Optimum Design Program for Shape of Arch Dams

Jalal Akbari*, Mohammad T. Ahmadi**, Hamid Moharrami***

*. PhD Student, Dept. of Civil Eng., Tarbiat Modares University, Tehran, Iran jakbari@modares.ac.ir

**. Professor of Civil Eng. , Tarbiat Modares University, Tehran, Iran mahmadi@modares.ac.ir

***. Assistant Professor of Civil Eng , Tarbiat Modares University , Tehran, Iran hamid@modares.ac.ir

Keywords: Concrete Arch Dam, Shape Optimization, Cubic Spline, Design

ABSTRACT

This paper presents an applicable and practical computer program(ODPSAD1) and suggests a new algorithm for geometry of concrete arch dams. In this program shape optimization of arch dams for static and dynamic loads and enhancement of its seismic behaviour is carried out. abutment excavation and dam body volumes have been considered as the objective function and in order to increase the consistency with practical conditions, a great number of geometrical and behavioural constraints have been included in the mathematical model. Three loading cases i.e., gravity loading, static loading and dynamic loading have been accounted for analysis. In seismic loading, longitudinal component of the OBE2 spectrum have been considered. For finite element analysis the base of program FEAPpv3 is used and extended for this purpose .Sequential nonlinear approximation(SQP4) procedure is used for the optimization analysis. Results show that the modification and optimization of shape can be very versatile and professional leading to considerable reduction in the dam body volume and costs in regular cases.

1. Introduction

In concrete arch dams, geometry of the structure has a great influence on the safety and economy of design. Geometry is also an important factor in the stability of dam. Nowadays, a high number of arch dams are constructed in seismic zones and despite their acceptable performance under dynamic loading, there is still overmuch need to improve their behaviour and to increase their safety during strong ground motion events. Generally, shape design of an arch dam is based mainly on the experience of the designer, model tests and trial and error procedures. In trial and error procedures, an initial scheme is selected or given and then analyzed. If it satisfies the demands of the design specifications, the scheme is adopted. Otherwise, the shape of the dam is modified and reanalyzed. This process continues until the whole demands of the design specifications are satisfied. To get a better shape, the designer should select several alternative schemes with various patterns and modify them such as the first scheme to obtain a number of feasible shapes. The best shape considering economy of design, structural considerations, safety, etc. is selected as the final shape. The shape of the dam obtained in this way is feasible but not necessarily optimal or even good. Moreover, the time of design is rather long. To overcome these difficulties, special efforts have begun by researchers for optimal shape design from the late 1960s.In the search of arch dams with optimal shape, early 1 -Optimum Design Program for Shape of Arch Dams 2 -Operating Basis Earthquake 3 -Finite Element Analysis Program for Personal Version 4 -Sequential Quadratic Program

