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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (4) : 852-862     https://doi.org/10.1007/s11709-019-0522-x
RESEARCH ARTICLE
Shape design of arch dams under load uncertainties with robust optimization
Fengjie TAN, Tom LAHMER()
Institute of Structural Mechanics, Bauhaus-Universität, Weimar D-99423, Germany
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Abstract

Due to an increased need in hydro-electricity, water storage, and flood protection, it is assumed that a series of new dams will be build throughout the world. The focus of this paper is on the non-probabilistic-based design of new arch-type dams by applying means of robust design optimization (RDO). This type of optimization takes into account uncertainties in the loads and in the material properties of the structure. As classical procedures of probabilistic-based optimization under uncertainties, such as RDO and reliability-based design optimization (RBDO), are in general computationally expensive and rely on estimates of the system’s response variance, we will not follow a full-probabilistic approach but work with predefined confidence levels. This leads to a bi-level optimization program where the volume of the dam is optimized under the worst combination of the uncertain parameters. As a result, robust and reliable designs are obtained and the result is independent from any assumptions on stochastic properties of the random variables in the model. The optimization of an arch-type dam is realized here by a robust optimization method under load uncertainty, where hydraulic and thermal loads are considered. The load uncertainty is modeled as an ellipsoidal expression. Comparing with any traditional deterministic optimization method, which only concerns the minimum objective value and offers a solution candidate close to limit-states, the RDO method provides a robust solution against uncertainty. To reduce the computational cost, a ranking strategy and an approximation model are further involved to do a preliminary screening. By this means, the robust design can generate an improved arch dam structure that ensures both safety and serviceability during its lifetime.

Keywords arch dam      shape optimization      robust optimization      load uncertainty      approximation model     
Corresponding Authors: Tom LAHMER   
Just Accepted Date: 07 March 2019   Online First Date: 23 April 2019    Issue Date: 10 July 2019
 Cite this article:   
Fengjie TAN,Tom LAHMER. Shape design of arch dams under load uncertainties with robust optimization[J]. Front. Struct. Civ. Eng., 2019, 13(4): 852-862.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-019-0522-x
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I4/852
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Fig.1  Description of the shape of arch type dam. (a) The central vertical section of the arch dam (b) the horizontal curves of the arch type dam
Fig.2  Temperature distributions in winter time and summer time
Fig.3  Comparison of DO-based optimum design with an RO-based optimum design
Fig.4  The procedure of the optimization
Fig.5  Approximation with Kriging metamodel. (a) The approximation of tensile stress with Kriging Metamodel; (b) the approximation of dam volume with Kriging metamodel
samplings leave-one-out error
tensile stress approximation LHS:800 9.0534e−2
volume approximation LHS:800 4.8018 e−2
Tab.1  Approximation model of tensile stress and dam volume
0.5α0.9 0.3 β10.7 0.3 β20.7
3 t110 10 t230 15 t135
25 ϕ170 25 ϕ270 15 ϕ340
2α1 2 2α22 2α3 2
Tab.2  The upper and lower boundaries of shape design variables
parameters nominal value
change of air temperature (°C ) −5.4
change of reservoir water temperature (°C ) −4.1
water level (m) 135
density (kg/m3) 2400
Young’s Modulus (Pa) 2.1e10
Tab.3  Loads and material parameters of the dam model
height thickness (m) central angle (°) coefficient ‘a’
140 5.59 115.45 −0.311
120 7.12 108.28 −0.302
105 8.29 101.88 −0.309
90 9.47 94.62 −0.329
75 10.68 86.49 −0.360
55 12.32 74.31 −0.421
35 13.99 60.58 −0.502
20 15.27 49.27 −0.579
0 17.01 32.85 −0.698
Tab.4  Geometrical parameters of the dam model according to the DO
height thickness (m) central angle (°) coefficient ‘a’
140 3.00 123.41 −0.271
120 8.04 117.99 −0.405
105 11.28 112.38 −0.530
90 14.07 105.45 −0.677
75 16.41 97.20 −0.844
55 18.81 84.15 −1.100
35 20.41 68.75 −1.395
20 21.07 55.66 −1.640
0 21.25 36.16 −2.00
Tab.5  Geometrical parameters of the dam model according to the RO
Fig.6  Comparison of the geometries between DO optimal design and RO optimal designs
Fig.7  The sum displacement of optimal arch dam based on DO
Fig.8  The sum displacement of optimal arch dam based on RO
volume (m3) sum displacement (m ) tensile stress (Pa) von Mises stress (Pa)
DO 3.2382e5 0.01256 1.48 2.89
RO 4.4173e5 0.00987 1.40 2.79
Tab.6  Comparison
Fig.9  Comparison of mechanical properties between the arch dams acquired with DO method and RO method. (a) Comparison of tensile stress; (b) comparison of von Mises stress; (c) comparison of displacement
parameters distribution range
change of air temperature (°C ) uniform [−4,−7]
change of reservoir water temperature (°C ) uniform [−2,−6]
water level (m) uniform [130,140]
density ((kg/m3) uniform [2200,2600]
Young’s Modulus (Pa) uniform [1.9e10,2.3e10]
Tab.7  Range of random samples
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