Reliability-based design optimization of a concrete dam

Siacara, Adrian Torrico; Pellizzer, Giovanni Pais; Beck, André Teófilo; Futai, Marcos Massao

doi:10.1590/S1983-41952022000500001

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ORIGINAL ARTICLE • Rev. IBRACON Estrut. Mater. 15 (5) • 2022 • https://doi.org/10.1590/S1983-41952022000500001 copy

Reliability-based design optimization of a concrete dam

Otimização de projeto baseada em confiabilidade de uma barragem de concreto

Authorship SCIMAGO INSTITUTIONS RANKINGS

abstract:

Performance and safety of geotechnical structures are affected by uncertainties. Yet, the design of dams is nowadays still made using deterministic methods and design codes. Dam optimization in a deterministic setting may lead to compromised safety margins. In this setting, Reliability-Based Design Optimization (RBDO) appears as an alternative, allowing one to optimize dam performance, but respecting specified reliability constraints. In this paper, we employ an efficient and accurate Single-Loop Approach (SLA) in the RBDO of a concrete dam. Considering dam equilibrium reliability constraints, we find the optimal dam base and optimal placement of drainage galleries, for different dam heights and different target reliability index ( $β_{T}$ ). We show how the governing failure mode changes for each optimal solution: for large $β_{T}$ , sliding limit state is the active constraint; for smaller $β_{T}$ values, the eccentricity limit state function is found to be the active constraint for the optimum dam. We also investigate how the importance of random parameters change for each optimum solution: for large $β_{T}$ and failure controlled by sliding, the cohesion and friction angle along dam base interface with foundation rock are the most relevant uncertain parameters for dam equilibrium. For smaller $β_{T}$ with failure controlled by eccentricity, the more relevant uncertain geotechnical parameters are the base length of the dam, the specific weight of concrete, and the coefficient of drainage gallery inefficiency.

Keywords:
reliability-based design optimization; concrete dam; reliability index; dam design; single-loop approach

Expected performance level	Reliability index (β)	Probability of failure (P_f)
High	5.0	3.0 x10^-7
Good	4.0	3.0 x10^-5
Above average	3.0	1.0 x10^-3
Below average	2.5	6.0 x10^-3
Poor	2.0	2.3 x10^-2
Unsatisfactory	1.5	0.07
Hazardous	1.0	0.16

Force	Moment arm
$W_{1} = L_{1} H_{1} γ_{c}$	$a_{1} = L_{2} - \frac{L_{1}}{2}$
$W_{2} = \frac{(L_{2} - L_{1}) (H_{1} - H_{2})}{2} γ_{c}$	$a_{2} = \frac{2 (L_{2} - L_{1})}{3}$
$W_{3} = a^{2} γ_{c}$	$a_{3} = L_{2} - L_{3}$
$F_{1} = \frac{H_{4}^{2}}{2} γ_{w}$	$a_{F 1} = \frac{H_{4}}{3}$
$F_{2} = \frac{H_{6}^{2} (1 - \sin ϕ_{s}^{'})}{2} γ_{s}$	$a_{F 2} = \frac{H_{6}}{3}$
$F_{3} = \frac{H_{4} L_{3} + H_{e} L_{2} + H_{5} L_{2} - H_{5} L_{3}}{2} γ_{w}$	$a_{F 3} = \frac{2 [H_{4} (\frac{L_{3} L_{2}}{2} - \frac{L_{3}^{2}}{6}) + H_{e} (\frac{L_{2}^{2}}{3} - \frac{L_{3} L_{2}}{6}) + H_{5} (\frac{L_{2}^{2}}{6} + \frac{L_{3}^{2}}{6} - \frac{L_{3} L_{2}}{3})]}{H_{4} L_{3} + H_{e} L_{2} + H_{5} (L_{2} - L_{3})}$
$F_{4} = \frac{H_{5}^{2} (L_{2} - L_{1})}{2 (H_{1} - H_{2})} γ_{w}$	$a_{F 4} = \frac{H_{5} (L_{2} - L_{1})}{3 (H_{1} - H_{2})}$
$F_{5} = \frac{H_{5}^{2}}{2} γ_{w}$	$a_{F 5} = \frac{H_{5}}{3}$

