Reliability analysis of built concrete dam

The conventional design of concrete gravity dams still follows the deterministic method, which does not directly quantify the effect of uncertainties on the safety of the structure. The theory of structural reliability allows the quantification of safety of these structures, from the quantification of the inherent uncertainties in resistance and loading parameters. This article illustrates application of structural reliability theory to the case study analysis of a built concrete gravity dam. Results show that reliability of the built structure is greater than that of the designed structure. The study compares reliability for design conditions, with the corresponding safety coefficients, illustrating a lack of linearity between safety coefficients and reliability. Furthermore, the study shows which are the failure modes and the design parameters with greater influence on dam safety.

Keywords:
structural safety assessment; concrete dams; reliability analysis

Authorship

K. O. PIRES

University of São Paulo, Polytechnic School, Post-graduation Program in Civil Engineering, São Paulo, SP, Brazil; kironipires@hotmail.com tbitten@gmail.com futai@usp.brUniversity of São PauloBrazilSão Paulo, SP, BrazilUniversity of São Paulo, Polytechnic School, Post-graduation Program in Civil Engineering, São Paulo, SP, Brazil; kironipires@hotmail.com tbitten@gmail.com futai@usp.br

https://orcid.org/0000-0002-6972-0183

A. T. BECK

University of São Paulo, School of Engineering of São Carlos, Structural Engineering Department, São Carlos, São Paulo, SP, Brazil. atbeck@sc.usp.brUniversity of São PauloBrazilSão Carlos, São Paulo, BrazilUniversity of São Paulo, School of Engineering of São Carlos, Structural Engineering Department, São Carlos, São Paulo, SP, Brazil. atbeck@sc.usp.br

https://orcid.org/0000-0003-4127-5337

T. N. BITTENCOURT

https://orcid.org/0000-0001-6523-2687

M. M. FUTAI

https://orcid.org/0000-0002-4969-3085

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (6)
Tables (8)

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Figure 1
Geometry of the cross section of block n. 30 of the dam

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Figure 2
Piezometers installed in block n. 30 of the dam

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Figure 3
Reliability index of scenarios related to the design and operation phase

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Figure 4
Safety factor and reliability index to sliding failure mode

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Figure 5
Safety factor and reliability index to overturning failure mode

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Figure 6
Safety factor and reliability index to flotation failure mode

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Table 1
Minimum values of safety factors

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Table 2
Safety factors obtained and required

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Table 3
Scenarios of loading and resistances combinations

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Table 4
Summary of random variables

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Table 5
Friction angle and cohesion parameters of concrete to rock interface

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Table 6
Maximum annual values of k

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Table 7
Values of (β), (P_f) and (α) for each failure mode and scenarios

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Table 8
Values of (β) e (P_f) conditional and unconditional for each failure mode and scenarios

Figure 1 Geometry of the cross section of block n. 30 of the dam

Figure 2 Piezometers installed in block n. 30 of the dam

Figure 3 Reliability index of scenarios related to the design and operation phase

Figure 4 Safety factor and reliability index to sliding failure mode

Figure 5 Safety factor and reliability index to overturning failure mode

Figure 6 Safety factor and reliability index to flotation failure mode

Table 1 Minimum values of safety factors

Safety factor	Loading conditions
Safety factor	Usual	Exceptional	Limit	Construction
f_sdc	3.0 (4.0)	1.5 (2.0)	1.3 (2.0)	2.0 (2.5)
f_sdϕ	1.5 (2.0)	1.1 (1.3)	1.1 (1.3)	1.3 (1.5)
Sliding Safety Factor (SSF-total)	≥ 1	≥ 1	≥ 1	≥ 1
Overturning Safety Factor (OSF)	1.5	1.2	1.1	1.3
Flotation Safety Factor (FSF)	1.3	1.1	1.1	1.2

Ref.: Eletrobrás [²⁸[28] Centrais Elétricas Brasileiras S.A. - ELETROBRÁS, “Critérios de Projeto Civil de Usinas Hidrelétricas.” Rio de Janeiro, 2003.];
f_sdc reduction factor of the cohesion resistance;
f_sdϕ reduction factor of the friction resistance;
The factors of reduction in brackets should be adopted when the knowledge of materials is poor.

