end of the overflow spillway. Connecting the second spline and the third spline is a straight line with the slope m = 0.75.

To eliminate the negative effect of a sudden discontinuity at the upstream face of the weir crest, a defined curve consisting of three arcs is introduced upstream before the top of the crest. The three arc upstream curve is the most common in the People’s Republic of China, although variants with one or two arcs, or even an elliptic curve, are equally valid. The three arc spline can be seen magnified in fig. 3, where the different arc radii and distances from the crest origin are defined.

The first arc closest to the origin has an radius of R 1 = 0.5H d and its arc end is placed a horizontal

y

x b 1 = 0.175H d

b 2 = 0.276H d

b ₃ = 0.282H _d

R ₁ = 0.50H _d R 2 = 0.20H d

R 3 = 0.04H d

y

x

Figure 3: Detailed view of the 3 arcs constructing the first spline of the overflow design.

distance of b 1 = 0.175H _d , where H d is the design water head. The second arc continues where the first arc ended with a radius of R 2 = 0.2H _d and with its end placed a horizontal distance b ₂ = 0.276H _d from the origin. The third and last arc has a radius of R 3 = 0.04H _d with its end placed a horizontal distance b 3 = 0.282H d from the crest origin according to the figure.

The second spline consist of a WES curve, and the shape of such a curve can be seen in fig. 2 (in blue), and the equation can be seen in eq. (8) (according to the design code of the State Economy and Trade Commission of the People’s Republic of China, [7]).

y = x ^η

KH d η−1 (8)

Here, H d is the design water head and K and η are constants which depends on the upstream inclination and the velocity of the approaching flow. In the case of a vertical upstream dam face, as is the case in this design, η = 1.850 and K = 2.0. The function of this spline is to keep the negative pressure within certain desired limits. More negative pressure usually allows for larger discharges but can also increase the risk for destructive cavitation damages on the spillway face.

The third spline makes up the flip bucket responsible for the trajectory of water at the end of the overflow spillway, and can be seen in fig. 4. The flip bucket consists of an anti-arc radius R which is dependent on h, which is the average water depth at the bottom of the arc. This water depth is determined by using the check flow discharge formula seen in eq. (9), which was handed to the authors by the supervisors.

Q c = Bhν = Bhφ p

2gH 0 (9)

Here, Q c is the check flow discharge, B is the orifice width of the spillway, φ is the velocity coef-

ficient, and H 0 is the height difference between the upstream water level and the top of the weir

crest. By solving this equation for h, the anti-arc radius R can be determined. A larger radius

improves the flow conditions but will increase the project cost since more concrete is needed, while

smaller radius instead can lead to non-smooth jet trajectory limiting the throwing distance from

the bucket. According to [2], a good compromise is a radius of (6 − 10)h and the elevation of the

bucket lip should be placed approximately 1 to 2 metres above the check tailwater level to provide

H l

θ R

Figure 4: Magnified view of the third spline making up the flip bucket. R is the anti-arc radius and θ is the bucket angle.

Tailwater level Maximum water head

L h 2

h 1

t k

H ν l

h

θ

Figure 5: Figure showing the trajectory distance L and the scour hole depth t k after the flip bucket (reproduced from Chen [2]).

a desirable aeration of the bottom jet nappe.

The bucket angle θ is the angle which the flip trajectory makes with the elevation line. A larger bucket angle moves the impact zone further away from the bucket, which is often desirable since it reduces retrogressive erosion. A larger bucket angle will however also increase the accompanied dip angle of the water jet, increasing the depth of the scouring pit. [2] suggests an angle between 15 ^◦ and 35 ^◦ , although the supervisors required an angle above 25 ^◦ .

2.2.3 Diversion walls and gate house

To contain the water flow and preventing it from sipping over to neighbouring spillways, diversion walls are built. These should be able to safely maintain the average water level height in the spillway, as well as the water in the flip bucket. The height of the diversion wall along the spillway side is given by the supervisors and is 10 metres, while it is slightly higher next to the flip bucket.

From design specifications the height of the gate coincides with the elevation of the normal wa- ter level which is 377.10 metres and the elevation of the pier (not including the wave wall) was calculated by [16] to be 381.00 metres. The width of the pier is 30 metres.

2.2.4 Trajectory distance and scour hole depth

The trajectory distance is defined as the distance L between the bucket lip and the deepest point

of the scour pit as seen in fig. 5, and a formula making a decent estimation of it can be seen in

θ by some power law, i.e. L ∝ θ ^p , where p < 1 [7].