Analysis	Parameters	µ	COV (%)	Observation
Concrete	γ_c (kN/m³)	24	4	[¹1 People’s Republic of China Industry Standard, Design specification for earth-rock fill dams, DL5395-2007. Beijing, China: China Railway Publishing House, 2007.]
Rock	q_u (MPa)	15	50	[²2 China Electric Council, The Standards Compilation of Water Power in China. Beijing, China: China Electric Power Press, 2000.]
Sedimentary material	γ_s (kN/m³)	19	7	[³3 U.S. Army Corps of Engineers – USACE, Evaluation and Comparison of Stability Analysis and Uplift Criteria for Concrete Gravity Dams by Three Federal Agencies. Washington DC, USA: USACE Engineer Manual, 2000.]
	ϕ´s (kN/m³)	28	20	[⁴4 U.S. Army Corps of Engineers – USACE, Gravity Dam Design. Washington DC, USA: USACE Engineer Manual, 1995.]
Interface Rock-Concrete	c´_rc (kPa)	250	40	[⁵5 ELETROBRÁS, Civil project criteria for hydroelectric plants (in portuguese). Brasília, Brasil: CBDB, 2003.]
Interface Rock-Concrete	ϕ´_rc (kN/m³)	35	30	[⁶6 Federal Energy Regulatory Commission – FERC, Engineering Guidelines for Evaluation of Hydropower Projects (Gravity Dams, Report No. FERC 0119-2). Washington, DC, USA: Office of Hydropower Licensing, 2016.]
Drainage gallery inefficiency	k_DG	0.33	20	[⁷7 United States Bureau of Reclamation – USBR, Design of Gravity Dams. Washington, DC, USA: US Government Printing Office, 1976.]

