Probabilistic Safety Evaluation of a Concrete arch dam Based on Finite Element Modeling and A Reliability L-R Approach

Abstract The safety assessment of the Pacoima arch dam is investigated in this paper. A Load – Resistance (L-R) method was used to ensure that the dam is safe or if it is at risk of failure. The “probabilistic design system” ANSYS finite element software was used to calculate the probability of failure. The Monte Carlo (MC) method with 50,000 iterations utilized for simulation and the Latin Hypercube method were used for Sampling. Input random variables with normal distribution and coefficient of variation of 15% due to uncertainties were considered and the six random variables used are the concrete modulus of elasticity, Poisson’s ratio of concrete, concrete mass, up-stream normal water level of the reservoir, and the allowable tensile and compressive strength of the concrete. Linear elastic behavior was assumed for the constitutive law of concrete material and if the stress exceeds the allowable stress of the concrete this is considered as a failure limit state. The maximum and minimum principal stresses were considered as the output parameter. Dam body safety was investigated only under self-weight and upstream hydrostatic pressure at the normal water level. The probability of failure of the dam body system was determined as βsystem=3.98, the safety index as pfsystem =3.42×10−15 and the dam is at risk of failure. The first and third principal stresses in the dam body were also S1max=2.03MPa and S3min=4.6MPa, respectively for the worst case of MC simulation.


INTRODUCTION
A dam is one of the infrastructures of any country whose structural failure can result in significant loss of life and irreparable financial damage.The acceptable risk of failure to such structures is, therefore, much lower than that of conventional residential and industrial buildings.The real world is a world of uncertainties.Routine risk and safety assessments of dams are carried out by using the determination approach (safety factor) [27], particularly in the static analysis of dams, in which the physical and mechanical properties of the dam body materials and the hydrostatic pressure are assumed to be constant values.The probabilistic analysis approach seeks to more accurately assess the behavior of dams and has been used in many studies [1][2][3][4][5]7,[9][10][11][12][13][14][15][16][17]26].In such studies, the Probabilistic Design System (PDS) tool from ANSYS software is used to evaluate the probabilistic and reliability analysis of structures [4,6,20,24].The ANSYS/PDS provides an efficient tool to assess the interactions, effects, and sensitivities between input parameters and output variability.However, none of the aforementioned studies looked at the reliability analysis of the concrete dams from the perspective of the physical and mechanical properties of the concrete and the up-stream hydrostatic pressure.The purpose of this paper is a probabilistic safety assessment of a dam under usual loads including dam body self-weight and up-stream hydrostatic pressure on the dam.The ANSYS/PDS tool was utilized to perform the reliability analysis of a concrete arch dam using Monte Carlo Simulation (MCS) and Latin Hypercube Sampling (LHS).The performance criteria were defined using tensile stress.The innovational aspect of this paper is the investigation of dam safety by considering uncertainty in specific gravity, concrete strength, and upstream hydrostatic pressure.The Pacoima concrete arch dam has been selected as the case study for this paper.

A DESCRIPTION OF THE PACOIMA CONCRETE DAM
The Pacoima double-curved concrete arch dam is located in Los Angeles, California.The dam was completed in 1928 A.D and is shown in Figure 1.The Pacoima dam is 113-meters high with a crest length of 180 meters.The thickness of the dam varies from about 3m at the crest to 30m at the base of its crown cantilever.The eleven contraction joints in the dam body have beveled keys that are 30cm deep.The finite element modeling (FEM) of the dam body was done by assuming a rigid foundation and simplifying the geometry of the dam on the left abutment. .Each node has three degrees of freedom comprised of translations in the nodal X, Y, and Z directions.243 nodes and 104 solid elements were used in the FE model.The minimum number of elements were utilized to save computational cost.The frequency of the first symmetric mode of the dam body is obtained at 5.42 Hz with a damping ratio of 5%.The FE model degrees of freedom (DOF) is 972 of which 135 are supported DOF.Nodal displacement constraints were applied to the nodes located on the dam base and left and right abutments of the dam body.The modulus of elasticity and Poisson's ratio of mass for concrete was taken as 21.9GPa and 0.2, respectively.The mass density of the concrete is chosen as 3 2230 m kg   [8].Loads applied to the dam include the dam body self-weight and up-stream hydrostatic pressure [18,[21][22][23].Thermal loading is not considered in this study.The hydrostatic pressure of the normal level of the reservoir is shown in Figure 2. The hydrostatic pressure applied to the upstream face is perpendicular to the surface of each solid element.In Figure 2, the contour of the hydrostatic pressure is visible.To verify the application of water level on the upstream, the equation of hydrostatic pressure can be controlled.By dividing the value of hydrostatic pressure at base level by Where NSIM is the number from the Monte Carlo simulation.In the design of concrete arch dams, the maximum tensile stress of the dam body must be less than its allowable stress values [18,[21][22][23].Dam failure is assumed to be due to cracking in the dam body.The maximum existing stress in the dam body under imposed loads should be limited to the allowable stress of its materials.Reliability analysis using Monte-Carlo simulation is done for 50000 time generations .The " "i subscription, assigned for the th i failure function as well as capacity and demand functions, relates to the th i simulation loop.In order to save computational time, the analysis is performed in batch mode.The procedure followed in the present research to assess the dam body safety level is shown by the flow chart in Figure 4.