L = 1

g ν _l ² sin θ cos θ + ν l cos θ q

ν _l ² sin ² θ + 2g(h ₁ cos θ + h 2 )

!

(10) Here, ν l is the velocity at the top of the bucket lip (approximately 1.1 times the average velocity ν ), θ is the bucket angle, h 1 is the vertical projection of the average water depth h defined as h 1 = h cos θ, and h 2 is the height from the bucket lip to the river bed. For jet velocities below 20 metres per second, the air resistance on the jet trajectory is negligible, while it for larger velocities can have a significant impact on the trajectory length.

Although many different formulae exists for approximating the scour depth, the most common one in the People’s Republic of China can be seen in eq. (11).

t k = αq ^1/2 H ^1/4 (11)

Here, t k is the distance from the scour pit bottom to the tailwater level (also seen in fig. 5), α is material dependent coefficient (1.2 for the Longtan dam), q is the unit discharge of the jet trajectory defined as (Q c − Q g )/B , where Q g is the flow through the generators (approximately 2500 m ³ s ⁻¹ ) and B is the channel width (105 m), and H is the difference in headwater level and tailwater level (also seen in fig. 5).

2.3 Loads

In order for the dam to operate safely, both now and in the future, every significant force acting on the dam structure and its effects has to be evaluated. In order to evaluate the dams ability to withstand the external forces it is also useful to calculate the moments that is generated from the different forces acting on the section about the midpoint of the dam axis. This makes it conve- nient to compare the overturning moments with the moments withstanding overturn in order to determine the stability of the dam. The forces in this thesis are not explicitly evaluated, rather, they are evaluated per unit meters width of the section, i.e. the analysis is two-dimensional. Also, the arm of the moments are evaluated using AutoCAD ^® if no readily known formula is available.

The loads included in this thesis are:

• Self-weight of the overflow section

• Hydrostatic pressure at the upstream and downstream face

• Uplift pressure

• Hydrodynamic pressure at the weir curve and flip bucket

• Silt pressure at the upstream face

• Seismic forces, i.e. earthquake inertia and hydrodynamic forces

A dam is also often affected by other forces, such as the thermal effects and ice pressure, however, only the forces listed above were taken into account in this thesis. This chapter covers the theory behind these forces and the partial coefficient method which was used to evaluate the design condition.

2.3.1 Self-weight

The primary opposing force of a gravity dam is generated by its self-weight. The self-weight is considered to act through the centroid of the section in question with a magnitude given by

G i = γ c A i = ρ c gA i (12)

x G ₁

G 2

G 3

G 4

G ₅ G ₆

L _G

L G

L _G

L G

L _G

L G

Figure 6: Conceptual figure describing the gravitational forces acting on the dam structure. The moment arms relative to the dam centre are also shown.

where γ c is the unit weight of concrete and A i is the area of the section in question. It is important to note that the value of the unit weight varies depending on the water content of the concrete, i.e.

the unit weights of dry- and saturated concrete is different, and also whether or not the section is submerged under water where buoyancy effects are present.

By dividing the geometry of the profile into several sections, one may gain insight about how the different sections affects the net moment, see Figure 6. Also, by carefully choosing the sections one can obtain the coordinates of the centroids and areas of each section using basic geometry, however, most CAD-software can obtain these directly. In this thesis, the geometry is divided into six subsections. Note that only the basic profile section was considered, e.g. the pier and diversion walls were not considered.

2.3.2 Hydrostatic pressure

The main external force acting on the dam is the hydrostatic pressure from the water in the reservoir applied on the upstream dam face. The hydrostatic pressure at a point is directly proportional to the height of the water surface, i.e.

¯

p = γ w h = ρgh (13)

where γ w = 9.81 kNm ⁻³ is the unit weight of water and h is the water surface height. Note that ¯p is a vector quantity, i.e. it has a direction. The hydrostatic pressure is always applied orthogonally to the surface in question and the total pressure force field acting on the dam section can be obtained by integrating eq. (13) over the bounding area Ω, i.e.