Table 2 Safety factors obtained and required

Safety assessment		Loading conditions
Safety assessment		Usual	Exceptional-1	Exceptional-2	Limit
Sliding Safety Factor (SSF)	f_sdc	3.0	2.0	2.0	1.3
	f_sdϕ	1.5	1.3	1.3	1.1
	Required	≥ 1	≥ 1	≥ 1	≥ 1
	Obtained	1.01	1.04	1.11	1.15
Overturning Safety Factor (OSF)	Required	1.5	1.2	1.2	1.1
Overturning Safety Factor (OSF)	Obtained	1.89	1.68	1.46	1.32
Flotation Safety Factor (FSF)	Required	1.3	1.1	1.1	1.1
Flotation Safety Factor (FSF)	Obtained	4.5	3.56	2.6	2.19

Table 3 Scenarios of loading and resistances combinations

Load and resistances combinations	Load conditions	h₁ (m)	h₂ (m)	L_c (m)	Mean of random variables
Load and resistances combinations	Load conditions	h₁ (m)	h₂ (m)	L_c (m)	γ_c (kN/m³)	φ_i (°)	c (kN/m²)	k	i_u
D1	Usual	70	0.96	-	25.80	37.00	250	0.33	-
D2	Exceptional 1	70	0.96	-	25.80	37.00	250	0.89	-
D3	Exceptional 2	71.5	8.66	5.55	25.80	37.00	250	0.33	1.00
D4	Limit	71.5	8.66	5.55	25.80	37.00	250	0.99	1.00
O1	Usual*	70	0.96	-	26.00	41.99	700	0.18	-
O2	Exceptional 1*	70	0.96	-	26.00	41.99	700	0.89	-
O3	Exceptional 2*	71.5	8.66	5.55	26.00	41.99	700	0.18	0.50
O4	Limit*	71.5	8.66	5.55	26.00	41.99	700	0.99	0.50

* Loading condition adjusted according to data obtained in the operation phase;
D1, D2, D3 and D4 are scenarios of the design phase;
O1, O2, O3 and O4 are scenarios of the operation phase;
h₁ headwater;
h₂ tailwater;
L_c crack length;
γ_c concrete specific weight;
φ_i internal friction angle;
c concrete-rock interface cohesion;
k coefficient of hydraulic inefficiency;
i_u coefficient of increase of uplift pressure.

Table 4 Summary of random variables

Random variables (R.V.)	Unit	Distribution	Mean	SD	COV (%)
Concrete specific weight (γ_c) ^a	kN/m³	Normal	25.80	1.29	5 ^{h, D1-4}
Concrete specific weight (γ_c) ^b	kN/m³	Normal	26.00	0.94	3,62 ^O1-4
Internal friction angle (φ_i) ^a	º	Normal	37	9.99	27 ^{i, D1-4}
Internal friction angle (φ_i) ^c	º	Normal	41.99	11.34	27 ^O1-4
Concrete-rock interface cohesion (c) ^a	kN/m²	Lognormal	250	100.00	40 ^{i, D1-4}
Concrete-rock interface cohesion (c) ^c	kN/m²	Lognormal	700	280.00	40 ^O1-4
Coefficient of hydraulic inefficiency (k) ^d	-	Normal	0.18	0.03	15 ^{O1, O3}
Coefficient of hydraulic inefficiency (k) ^a	-	Normal	0.33	0.10	30 ^{j, D1, D3}
Coefficient of hydraulic inefficiency (k) ^e	-	Normal	0.89	0.13	15 ^{k, O2}
Coefficient of hydraulic inefficiency (k) ^e	-	Normal	0.89	0.27	30 ^{j, D2}
Coefficient of hydraulic inefficiency (k) ^e	-	Normal	0.99	0.15	15 ^{k, O4}
Coefficient of hydraulic inefficiency (k) ^e	-	Normal	0.99	0.30	30 ^{j, D4}
Coefficient of increase of uplift pressures (i_u) ^f	-	Uniform	0.5	0.29	58 ^{O3, O4}
Coefficient of increase of uplift pressures (i_u) ^g	-	Deterministic	1	-	- ^{D3, D4}

^a
based on the dam design parameters;
^b
based on the measurement of specimens extracted from the concrete of the dam;
^c
based on the China Electricity Council [³¹[31] China Electricity Council, “The Standards Compilation of Water Power in China.” 2000.];
^d
based on the measurement of the PZC01 piezometer during a 4-year period;
^e
based on the dam design parameters and theoretical uplift pressure;
^f
assumed;
^g
considered with deterministic value 1;
^h
based on the measurement of specimens made during the construction of the dam;
ⁱ
considered the same COV described in the China Electricity Council [³¹[31] China Electricity Council, “The Standards Compilation of Water Power in China.” 2000.];
^j
based on the measurement of others piezometers installed in block 30 of the dam foundation;
^k
considered the same COV of the piezometer PZC01;
^O
^{1-4, O1, O2, O3 and O4} operation phase scenarios;
^D
^{1-4, D1, D2, D3 and D4} design phase scenarios.