research works (Fialho (1955) & Serafim (1966)) [1,2] dealt mainly with membrane-type solutions that ignored foundation elasticity and bending stresses, and considered a single, simple loading condition (Hydrostatic pressure and the weight of concrete). These methods can provide only useful starting points for more comprehensive studies. Later, Rajan, M.K.S. (1968) [3], Mohr, G.A. (1979) [4] and Sharma, R.L. (1983) [5], considerably developed membrane shell theory based solutions. Sharpe, R. (1969) [6] was the first one to formulate the optimization problem as a mathematical programming problem. A similar method was also adopted by Rickeetts, R.E. & Zienkiewicz, O.C. (1975) [7] who used finite element method for stress analysis and Sequential Linear Programming (SLP) for the shape optimization of arch and buttress dams under static loads. Wassermann (1983-84) [8,9] used a mathematical formulation for arch dam design, in which the arch dam and part of its foundation were described by three-dimensional 8-node isoparametric hyper-elements. Design variables were the geometrical parameters of these elements and the objective function was the dam volume. The resulting optimization problem was solved by SLP method. From other works in this field, the works done by Rahim, A.S. (1983) [10], Samy, M.S. (1987) [11,12], Yao, T.M. & Choi, K.K. (1989) [13], Guohua, L. & Shuyu, W. (1990) [14] can be mentioned. Fanelli, A., Fanelli, M. & Salvaneschi, P. (1993) [15] used neural network approach to the definition of near optimal arch dam shape. Nevertheless, one of the most important investigations in the field of shape optimization of concrete arch dams carried out by Bofang, Z. et al. [16,17,18,19,20] in China since the middle 1970s. The geometrical model used by Bofang, Z. is a continuous, rational and practical model and thus could draw the interests of many researchers. It has been the basis of majority of later works in the field of shape optimization of arch dams. From among the works done up to early 1990s in the field of shape optimization of concrete arch dams, the effects of seismic dynamic loads in determination of the optimum shape of arch dams has rarely been considered. Bofang, Z. et al., considered dynamic loading in trial load method to optimize the shape of arch dams. Later on, Simes, L.M.C. (1994) [21,22] continued his works using the finite element method for stress analysis. He accounted the seismic dynamic loads considering dam-foundation interaction and the reservoir effects. In recent years, particular attention is paid to this subject and specially in China, a number of articles in this field has been published in Chinese language. Last work is done by Tajalli & Ahmadi(2005) [23] using Bofang formulation for parabolic arch dam. The finite element analysis is done by a commercial program and optimization procedure is carried out with DOT software. In the latter seismic loads , dam-foundation- reservoir interaction have been considered and method is applied successfully in Shahid Rajaee Arch dam in Iran. In this research we use new algorithm for modelling geometry of arch dams [24].As a high number of design variables and large dimensions of shape optimization of concrete arch dams problem, make its nature very complicated, preparing a suitable method for this problem is highly demanding. This research employs Bofangs dam body geometrical expression but is quite improved in modelling the Dam-Foundation (DF) system, dynamic load definition, stress constraints, numerical analysis method and method of optimization. The main aim has been an attempt to develop the practical and professional program for shape optimization of arch dams[25].

2. Dam Body Geometry

2.1.Coordinate Systems

Site geometry is described in the cartesian site coordinate system 210 ,, ttt .This global coordinate system is chosen at the beginning of the project and remains unchanged throughout the design procedure. The axes 0t and 1t lie in a horizontal plane. The axis 0t points to the right side of the dam body. The axis 2t points vertically upward. The geometry of the arch dam is described in the cartesian dam coordinate system cxxx ,,, 210 whose origin and orientation change for each design with the location of the dam and the orientation of the axis of the dam. Fig (1).

Fig .1.Site & Dam Coordinate Systems

2.2.Crown Shape

For crown shape It is assumed that design variables are the dam thicknesses cumb TTTT ,,, and the overhang parameters cumb pppp ,,, at the interpolation stations Fig(2). Shape of crown is defined via cubic Spline equations of upstream and downstream faces. Thickness of crown cantilever is obtained with the following equation:

),(),( ,, ZZYSplineZZYSplineYYT iuiidiud == (1)

Fig .2. Crown Cantilever Profile

2.3. Thickness of the Dam Body

The thickness variation in the crown is specified by cumb TTTT ,,, design variables. The upper and middle levels are specified relative to cH with the level ratios 40.,75. == mu hh . The thickness is interpolated with Splines over the height of the dam. The upstream face of the dam is divided into a central zone of constant thickness, which will be denoted by CZ , and zones of variable thickness at the left and right abutments, which will be denoted by LZ and RZ . The zones are shown in Fig.(3).

In zone CZ the thickness of the dam body in each elevation equals the thickness in the crown joint. In the zones LZ and RZ , the thickness in each elevation varies parabolically from the thickness of CZ to the abutment thickness. The start points of variable zone in 4 stations on the right and left are defined with 8 design variables .i.e. cumbcumb sssstttt ,,,,,,, .The thickness of the dam body at the abutment is specified by the thicknesses rcrurmrb TTTT ,,, and lclulmlb TTTT ,,, , which are design variables. The thickness is interpolated with Splines over the height of the dam.

Fig .3. Thickness Zones of the Dam Body

2.4. Shape of Elevation Arches

Consider the elevation of a dam body at level 2x as shown in Fig.(4). The upstream edge of the elevation is a parabola whose vertex lies on the upstream edge of the crest crown cantilever profile. The radius of curvature r of the parabola is specified at levels cumb HHHH ,,, by the cumb RRRR ,,, design variables. r is interpolated between these levels with the following Spline

),,( 2 ii RHxSpliner = (2) The equation of the parabola AB with vertex ( 10 ,aa ) and radius r is :

raxaX 2/)( 20011 += (3)

Fig .4. An elevation Arch of the Dam Body

2.5. 3D Shape of Dam & Abutment

With the above defined design variables a 3D shape of system is created .Typical shape of the system is presented in Fig.(5).