Observation [¹1 People’s Republic of China Industry Standard, Design specification for earth-rock fill dams, DL5395-2007. Beijing, China: China Railway Publishing House, 2007.]: The COV of _C is 4 % [⁷³73 JCSS, “Probabilistic Model Code, Part I - Basis of design”. https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed Aug. 26, 2021).
https://www.jcss-lc.org/jcss-probabilist... ] and other authors have related a COV about 3 % to 5 % [⁷⁴74 K. O. Pires, A. T. Beck, T. N. Bittencourt, and M. M. Futai, “Reliability analysis of built concrete dam,” Rev. IBRACON Estrut. Mater., vol. 12, no. 3, pp. 551–579, Jun 2019, http://dx.doi.org/10.1590/s1983-41952019000300007.
http://dx.doi.org/10.1590/s1983-41952019... ]. Nominal values were based on CFBR [⁸8 CFBR – Comité Français des Barrages et Réservoirs, Recommendations for Justification of Stability of Gravity Dams. France: CFBR, 2012.] and JCSS [⁷³73 JCSS, “Probabilistic Model Code, Part I - Basis of design”. https://www.jcss-lc.org/jcss-probabilistic-model-code/ (accessed Aug. 26, 2021).
https://www.jcss-lc.org/jcss-probabilist... ]. [²2 China Electric Council, The Standards Compilation of Water Power in China. Beijing, China: China Electric Power Press, 2000.]: The COV of _u for a rock foundation varies from 25.9 % to 44.5 % [⁷⁵75 N. Mao, T. Al-Bittar, and A.-H. Soubra, “Probabilistic analysis and design of strip foundations resting on rocks obeying Hoek–Brown failure criterion,” Int. J. Rock Mech. Min. Sci., vol. 49, pp. 45–58, Jan 2012, http://dx.doi.org/10.1016/j.ijrmms.2011.11.005.
http://dx.doi.org/10.1016/j.ijrmms.2011.... ], 127 % [⁷⁶76 A. H. Alavi and E. Sadrossadat, “New design equations for estimation of ultimate bearing capacity of shallow foundations resting on rock masses,” Geosci. Frontiers, vol. 7, no. 1, pp. 91–99, Jan 2016, http://dx.doi.org/10.1016/j.gsf.2014.12.005.
http://dx.doi.org/10.1016/j.gsf.2014.12.... ], and for soil foundations, 20 % [¹¹11 Z. Mahmood, “Reliability-based optimization of geotechnical design using a constrained optimization technique,” SN Appl. Sci., vol. 2, no. 2, 2020, http://dx.doi.org/10.1007/s42452-020-1948-4.
http://dx.doi.org/10.1007/s42452-020-194... ]. Nominal values for a sandstone rock was taken from Duncan [⁷⁷77 C. W. Duncan, Foundations on Rock. 2nd ed. London/New York: E&FN SPON, 1999.]. [³3 U.S. Army Corps of Engineers – USACE, Evaluation and Comparison of Stability Analysis and Uplift Criteria for Concrete Gravity Dams by Three Federal Agencies. Washington DC, USA: USACE Engineer Manual, 2000.]: The COV of _S for all the different soils varies from 3 % to 11 % [⁷⁸78 A. T. Siacara, A. T. Beck, and M. M. Futai, “Reliability analysis of rapid drawdown of an earth dam using direct coupling,” Comput. Geotech., vol. 118, pp. 103336, Feb 2020, http://dx.doi.org/10.1016/j.compgeo.2019.103336.
http://dx.doi.org/10.1016/j.compgeo.2019... ], [⁷⁹79 A. T. Siacara, G. F. Napa-García, A. T. Beck, and M. M. Futai, “Reliability analysis of earth dams using direct coupling,” J. Rock Mech. Geotech. Eng., vol. 12, no. 2, pp. 366–380, Apr 2020, http://dx.doi.org/10.1016/j.jrmge.2019.07.012.
http://dx.doi.org/10.1016/j.jrmge.2019.0... ]. Nominal values were based on Fredlund et al. [⁸⁰80 D. G. Fredlund, H. Rahardjo, and M. D. Fredlund, Unsaturated Soil Mechanics in Engineering Practice. Hoboken, NJ, USA: Wiley, 2012.] and USBR [⁸¹81 United States Bureau of Reclamation – USBR, Design of Small Dams. 3rd ed. Washington DC, USA: US Government Printing Office, 1987.]. [⁴4 U.S. Army Corps of Engineers – USACE, Gravity Dam Design. Washington DC, USA: USACE Engineer Manual, 1995.]: The COV of ϕ´ for all the different soils varies from 10 % to 30 % in clays [⁷⁸78 A. T. Siacara, A. T. Beck, and M. M. Futai, “Reliability analysis of rapid drawdown of an earth dam using direct coupling,” Comput. Geotech., vol. 118, pp. 103336, Feb 2020, http://dx.doi.org/10.1016/j.compgeo.2019.103336.
http://dx.doi.org/10.1016/j.compgeo.2019... ], [⁷⁹79 A. T. Siacara, G. F. Napa-García, A. T. Beck, and M. M. Futai, “Reliability analysis of earth dams using direct coupling,” J. Rock Mech. Geotech. Eng., vol. 12, no. 2, pp. 366–380, Apr 2020, http://dx.doi.org/10.1016/j.jrmge.2019.07.012.
http://dx.doi.org/10.1016/j.jrmge.2019.0... ]. Nominal values were taken from Fredlund et al. [⁸⁰80 D. G. Fredlund, H. Rahardjo, and M. D. Fredlund, Unsaturated Soil Mechanics in Engineering Practice. Hoboken, NJ, USA: Wiley, 2012.] and USBR [⁸¹81 United States Bureau of Reclamation – USBR, Design of Small Dams. 3rd ed. Washington DC, USA: US Government Printing Office, 1987.]. [⁵5 ELETROBRÁS, Civil project criteria for hydroelectric plants (in portuguese). Brasília, Brasil: CBDB, 2003.]: The COV of _rc for all the different rocks varies from 36 % to 40 % [²2 China Electric Council, The Standards Compilation of Water Power in China. Beijing, China: China Electric Power Press, 2000.], 64 % to 69 % [⁸²82 K. Y. Lo and J. D. Grass, “Recent experience with safety assessment of concrete dams on rock foundations,” in Canadian Dam Safety Conf., 1994, pp. 231–250.], 65 % [⁸³83 Y. A. Fishman, “Stability of concrete retaining structures and their interface with rock foundations,” Int. J. Rock Mech. Min. Sci., vol. 46, no. 6, pp. 957–966, Sep 2009, http://dx.doi.org/10.1016/j.ijrmms.2009.05.006.
http://dx.doi.org/10.1016/j.ijrmms.2009.... ] and 44 % [⁸⁴84 M. W. Wilde and F. Johansson, “System reliability of concrete dams with respect to foundation stability: Application to a spillway,” J. Geotech. Geoenviron. Eng., vol. 139, no. 2, pp. 308–319, 2013, http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000761.
http://dx.doi.org/10.1061/(ASCE)GT.1943-... ]. Nominal values were based on Pires et al. [⁷⁴74 K. O. Pires, A. T. Beck, T. N. Bittencourt, and M. M. Futai, “Reliability analysis of built concrete dam,” Rev. IBRACON Estrut. Mater., vol. 12, no. 3, pp. 551–579, Jun 2019, http://dx.doi.org/10.1590/s1983-41952019000300007.
http://dx.doi.org/10.1590/s1983-41952019... ]. [⁶6 Federal Energy Regulatory Commission – FERC, Engineering Guidelines for Evaluation of Hydropower Projects (Gravity Dams, Report No. FERC 0119-2). Washington, DC, USA: Office of Hydropower Licensing, 2016.]: The COV of _RC for all the different rocks varies from 25 % to 30 % [²2 China Electric Council, The Standards Compilation of Water Power in China. Beijing, China: China Electric Power Press, 2000.], 15 % [⁸⁴84 M. W. Wilde and F. Johansson, “System reliability of concrete dams with respect to foundation stability: Application to a spillway,” J. Geotech. Geoenviron. Eng., vol. 139, no. 2, pp. 308–319, 2013, http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000761.
http://dx.doi.org/10.1061/(ASCE)GT.1943-... ], and 40 % [⁸³83 Y. A. Fishman, “Stability of concrete retaining structures and their interface with rock foundations,” Int. J. Rock Mech. Min. Sci., vol. 46, no. 6, pp. 957–966, Sep 2009, http://dx.doi.org/10.1016/j.ijrmms.2009.05.006.
http://dx.doi.org/10.1016/j.ijrmms.2009.... ]. Nominal values were taken from NAVFAC DM7-02 [⁸⁵85 Naval Facilities Engineering Command, Foundations and earth structures, NAVFAC DM7-02, 1986.]. [⁷7 United States Bureau of Reclamation – USBR, Design of Gravity Dams. Washington, DC, USA: US Government Printing Office, 1976.]: The COV of k in a concrete dam varies from 15 % to 30 % [⁷⁴74 K. O. Pires, A. T. Beck, T. N. Bittencourt, and M. M. Futai, “Reliability analysis of built concrete dam,” Rev. IBRACON Estrut. Mater., vol. 12, no. 3, pp. 551–579, Jun 2019, http://dx.doi.org/10.1590/s1983-41952019000300007.
http://dx.doi.org/10.1590/s1983-41952019... ]. Nominal values were based on FERC [⁶6 Federal Energy Regulatory Commission – FERC, Engineering Guidelines for Evaluation of Hydropower Projects (Gravity Dams, Report No. FERC 0119-2). Washington, DC, USA: Office of Hydropower Licensing, 2016.] and USACE [³3 U.S. Army Corps of Engineers – USACE, Evaluation and Comparison of Stability Analysis and Uplift Criteria for Concrete Gravity Dams by Three Federal Agencies. Washington DC, USA: USACE Engineer Manual, 2000.].