Table 5 Friction angle and cohesion parameters of concrete to rock interface

Rock properties of dam foundation	Friction angle (N distribution)		Cohesion (LN distribution)
Rock properties of dam foundation	Mean (º)	SD (º)	Mean (MPa)	SD (MPa)
Category I: dense and sound, distance between cracks >1 m	56.31	16.70	1.5	0.54
Category I: dense and sound, distance between cracks >1 m	52.43	14.57	1.3	0.47
Category II: sound, weakly weathered massive rock with crack spaces 0.5-1 m	52.43	14.57	1.3	0.47
	47.73	11.86	1.1	0.40
Category III: rock mass of medium sound, crack spaces 0.3-0.5 m	47.73	11.86	1.1	0.40
	41.99	11.31	0.7	0.28

Ref.: China Electricity Council [³¹[31] China Electricity Council, “The Standards Compilation of Water Power in China.” 2000.]

Table 6 Maximum annual values of k

Year	Headwater (m)	Maximum pressure (m.w.c.)	k
2016	70.01	11.21	0.15
2015	70.01	14.34	0.19
2014	70.01	15.71	0.21
2013	70.04	13.51	0.18
Mean			0.18
SD			0.03
COV			0.15

m.w.c.: meters of water column

Table 7 Values of (β), (P_f) and (α) for each failure mode and scenarios

Scenarios		D1	D2	O1	O2	D3	D4	O3	O4
Scenarios		(design)		(dam built)		(design)		(dam built)
Headwater	h₁(m)	70	70	70	70	71.5	71.5	71.5	71.5
Tailwater	h₂(m)	0.96	0.96	0.96	0.96	8.66	8.66	8.66	8.66
Mean R.V.	γ_c	25.80	25.80	26.00	26.00	25.80	25.80	26.00	26.00
	φ_i	37.00	37.00	41.99	41.99	37.00	37.00	41.99	41.99
	c	250	250	700	700	250	250	700	700
	k	0.33	0.89	0.18	0.89	0.33	0.99	0.18	0.99
	i_u	-	-	-	-	1.00	1.00	0.50	0.50
Sliding	SSF	1.01	1.03	-	-	1.11	1.15	-	-
	β_FORM	1.81	1.35	3.42	3.17	1.48	0.88	3.15	2.82
	P_fFORM	3.50E-02	8.91E-02	3.15E-04	7.67E-04	6.92E-02	1.91E-01	8.15E-04	2.43E-03
	β_MC	1.87	1.40	3.47	3.23	1.55	0.91	3.24	2.89
	P_fMC	3.06E-02	8.13E-02	2.65E-04	6.11E-04	6.06E-02	1.80E-01	5.91E-04	1.90E-03
α values	γ_c	0.01	0.03	0.00	0.00	0.02	0.04	0.00	0.01
	φ_i	0.81	0.68	0.66	0.55	0.81	0.64	0.66	0.53
	c	0.17	0.24	0.34	0.45	0.17	0.23	0.34	0.46
	k	0.00	0.06	0.00	0.00	0.01	0.09	0.00	0.01
	i_u	-	-	-	-	0.00	0.00	0.00	0.00
Overturning	OSF	1.89	1.46	-	-	1.68	1.32	-	-
	β_FORM	> 8	3.57	> 8	6.28	7.28	2.82	> 8	5.14
	P_fFORM	< 6E-16	1.76E-04	< 6E-16	1.67E-10	1.67E-13	2.43E-03	< 6E-16	1.40E-07
	β_MC	> 8	3.57	> 8	6.27	7.30	2.78	> 8	5.22
	P_fMC	< 6E-16	1.79E-04	< 6E-16	1.81E-10	1.47E-13	2.75E-03	< 6E-16	8.78E-08
α values	γ_c	0.77	0.32	0.96	0.50	0.81	0.33	0.97	0.50
	k	0.23	0.68	0.04	0.50	0.19	0.67	0.03	0.48
	i_u	-	-	-	-	0.00	0.00	0.00	0.02
Flotation	FSF	4.5	2.6	-	-	3.56	2.19	-	-
	β_FORM	> 8	6.68	> 8	> 8	> 8	5.81	> 8	> 8
	P_fFORM	< 6E-16	1.18E-11	< 6E-16	< 6E-16	< 6E-16	3.15E-09	< 6E-16	< 6E-16
	β_MC	> 8	6.68	> 8	> 8	> 8	5.85	> 8	> 8
	P_fMC	< 6E-16	1.17E-11	< 6E-16	< 6E-16	< 6E-16	2.50E-09	< 6E-16	< 6E-16
α values	γ_c	0.75	0.29	0.96	0.47	0.78	0.28	0.95	0.45
	k	0.25	0.71	0.04	0.53	0.22	0.72	0.04	0.54
	i_u	-	-	-	-	0.00	0.00	0.02	0.01