Fig .5.Shape of the Dam and Its Foundation

3. Loading Cases

Three basic loading cases have been considered in the process of shape optimization. Gravity load (case 1) is applied to free cantilevers (even and odd cantilevers) once. This load is applied for three stages grouting . in Dominant static loading (case 2) monolithic shell structure of dam body is subjected to the action of hydrostatic pressure (with full reservoir) plus load case 1 as described before.In dynamic seismic loading (case 3) loading case 2 is superposed by the effects of longitudinal component of the Operating Basis Earthquake defined as a target spectrum.

4. Properties of the Mathematical Model

The mathematical model consists of dam-foundation for loading case 1. In loading case 2, the hydrostatic pressure is added to the dam-foundation model. In loading case 3, the mathematical model consists of dam-foundation and reservoir. The foundation is weightless and massless in static and dynamic loading cases respectively. The dam body is modelled by four layers of 8-node brick elements. The foundation have 4 layers of 8-node brick elements that extends about 1.5 times the dam body height into the rock mass. The governing equations are linear and the displacements are small. Also initial condition for the dynamic loading is the dominant static response (due to loading case 2).

5. Objective Function

The objective function is the volume of the dam and foundation excavation, which expressed as follows:

)()()( 21 xVxVxf += (4) In which )(xf = the sum of volumes, )(1 xV = volume of concrete in the dam body, )(2 xV = volume of foundation excavation and the x is the vector of design variables.

6. Constraints

In shape optimization of concrete arch dams, the three following types of constraints should satisfy the demands of design and construction requirements: 1)-Geometrical constraints 2)- Stress constraints 3)- Stability constraints The constraints are shown by a set of 0)( xgi conditions

1.6. Geometrical Constraints

To facilitate construction, the maximum slope of overhang at the upstream and at the downstream faces ( ..max

SUS , ..maxSDS ) and the maximum overhang at the upstream face relative to the toe (Undercut)

should be limited to equations (5),(6) Fig.(7). For constructing facilities and having smooth cantilevers over the height the start points of the variable thickness zone must satisfy the equations (8),(9).Fig.(6)

Fig.6. Inter.Angle of Start Points with Crest & Bottom Fig. 7. Slope of Overhangs

....max

SUalw

SU SS (5) ....

maxSD

alwSD SS (6)

alwUndercutUndercut (7) alwclcr , (8) alwblbr , (9)

where SUalwS . , ..SDalwS , alwUndercut and alw are the allowable absolute values of the aforementioned parameters. According to the requirements of traffic and gaining access to the equipments located at different points of the dam and so on, the minimum thickness of the crest is then decided. Sometimes the maximum thickness of the base of the dam is also limited.

6.2. Stress Constraints

The main stress constraints for both faces of arch dam body, which is primarily subjected to a biaxial stress state, result from the condition that the principal stresses are within the biaxial strength

envelope for unreinforced mass concrete. It could be constructed considering appropriate safety factors. In general, the tensile stresses in either pure tension or combined tension/compression states are important and control the design. In the static and dynamic loading conditions, stress constraints are expressed by biaxial stress envelope [26]. Modified envelope curve for biaxial dynamic strength of concrete is expressed as Shown in Fig.(8):In static and dynamic analysis a safety factor concept is adopted. at any arbitrary stress point ),( rB is defined as follows:

rRFS =.. (10)

The latter value should not be less than the defined limit at any arbitrary stress point under the total (static + dynamic) principal stresses.

Fig. 8. Modified Envelope Curve for Biaxial Strength of Concrete

3.6. Stability Constraints

Constraints ensuring the sliding stability of the dam may be expressed by at least one of the following three conditions: a) Constraint of coefficient of sliding stability (adequately defined for large dams abutments):

min)( ii KK (11) where iK is the coefficient of sliding stability at point i and min)( iK is the allowable minimum value of iK .

b) Constraint of the angle between the arch thrust and sound rock contour lines at each elevation of the abutments (usually for small dams):

min (12) where is the angle between the arch thrust direction and the sound rock contour lines at each elevation of the abutments and min is the allowable minimum value of .

c) Constraint of central angle of the arch: max (13)

where is the sum of central angles at the right and the left wings and max is the allowable maximum value of central angle which is defined considering dam body curvature, canyon shape, strength and stability of the abutments and so on.

