Gravity dam and earthquake Tardieu s Dynamic simplified method Patrick LIGNIER, Tractebel Engineering Coyne et Bellier Château des Comtes de Challes 9 octobre 2014 CONTENTS 2 Vulnerability of gravity dam against earthquake Which approach and method to verify the dam stability? Tardieu s dynamic simplified method 1
3 VULNERABILITY OF GRAVITY DAMS TO EARTHQUAKES No actual dam failures except for Shih-Kang Dam (Taiwan) built on an active fault Local failure (cracks in the upper part of the dam, opening or displacement at vertical construction joints) for PGA up to 0.6 g 4 VULNERABILITY OF GRAVITY DAM AGAINST EARTHQUAKE Shih-Kang Dam (Taiwan) 2
5 VULNERABILITY OF GRAVITY DAM AGAINST EARTHQUAKE Sefi Rud Dam (Iran) 6 VULNERABILITY OF GRAVITY DAM AGAINST EARTHQUAKE Main conclusions - Generally satisfactory behaviour of gravity dams (no failure but damage) up to PGA of 0.6 g - Amplification of the acceleration (PGA) due to the dynamic response of the dam can lead to excessive stresses in the upper part of the dam - We do not know the actual safety margin of gravity dams under seismic loading (dynamic resistance, increase of damping with higher oscillations) 3
7 WHICH APPROACH AND METHOD TO VERIFY THE DAM STABILITY? - Imagine a failure scenario - Demonstrate that for this scenario the dam is stable during and after the earthquake 8 WHICH APPROACH AND METHOD TO VERIFY THE DAM STABILITY? FAILURE MECANISM - Excessive cracking can lead to sliding or overturning of the structure 4
9 FAILURE MECANISM 1 2 10 WHICH METHOD? - Pseudo-static method - Simplified dynamic method or dynamic method FEA taking into account the dynamic response of the dam 5
11 TARDIEU S DYNAMIC SIMPLIFIED METHOD - GENERAL - This method estimates the maximum acceleration at each point of the dam then the maximum stresses. - This method is based on two assumptions: - 1) the acceleration depends on the seismic spectrum and on the shape of the dam - 2) Gravity dams always have the same triangular shape 12 TARDIEU S DYNAMIC SIMPLIFIED METHOD - GENERAL 6
13 TARDIEU S DYNAMIC SIMPLIFIED METHOD - HYPOTHESES The dam is founded on sound rock The effect of the bank to bank acceleration is negligible The dam behaves as a triangle where the height is equivalent to the upstream water height for a reservoir at Full Supply Level. The maximum acceleration is supposed to be reached for the most critical fundamental frequency of the dam The hydrodynamic effect is taken into account by Westergaard analysis. 14 TARDIEU S DYNAMIC SIMPLIFIED METHOD IN 4 STAGES 1) Determine the first, and most critical natural frequency of the gravity dam 2) Compare with spectrum to know the spectral amplification of the dam 3) Evaluate the shape of the envelope of the maximum acceleration in the dam 4) Assess the maximum stresses at the u/s and d/s faces 7
15 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 1 N=0,23 S/H for empty reservoir N=0,17 S/H for full reservoir where S=(G/ ) 0,5 (Shear wave velocity) with G=E/2(1+ ) For example, an RCC dam 110 m meters high (Case 1) N =2,9 Hz (E = 20 GPa, 2400kg/m3, v=0,2) A masonry dam 40 meters high (Case 2) N = 7,2 Hz (E= 15 GPa, 2200 kg/m3, v=0,2 ) 16 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 2 This frequency is compared with the spectrum of the site to know the spectral acceleration of the dam for a certain level of damping which increase when the oscillation increases. For example, for NF EN 1998-1 spectrum and Case 1 Concrete dam or Case 2 Masonry Dam the spectral amplification is 1,5 or 2,0 8
17 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 2 Spectral amplification Periode in seconds 18 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 3 Maximum acceleration in all part of the dam can also be calculated with regards to the relative height of the dam according to a diagram presenting the following characteristics. At the bottom, the PGA At level corresponding to 0.6 H, the spectral acceleration multiplied by 1 for concrete dams and 0.9 for masonry dams At the crest, the spectral acceleration multiplied by 2.5 for concrete dams and 1.9 for masonry dams For example, the maximum amplification for Case 1 is 3,75 and 2,85 for Case 2 9
19 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 3 Relative elevation (total dam height) Acceleration amplification 20 TARDIEU S DYNAMIC SIMPLIFIED METHOD STAGE 4 - Maximum stresses can be calculated at each level with regards to the forces and moments by assuming a linear distribution between the u/s and d/s face of the dam. 10
21 ANALYSIS OF THE RESULTS - To evaluate of the risk of cracking, the maximum tensile stress shall be compared with the maximum dynamic tensile resistance of the material. - If cracking does occur, check the stability post-earthquake taking into account the pressure inside the cracks. - Another approach is to compare the maximum acceleration to the critical acceleration at any level on an horizontal joint (g. tan ). 22 ANALYSIS OF THE CRITICAL ACCELERATION TO CHECK THE STABILITY OF THE UPPER PART OF THE DAM - For a concrete dam, we can assume the friction angle at the beginning of the movement to be as high as 50 ; tan = 1,19 - For a masonry dam, we can assume the friction angle at the beginning of the movement to be as high as 45 ; tan = 1 11
23 ANALYSIS OF THE CRITICAL ACCELERATION TO CHECK THE STABILITY OF THE UPPER PART OF THE DAM - At the upper part of the dam, above the full supply level, there is no water effect. - For a concrete dam, we can assume an amplification equal to 6,25 x PGA. Risk of irreversible displacement is unlikely to occur for PGA below 0,2 g - For a masonry dam, we can assume an amplification equal to 3,8 x PGA. Risk of irreversible displacement is unlikely to occur for PGA below 0,25 g. 24 ASSESSMENT OF IRREVERSIBLE DISPLACEMENT - Irreversible displacement can be assessed by integrating twice the part of the accelerogram which overpass the critical acceleration. - Assuming that the curve of the accelerogram is a sinusoid with a period T, the irrevesible displacement is equal to - Dirr = A.T² / 4 x (1-2/. Asin(g.tan / A) with T period and A maximum acceleration of the accelogramm at the upper part of the dam. 12
25 ASSESSMENT OF IRREVERSIBLE DISPLACEMENT 160 Irreversible displacement 140 120 Dam height m 100 80 60 5 cm 1 cm 2 mm 40 20 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 PGA (g) 26 COMPARISON WITH DATA GIVEN BY THE JCOLD - JCOLD provides data showing amplification versus PGA. - These data are compared with the amplification calculated with Tardieu s method versus damping. 13
27 COMPARISON WITH DATA GIVEN BY THE JCOLD 2 % 5 % 10 % Damping Amplification 15 10 5 Tardieu versus damping JCOLD versus PGA 0,1g 0,2g 0,3g 0,4g 0,5g P.G.A PUBLIC PUBLIC 14