P = ¯ Z

Ω

¯

pdΩ (14)

The hydrostatic pressure is often divided into two components; one horizontal and one vertical

part. The horizontal pressure is mainly applied at the upstream face, however, the downstream

x L _P

L P

L P

P u (y)

P _d (y) P _v (x)

Figure 7: Conceptual figure describing the hydrostatic pressure acting on the dam structure. The arms of the moments can also be seen here.

face also experiences some pressure, see Figure 7. These horizontal pressure distributions can be viewed as an resultant force with the magnitude given by

P u = 1

2 γ w H _u ² (15)

and

P d = 1

2 γ w H _d ² (16)

which is the area of the force distributions. Here, P u corresponds to upstream water pressure and P d corresponds to the downstream water pressure. H u and H d corresponds the upstream and downstream water depths, respectively. This resultant horizontal force is acting through the centroid of the force distribution and thus, from knowledge about the geometry of a triangle, the arms of the moments is given by

L P

= 1

3 H u (17)

and

L P

= 1

3 H d (18)

The vertical pressure for the dam section in this report is only being applied at the upstream face where, in contrast to the downstream face, there is a lip at the toe, see Figure 7. By obtaining the area of the water directly above the dam lip it is possible to calculate the weight of water, i.e.

P v = A ² _v γ w (19)

where A v is the area of the water above the heel lip.

x

H up

α ₁ H _up

H s n + 7

H down

α 1 H down

7

7 H _s

Figure 8: Conceptual figure depicting the uplift pressure acting on the overflow spillway section.

2.3.3 Uplift

As water seeps through the foundation material of the dam, e.g. through fractured rock joints, an uplift force will be generated. The uplift pressure exists in the boundary between the dam struc- ture and the foundation, as well as in the cracks of the foundation and dam structure themselves.

A common method to reduce the uplift is to build a drainage system, as has been done for the dam covered in this report. Another interesting method for decreasing the seepage through the foundation and consequently the uplift is to let silt with low seepage permeability accumulate at the upstream face [21].

The uplift pressure is assumed to act on the whole bottom part of the structure, and if no drainage system is available, the uplift pressure can be assumed to vary linearly from the water pressure at the upstream face heel to the water pressure at the downstream face heel [10] [21]. If a drainage system is in place the uplift can be reduced at these locations, governed by a effectiveness scale factor α, usually ranging from 0.25-0.50 depending on size, depth, and scaling of the drains [10]

(even though other values are possible if the dam foundation and data supports it). Figure 8 is showing the overflow spillway design in this thesis with two drains, one in at the upstream level and one at the downstream level, with drain factors of α 1 = 0.2 and α 2 = 0.5 , respectively.

When calculating the resulting uplift forces on the structure, the equivalent static load is deter-

mined from the area of the line load, and its action is determined from the combined centroid of

the line load. From this, a resulting moment acting on the dam structure relative to bottom centre

can be calculated. Doing it this way, there will be three resulting forces from the uplift, one acting

on the load centroid between the dam heel and the first grout curtain, the second acting on the

load centroid between the curtains, and a third acting on the load centroid between the second

curtain and the dam toe.

x AB

C

D E

φ d

d

θ ⁰ θ F P

Figure 9: Conceptual figure depicting the hydrodynamic pressures acting on the overflow section.

Note that the spillway is divided into several sections between point A and E.

2.3.4 Hydrodynamic pressure

In contrast to the hydrostatic pressure, the hydrodynamic pressure concerns the pressure exerted by a fluid in motion. There is, essentially, no difference between the static and dynamic pressure when the flow is uniform. Even for small flow variations (for example convey channels) the dif- ference is insignificant. However, in the presence of strong, concave curvature of streamlines the dynamic pressure will dominate.

For the first part of the weir curve, i.e. section AB in Figure 9, the hydrodynamic pressure can confidently be represented as a vertical hydrostatic pressure. The section BC is described by the WES weir profile seen earlier in eq. (8). When using this profile, the resulting pressure can be negative, zero or positive depending on whether or not the working head is greater, equal to or smaller than the design head, respectively. Thus, provided a proper choice of the design head, the pressure acting on section BC can be assumed to be small enough to be ignored. For the straight line section CD the flow variations can be assumed to be small, i.e. the hydrodynamic pressure can be described by a hydrostatic pressure. The magnitude of this pressure can be expressed as

F = Aγ w sin α = hΓγ w sin α (20)

where A is the area of the water above the section, Γ is the length of the line segment and α is the angle between the straight line and a vertical reference line. The average flow depth h is assumed to be the same as the average height in the bucket, which is given by eq. (9).