SSF (Sliding Safety Factor); OSF (Overturning Safety Factor); FSF (Flotation Safety Factor).

Table 8 Values of (β) e (P_f) conditional and unconditional for each failure mode and scenarios

Scenarios		D1	D2	O1	O2	D3	D4	O3	O4
Scenarios		(design)		(dam built)		(design)		(dam built)
Headwater	h₁(m)	70	70	70	70	71.5	71.5	71.5	71.5
Tailwater	h₂(m)	0.96	0.96	0.96	0.96	8.66	8.66	8.66	8.66
Mean R.V.	γ_c	25.8	25.8	26	26	25.8	25.8	26	26
	φ_i	37	37	41.99	41.99	37	37	41.99	41.99
	c	250	250	700	700	250	250	700	700
	k	0.33	0.89	0.18	0.89	0.33	0.99	0.18	0.99
	i_u	-	-	-	-	1	1	0.5	0.5
Sliding	SSF	1.01	1.03	-	-	1.11	1.15	-	-
C	β_MC	1.87	1.4	3.47	3.23	1.55	0.91	3.24	2.89
C	P_fMC	3.06E-02	8.13E-02	2.65E-04	6.11E-04	6.06E-02	1.80E-01	5.91E-04	1.90E-03
U	β_MC	1.87	2.94	3.47	4.22	4.37	4.96	5.30	5.77
U	P_fMC	3.06E-02	1.63E-03	2.65E-04	1.22E-05	6.06E-06	3.60E-07	5.91E-08	3.81E-09
Overturning	OSF	1.89	1.46	-	-	1.68	1.32	-	-
C	β_MC	> 8	3.57	> 8	6.27	7.3	2.78	> 8	5.22
C	P_fMC	< 6E-16	1.79E-04	< 6E-16	1.81E-10	1.47E-13	2.75E-03	< 6E-16	8.78E-08
U	β_MC	> 8	4.49	> 8	6.85	> 8	5.71	> 8	7.27
U	P_fMC	< 6E-16	3.58E-06	< 6E-16	3.63E-12	< 6E-16	5.49E-09	< 6E-16	1.76E-13
Flotation	FSF	4.5	2.6	-	-	3.56	2.19	-	-
C	β_MC	> 8	6.68	> 8	> 8	> 8	5.85	> 8	> 8
C	P_fMC	< 6E-16	1.17E-11	< 6E-16	< 6E-16	< 6E-16	2.50E-09	< 6E-16	< 6E-16
U	β_MC	> 8	7.23	> 8	> 8	> 8	7.74	> 8	> 8
U	P_fMC	< 6E-16	2.33E-13	< 6E-16	< 6E-16	< 6E-16	5.01E-15	< 6E-16	< 6E-16

C conditional probability values; U unconditional probability values.

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Reliability analysis of built concrete dam

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Mensagem

[TFN1] Ref.: Eletrobrás [²⁸[28] Centrais Elétricas Brasileiras S.A. - ELETROBRÁS, “Critérios de Projeto Civil de Usinas Hidrelétricas.” Rio de Janeiro, 2003.];

[TFN2] f_sdc reduction factor of the cohesion resistance;

[TFN3] f_sdϕ reduction factor of the friction resistance;

[TFN4] The factors of reduction in brackets should be adopted when the knowledge of materials is poor.