7. Optimization Problem

The shape optimization problem of an arch dam can be expressed as a nonlinear constrained optimization problem of the form:

q)1,2,...,(j , 0(X)g :Subject tof(X)C :Minimize

j ==

(14)

In which q is the total number of constraints .Because there is not any explicit formulation between design variables and the above functions , these functions are approximated in 0x via Taylor expansion series, so the nonlinear functions replaced with sequential approximated quadratic and linear functions as equations (15),(16)

)*(*)(*5.*)()()( 0000 iidfT

iiii xxHxxfxxxfxf ++= (15) )(*)()()( 000 ijiijj xgxxxgxg += (16)

where )(xf is approximated objective function , )( 0xf is objective function in 0x , f is objective function gradients in 0x , dfH is quasi Hessian matrix of objective function , )(xg j is

thj linear

approximated constraint, )( 0xg j is value of thj constraint in 0x design point and jg represents

gradient of thj constraint in 0x . To increase the rate of convergence, design variables are modified as equation (17) before approximating the objective function and the constraints.So approximated functions are replaced with the equations ( 18),(19) then optimization is carried out.

iii xxx 0/= (17) )1*(*)1(*5.*)1()()( 0 ++= idfTii xHxfxxfxf (18)

)(*)1()()( 00 ijijj xgxxgxg += (19) 8.Case Study

The proposed method is applied to shape optimization of the 130 m-high concrete arch dam. The canyon has trapezoidal shape with 100 m base and slopes 1:1 at left and right. The main characteristics of the dam and the considerations related to its shape optimization process are described at tables (1),(2),(3).Also seismic input load is illustrated in Fig(9).

Table.1. Characteristics of the Dam

Table.2. Allowable value of Constraints

Height(m) cf (Mpa) cE (Gpa) rE (Gpa) C (kN/m3) C r 130 35 22 8 24 .18 .22

..SUalwS (degree)

..SDalwS (degree) alwUndercut (m) alw (degree) stSF dySF

25 20 6.5 70 3 1

Table.3. Initial Values of Design Variables

up mp bp cT uT mT bT cR uR mR bR rcT ruT rmT

0.75 0.65 0.55 8.00 14.0 19.0 28.0 205 145 110 85 8.80 15.4 20.9

rbT lcT luT lmT lbT ct ut mt bt cs us ms bs

30.8 8.80 15.4 20.9 30.8 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50

Fig.9.Seicmic Input Load

9. Typical Results

The optimization process of aforementioned arch dam according to the above formulation converged after 12 iterations. Convergence rate . Notwithstanding the value of the objective function at the end of the 11th optimization cycle is minimum, but due to a partial violation of stress constraints in this cycle, results obtained from the 12th iteration is considered as the valid optimum solution of the objective function Fig.(10). After performing the optimization process, dam volume has decreased by 37% in comparison with the initial design.

Fig. 10. Convergence Rate of the Objective Function

Objective function convergence

350000

400000

450000

500000

550000

600000

650000

1 2 3 4 5 6 7 8 9 10 11 12 13Iterations

Volu

me(

m3)

Response Specturm

y = 2E-05x5 - 0.0015x4 + 0.0457x3 - 0.6588x2 + 3.9044x - 0.9004

-

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

- 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000

W(rad/sec)

Sa(

m/s

2)

Fig. 11. Crown Cantilever Evolution During the Optimization Cycles

10. Conclusion

In this paper a program(ODPSAD) is used to shape optimization of an arch dam . it suggests a new algorithm for geometry of concrete arch dams. Abutment excavation and dam body volumes have been considered as the objective function and in order to increase the consistency with practical conditions, a great number of geometrical and behavioural constraints have been included in the mathematical model. Three loading cases i.e., gravity loading, static loading and dynamic loading have been accounted . Results show that the optimization of shape can reduce the dam body volume 37 %.

Acknowledgment

The authors would like to express their gratitude to Professor P.J. Pahl, professor emeritus of TU-Berlin for his ideas and encouragements.

References

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