As aforementioned, the force can advantageously be divided into two parts; one horizontal force given by

F x = F cos α (21)

and one vertical force given by

F y = F sin α (22)

x L _Psk

L _Psk

P sk

P _sk ⁰

Figure 10: The overflow spillway design with the horizontal and vertical silt pressures acting on the structure.

These forces can be viewed as acting through the midpoint of CD, giving them the arm of moments L _F

and L F

. For the bucket section, i.e. section DE, the curvatures of the streamlines are strong and concave, thus, the dynamic pressure will dominate. According to [2], the hydrodynamic force exerted by the fluid on the bucket is given by

P x = qνγ w

g (cos θ ⁰ − cos θ) (23)

and

P y = qνγ w

g (sin θ ⁰ + sin θ) (24)

where P x is the horizontal part, P y is the vertical part, q is the unit discharge, θ ⁰ is the angle between the bottom of the bucket and the lip and θ is the angle between the bottom of the bucket and the upstream endpoint of the bucket, see Figure 9.

2.3.5 Silt pressure

During operation of the dam the river will bring with it silt. Due to the decreasing velocity profile close to the dam, the coarse grained silt will be deposited further away from the dam compared to the fine grained silt which builds up at the upstream dam face. As the amount of deposited silt is increased the force generated is also increased. By using the active earth pressure formulation [2], the horizontal silt pressure can be evaluated as

P sk = γ s h s tan ² (45 ^◦ − ϕ _s

2 ) (25)

where γ s is the buoyant unit weight of the silt, h s is the height of the deposited silt in front of

the dam and ϕ s is the internal friction angle of the sediment. The height of the deposited silt is a

complex function of various parameters, but it can be approximated by physical and mathematical

models based on information obtained from field studies. The internal friction angle is dependent

on the diameter, particle shape and graduation of the silt grain, as well as silt depth and unit

x F 1

F 2

F 3

F ₄ F 5

H t

0.54H t

0.46H t

F 0

θ f

L F

L _F

L F

L _F

L _F

Figure 11: The overflow spillway design with resulting earthquake forces and moment arms.

weight. It is also worth noting that, during the silt accumulation process, the unit weight and silt depth, and by extension the internal friction angle, will vary. More on this can be found in e.g. [2].

The values of the buoyant unit weight, height and internal friction angle was given to the authors by the supervisors beforehand and were 12 kNm ⁻³ , 92.6 m and 24 ^◦ , respectively.

For completely vertical upstream faces, the silt can be considered to act only in the horizontal direction. However, for inclined upstream faces, as for the overflow section covered in this thesis, the vertical silt pressure is calculated in a similar manner as the hydrostatic pressure, i.e.

P _sk

= A _s γ _s = 1

2 h ² _s nγ _s (26)

where A s is the area of the silt force distribution and n is the slope of the toe, see Figure 9.

2.3.6 Earthquake forces

Since gravity dams can harbour an enormous amount of water with its appurtenant potential energy, an earthquake can obviously have catastrophic effects on these structures if they are not designed accordingly. The dynamic forces acting on a gravity dam during an earthquake are:

• Seismic inertia forces

• Seismic dynamic water pressure

• Seismic dynamic earth pressure

The effect of the earthquake on additional forces such as uplift pressure, wave pressure, and silt pressure, are usually ignored in dam design [2].

In the People’s Republic of China, the seismic and fortification design criteria for hydraulic struc- tures are determined according to their importance and the basic intensity of the dam site [8].

For purposes of preliminary analysis, the pseudo-static method, or equivalent static load method,

are usually applied when designing hydraulic structures in China, even though a more thorough

or in more earthquake intensive areas.

With the pseudo-static method, the inertia forces are calculated according to a chosen maximum design acceleration, directed in either horizontal or vertical direction (or both, depending on the dam site), and are considered equal to static forces acting through the centroid of a mass element.

For dams in areas of moderate earthquakes, the inertia force in the horizontal direction is sufficient, and a free-body diagram for the spillway design scenario with five mass elements, desired by the supervisors, can be seen in Figure 11.

Each mass element in Figure 11 has an equivalent static force through its centroid given by eq. (27) [2].