Description	β_T	L₂	L₃	Cross-section Area	Difference (%)
Initial design	-	40.0	5.0	1088.50
Optimal design	1.5	40.3	3.0	1095.60	0.65
	2	41.9	3.0	1131.16	3.77
	2.5	50.9	3.0	1332.94	18.34
	3	77.4	3.0	1929.74	43.59

Failure mode	Target reliability
Failure mode	$β_{T} = 1.5$	$β_{T} = 2.0$	$β_{T} = 2.5$	$β_{T} = 3.0$
Overturning $(g_{O})$	306752.73	313218.48	485929.04	1196889.79
Sliding $(g_{S})$	3842.72	1244.79	-5.97 x10^-13	-8.53 x10^-13
Flotation $(g_{F})$	19236.12	19121.93	21595.01	29776.17
Eccentricity $(g_{E})$	-5.21 x10^-13	5.55 x10^-16	0.27	0.64

$H_{1}$	$β_{T}$	Analysis	$L_{2}$	$L_{3}$	Overturning $(g_{O})$	Sliding $(g_{S})$	Flotation $(g_{F})$	Eccentricity $(g_{E})$
70	1.8	RBDO	58.40	3.00	845917.08	869.37	37470.00	-5.10 x10^-9
	1.8	DDO	58.40	3.00	2.34	1.96	5.75	1.41
	1.9	RBDO	58.90	3.00	851773.89	2.61 x10^-12	37461.54	1.93 x10^-3
	1.9	DDO	58.90	3.00	2.37	1.98	5.75	1.44
	2	RBDO	61.99	3.00	972406.90	-3.41 x10^-12	38868.02	9.51 x10^-2
	2	DDO	61.99	3.00	2.50	2.07	5.74	1.67
	2.1	RBDO	65.37	3.00	1109851.24	-8.53 x10^-13	40404.58	0.18
	2.1	DDO	65.37	3.00	2.65	2.18	5.74	1.96
	2.2	RBDO	69.08	3.00	1267534.89	1.19 x10^-12	42091.45	0.27
	2.2	DDO	69.08	3.00	2.81	2.29	5.73	2.35
	2.5	RBDO	25.42	3.00	70483.94	265.11	6955.87	-2.22 x10^-16
	2.5	DDO	25.42	3.00	2.44	3.15	4.71	1.51
	2.6	RBDO	25.76	3.00	71804.29	-1.42 x10^-13	6968.93	9.81 x10^-3
	2.6	DDO	25.76	3.00	2.47	3.18	4.71	1.56
	2.7	RBDO	28.72	3.00	93075.74	4.55 x10^-13	7456.92	0.17
	2.7	DDO	28.72	3.00	2.70	3.50	4.66	2.03
	2.8	RBDO	32.39	3.00	121861.48	8.53 x10^-14	8063.73	0.32
	2.8	DDO	32.39	3.00	2.95	3.89	4.60	2.78
	2.9	RBDO	37.07	3.00	162410.96	-5.68 x10^-14	8838.91	0.444
	2.9	DDO	37.07	3.00	3.21	4.38	4.55	4.14

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E-mail: arlene@ibracon.org.br

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[1] Corresponding author: Adrian Torrico Siacara. E-mail: adriantorricosiacara@gmail.com