Random Variables
In this paper, the parameters listed in Figure 5 are those whose epistemic uncertainties were included in the reliability study as random variables.For probabilistic analysis, it is necessary to define the resistance and load parameters (modulus of elasticity of concrete, Poisson ratio of concrete, concrete density, and up-stream hydro-static pressure), which can be seen in Table 1.The normal distribution (N) is considered for the random variables in the present study.To generate random variables with the Gaussian distribution, there is a need to define mean value ) (  and standard deviation ) ( for each variable.The values in Table 1 are selected according to available literature [8,24].The values of allowable tensile stresses were then calculated according to the compressive strength of the concrete and standard deviation of 15%.To simplify, the standard deviation value for all random variables was considered equal and in the reliability analysis with PDS, the confidence and significance levels were set as 95 and 2.5% respectively.The number of cloud points in this figure is equal to the number of The results history for the maximum existing tensile stress in the dam body in each simulation loop is shown in Figure 7.The L-R model for maximum stress is shown in Figure 8.Both Load and Resistance were considered as random parameters.As shown in Figure 8, most of the resistance model samples were above the load effect samples.In Figure 9, the tensile failure function is plotted and the value of the two samples of all simulations was negative   The Failure tree of the dam body system is shown in Fig. 11.Therefore, the system probability of failure and the reliability index were calculated as The probabilistic analysis of the Pacoima dam has shown that the required safety is not provided for the limit state and the dam body is at risk of failure.The most vulnerable elements in tensile and compressive stress were located in the left abutment at mid-height of the dam body.

RESULTS AND DISCUSSION
. The reliability index value obtained in this study was not within this range, so the dam is at risk of failure.Cumulative Distribution Functions (CDF) of failure functions (first and third principal stresses) are shown in Figures 14 and 15

CONCLUSION
The purpose of this paper was a probabilistic safety assessment of the Pacoima arch dam under usual loading.In the structural analysis, the dam body self-weight and up-stream hydrostatic pressure were considered as usual loading and the foundation was considered rigid.A methodology to determine the reliability index of concrete dams was presented.The load-resistance method was used to calculate the probability of failure functions then, using the Gauss standard function and probability of failure, the reliability index was obtained.To ensure the dam is safe, the Reliability Index must be larger than the Target Reliability Index.
A normal distribution and a coefficient of variation of 15% were chosen for the production of the random variables.In the deterministic analysis of the Pacoima dam, safety was confirmed, while in the probabilistic analysis, the dam was at risk of failure.This failure presented as cracking of the left abutment elements at the mid-height of the dam body.These are the most vulnerable regions of the dam., respectively.The negative value of the failure function means that cracking will occur in vulnerable areas.The probability of failure was obtained by Monte Carlo simulation of value .It is clear that the POURAMINIAN, Somayyeh POURBAKHSHIAN, Ehsan NOROOZINEJAD FARSANGI, Reza FOTOUKIAN dam safety margin under the combination of its self-weight and the reservoir hydrostatic pressure loads is insufficient.Therefore, the Pacoima Dam is at risk of failure under the assumed conditions.From the results of the statistical study, important decisions must be made regarding retrofitting the dam so long as the reliability index of the dam reaches the target confidence level.

Fig. 1 .
Fig. 1.View of Pacoima dam of the upstream water is obtained and is equal to 90m.

Fig. 2 .Fig. 3 .
Fig. 2. Hydro-static pressure in upstream (US) face of dam body in normal water level Next, the dam was analyzed under usual loads, dam body self-weight, and upstream hydrostatic pressure.The dam structure is also statically analyzed.The distribution of minimum (S3) and maximum principal (S1) stress under the assumed loading is shown in Figure 3.The minimal value of principal stress 18 .2 3 min   S MPa and maximum principal stress 94 .0 1 max  S MPa are lower than

Fig. 4 .
Fig. 4. Flow chart of the procedure of Reliability Analysis used in the present study suggested to obtain the probability of failure.In the present study, with fewer simulations, the probability of failure was obtained and 000 .50  sim N simulations were considered for the probability of failure computing.The simulated samples of six random variables are shown in Figure 6.

Fig. 6 .
Fig. 6.Samples history of random variables simulation loops, minimum existing compressive stress being less than allowable compressive stress means the probability of crushing failure in the dam body was zero ( 0  c f p ).The compressive failure function is also shown in Fig. 10 with no negative values for samples   .

Table 1 .
Random variables defined in the finite element model of the dam i  

Table 2 .
Random values for variables in MCS number 44400 Random Parameters The worst value for the tension failure function obtained was