F _i = a _h ξG _E

α _i

g (27)

where a h is the standard value of the horisontal earthquake acceleration, G E

is the weight of the mass element i, ξ is the comprehensive influencing coefficient, and α i is the dynamic distribution coefficient at mass element i. The comprehensive influencing coefficient ξ is a necessary scale factor in order for the calculations to agree with reality, since many of the parameters used in the calculations are, at least partly, based on experience. The dynamic distribution coefficient α i of the mass element i for gravity dams is given by eq. (28) from [2]

α i = 1.4 1 + 4(h _i /H _T ) ⁴ 1 + 4 P N

j=1 G

G

(h j /H T ) ⁴ (28)

where N is the number of mass elements, H T is the total height of the dam, h i and h j are the height of the mass element i and j, respectively, G E is the standard value of the total weight of the dam, and G E

is the standard value of the weight of each mass element i. eq. (28) together with eq. (27) along with appropriate parameter values will yield the inertia forces on the hydraulic structure and the moment arms can be determined from the distance from the dam bottom centre to the respective centroid of the structure.

When a dam structure is moving due to an earthquake, the water reservoir tends to remain at rest causing a complex hydrodynamic pressure on the structure. According to Chinese design specifications [8], this seismic hydrodynamic pressure is calculated using a quasi-static method and is the non-linear function with regard to water depth given by eq. (29)

P w (y) = a h ξψ(y)ρ w H t (29)

where ψ(y) is the distribution coefficient of water pressure, ρ w is the standard value of water density, H t is the upstream water depth, ξ is the comprehensive influencing coefficient (same as in eq. (27)), and a h is the standard value of the horisontal earthquake acceleration (same as in eq. (27)). The total hydrodynamic force per unit length at a upstream water depth of (force of action) y = 0.54H 0 is given by eq. (30)

P ₀ = 0.65a _h ξρ _w H _t ² (30)

For the current project, a h = 0.1g , ξ = 0.25, ρ w = 1000 kgm ⁻³ , and the water depth is the normal or check water depth. If the upstream dam face has an inclination angle θ degrees according to Figure 11, eq. (30) should be multiplied with an reduction factor η c = θ/90 ^◦ .

In addition to the above, the seismic earth pressure is also an influencing factor during an earth-

quake. It consists of a dynamic part and a static part, and even though the dynamic part can

be estimated using complex analytic formulae with many unknowns (see [2] for brief explanation),

the hydrodynamic passive component has to be decided through special studies. The seismic earth

pressure is considered out of scope for this project will not be taken into account here.

Z > 0 (Safe state)

Z < 0 (Unsafe state)

Z = 0 (Limit state)

X ₁ X 2

Figure 12: Working state of a design model. The limit state function Z = 0 divides the model domain into three sets (reproduced from Chen [2]).

2.4 Stability

This section covers the theory of stability of hydraulic structures. The topics are reliability, state functions and the limiting state for compressive stress at dam toe, tensile stress at dam heel, and sliding at the dam base. Also covered are failure probability and partial safety factors and the corresponding actions.

2.4.1 Reliability of hydraulic structures

The reliability of a structure is basically defined as the probability of the structure to complete the functions it was designed for in the design reference period, and depends on the type of hydraulic structure and the service period. The design reference period is not necessarily equal to the service life of the hydraulic structure. The structure can often be used longer, but with reduced reliability as a consequence [2].

2.4.2 State functions and the limiting state

The limiting state is the state where the structure is on the verge of not satisfying its intended functions. In probabilistic terms, this can be expressed with a uncertainty variable Z, which is dependent on the most important uncertainties coupled to the performance of the structure. These uncertainties are represented in the form of independent random variables X n according to eq. (31), which are related to the resistance and load properties of the structure.

Z = g(X ₁ , X ₂ , X ₃ , ..., X _n ) (31)

[2] shows that the state function Z given in eq. (31) divides the stochastic domain of the working state into three sets which is illustrated in Figure 12. Z < 0 indicates an unsafe state, Z = 0 indicates a limiting safe/failure state, and Z > 0 indicates a safe state.

The limit state Z = 0 can be divided into two parts:

• Collapse limit state: The structure is at a critical point of losing integrity, demanding repair

or reconstruction of the site. This could be due to fatigue failure, seepage, or sliding.

normal operation. The structure will return to a safe state after unloading. This could be due to local (isolated) failures or strong vibrations.

2.4.3 Probability of failure

The structural reliability in relation to a definite state function Z can be expressed a distribution function F _s = P(Z ≥ 0) , and since the failure probability is complementary to reliability distribu- tion, it is defined as F f = 1 − F _s = P(Z < 0) . The probability P f of the limit state to be reached can be defined as eq. (32) (see for example [9] and [2]).

P f = Z

· · · Z

f X (X 1 , X 2 , . . . , X n )dX 1 dX 2 · · · dX n (32) Here, f X is the joint probability density function and X N are the variables defined in eq. (31).

The integration is made over the unsafe state region where Z < 0. Although this provides a quan- titative basis for the probability of failure, this integral is in general very computational intensive to solve except for a few and simple cases such as linear state functions. It also relies on that the uncertainties in designs are contained in the joint probability density f X and that f X is known (which is rarely the case, at least not to a greater extent).

To make life easier, linearisation techniques can be applied to Z around a mean value linearisation point, which will yield the linear limit function given by

Z = R − S (33)

where R is the structure resistance and S is the effect of external actions. This can later be used in the stability analysis of the structure.

2.4.4 Partial safety factors

Because of the complexity of the analysis and thus the inconvenience when implementing a design, partial safety factors are commonly used by engineers in the People’s Republic of China. They are a deterministic way of analysing the limit state equation Z = 0 by replacing stochastic variables by multiplying or dividing characteristic parameters by certain safety factors. This could be to reduce material strengths or amplify loads, and thus produce design values, which are the structural forces on the dam in case of a catastrophic event. Partial safety factors are determined using calibration methods accompanied with realistic stochastic models to achieve satisfactory safety margins and is described in full detail in [2].

Partial safety factors are used in different design conditions. According to "Unified design standard for reliability of hydraulic engineering structures" [13], there are three structural design conditions for hydraulic structures during building, operation, and overhaul:

• Permanent actions: Long term conditions which usually coincides with the design period.

The normal operation of the dam is such a condition.

• Temporary actions: Short term conditions, such as construction, overhaul, and repair of the structure.

• Accidents: Rare events that may have catastrophic consequences, such as check floods, design earthquakes, or insufficient drainage.

Depending on the probability of events and environmental conditions, two different design situa- tions are considered for the mandatory collapse limit state.

• Basic combinations: Permanent and alterable loads. Basic actions during normal operation.

(the probability for two or more accidents happening is considered negligible). A check flood and an earthquake, for example, are thus not combined.

Two further serviceability combinations are also defined and are presented below for completeness.

• Short-term combinations: Short-term effects of variable actions together with permanent and temporary loads.

• Long-term combinations: Long-term effects of variable actions together with permanent and temporary loads.

There are a few different partial safety factor in use [2]:

• Structure importance factor γ 0 : This factor is employed to represent the reliability level of the structural safety grades. γ 0 can have values of 1.1, 1.0, or 0.9, which corresponds to safety grade 1, 2, and 3, respectively.

• Design situation factor ψ: Used to represent reliability levels for different design situations. ψ can have values of 1.0, 0.95, and 0.85, which corresponds to permanent situations, temporary situations, and accidental situations, respectively. Note that ψ = 0.85 is not employed in an earthquake situation.

• Load factor γ f : Represents unfavourable variations to characteristic values of loads and is defined as the ratio between an characteristic value of load often specified in standards, and a design value defined near the check point of the limit state equation Z = 0.

• Resistance factor γ m : A material factor defined in much the same way as the load factor, namely a ratio between a characteristic value and a design value.

• Structural coefficient γ d : Employed to consider uncertainties in load and resistance effect calculations.

• Material property factor γ m : Produces design values regarding shear friction strengths, such as the friction and cohesion coefficients.

• Factor for permanent actions γ G : Employed to consider permanent actions on the structure.

• Factor for alterable actions γ Q : Used to scale the different forces acting on the structure, such as hydrostatic and hydrodynamic forces.

The factor for alterable actions γ Q is used in the dimensioning of the overflow spillway in order to amplify the loads and forces acting on the structure to create a safety margin, and their values for the different forces when designing the overflow spillway according to Chinese specifications can be seen in Table 1 [2].

The limit state equation using partial safety factors can now be expressed for both the basic combinations (normal actions) and accidental combinations. For the basic combinations of forces and collapse limit state, the limit state expression is given by eq. (34).

γ 0 ψS(γ G G K , γ Q Q K , a K ) ≤ 1 γ d1

R  f K

γ m , a K

 (34)

Here, S(· · ·) is the load effects function, depending on partial safety factors relating to permanent loads and geometry, R(· · ·) is the resistance function, depending on partial safety factors relating to resistance and geometry. γ 0 , ψ, and γ d1 is the structure importance factor, design situation factor, and structural coefficient for the basic combinations and collapse limit, respectively, as described earlier. a K is simply a characteristic value of a geometry parameter. For the accident combinations of forces and collapse limit state, the limit state expression is instead given by eq. (35) [2].

γ ₀ ψS(γ _G G _K , γ _Q Q _K , A _K , a _K ) ≤ 1

γ _d2 R  f K

γ _m , a _K

 (35)

Type of action Partial safety factors

Self weight 1.0

Hydraulic pressure Static pressure 1.0

Dynamic pressure: time aver- age, centrifugal, impact, fluc- tuation

1.05 , 1.1, 1.1, 1.3

Uplift pressure

Seepage pressure 1.2 (solid gravity dam), 1.1 (slotted or hollow gravity dam)

Uplift pressure 1.0

Uplift with pump drainage 1.1 (before the main drains) Residual uplift with pump

drainage

1.2 (after the main drains)

Silt pressure 1.2

Wave pressure 1.2

This equation is very similar to eq. (34), with the exception of A K , which is a characteristic value for the accident load, and γ d2 which is a different (larger) structural coefficient for the accident combinations and collapse limit. Besides collapse limit states, serviceability limit states for short- term and long-term combinations are also available (and differ slightly). For these states, the partial safety factors regarding loads and materials are all unity. The limit state expression for this serviceability state is given by eq. (36).

γ ₀ S(· · ·) ≤ c

γ d3 (36)

Here, γ d3 is the structural coefficient which is slightly different for short-term and long-term com- bination, and c is the limit value of structural function. With eq. (34), eq. (35), and eq. (36) can now be used to calculate the anti-sliding stability of the gravity dam (collapse limit state), the compressive strength at the dam toe (collapse limit state), and tensile strength at the dam heel (serviceability limit state).

2.4.5 Sliding at the dam base

Sliding for a gravity dam concerns the case where the base of the dam structure moves relative to the foundation. The boundary at which the sliding occurs is often referred to as the sliding surface and, in this case, it is the interface between the dam base and the foundation, i.e. a concrete-rock interface. Sliding of this type could have catastrophic consequences for a gravity dam where irre- versible changes to the structural integrity and location of the dam could occur.

Figure 13 shows the overflow spillway section with resulting forces and moments acting on it. Due to the severity of sliding, according to [2], the limit state for the sliding case is considered to be a part of the collapse limit state. The resistance R s of the structure and the effect of actions S s for this collapse limit state is given by eq. (37) and eq. (38)

R s = f ⁰ X

F V + c ⁰ A (37)

S s = X

F H (38)

where P F V is the net sum of the vertical forces, P F H is the net sum of horizontal forces, A is

the sliding surface area and f ⁰ and c ⁰ are the coefficients describing the shear friction and cohesion

x P F H

P F V

P M ~

Heel Toe

Figure 13: Conceptual figure of the overflow spillway showing the summation of horizontal forces F _H and vertical forces F V , along with the resulting moment ~ M with respect the centroid of the base.

of the concrete-rock boundary, respectively. The limit function is obtained by combining eq. (33) with eq. (37) and eq. (38) which gives

Z a = f ⁰ X

F V + c ⁰ A − X

F H = 0 (39)

The values of the shear friction coefficient and the cohesion coefficient that was used in this report was 0.85 kNm ⁻² and 0.40 kNm ⁻² , which are the values for the RCC concrete and rock boundary in Longtan Dam.

2.4.6 Compressive stress at the dam toe

The compressive strength is the quantification of a structures ability to withstand external loads which tends to decrease its size, i.e. loads that compresses the structure. The compressive strength is highly dependant on the material and is associated with cracking in a brittle structure [10], e.g.

crushing of the concrete structure.

The overflow section covered in this report experiences its maximum compressive stress at the dam toe due to the overturning moment. As for the sliding case, the compressive strength at the dam toe is considered to be a collapse limit state. The resistance R c of the structure and the effect of actions S c for this collapse limit state is given by [2]

R _c = σ _c (40)

S c =  P F V

W − 6 P M ~ W ²



(1 + m ² ) (41)

where σ c is the compressive strength of the concrete, P ~ M is the sum of the relevant moments

acting on the structure, m is the slope of the downstream face and W is the width of the dam

base. As before, combining eq. (33) with eq. (40) and eq. (41) gives the limit function as