By Masato Nakajima and Kosuke Yamamoto
Spillway gates are important structures for ensuring the safety of a dam during a flood. There are about 1,500 gates at the 400 dams maintained by major electric power companies in Japan. Most of these gates were installed between 1945 and 1970. Over that time period, drastic improvements were made in gate design, manufacturing technology, and materials. As a result, there is a great deal of variation in the type and safety performance of the gates installed at these dams.
Although accidents due to failure of dam spillway gates are rare, attention must be given to those that have occurred, including Wachi Dam in Japan and Folsom Dam in the United States.^{1,2} Studies of these accidents suggest that buckling of the strut arms could cause structural failure in aging radial gates. However, methods for evaluating aged and deteriorated radial gates are limited. Thus, researchers in Japan conducted a study to develop a more effective structural reliability evaluation method for deteriorated radial gates. The resulting method provides dam owners and operators with an opportunity to develop a riskbased maintenance program for spillway gates.
Assessing gate performance
Performance assessments of radial gates should be based on the structure’s actual load conditions, as determined by historical operation and maintenance records. The loads are considered as either normal or earthquake.
Normal loads include selfweight; hydrostatic, sediment, wave, and ice pressure; buoyancy; wind and snow load; temperature change; hydraulic pressure change as a result of flowing water, primarily when the gates are closed; load increase due to vibrations caused by a change in hydraulic pressure; and gate operating force.
Earthquake loads include selfweight; hydrostatic, sediment, wave, and ice pressure; buoyancy; snow load; and dynamic pressure and inertia force during an earthquake.
Deterioration is an abnormal condition that increases every year and requires accurate evaluation to allow an adequate performance assessment throughout the service period of the gate. Primary deterioration factors that affect structural safety are corrosion, wear, and fatigue. Corrosion is particularly important because it reduces the thickness of the structural members, thus reducing the loadcarrying performance of the gate. Therefore, the extent of corrosion must be carefully measured. Special attention also is required for corrosion involving the accumulation of rust on support bearings (trunnion pins), which causes frictional resistance. This resistance can result in increased hoisting load as a gate operating force.
Developing a structural reliability method
Most hydropower structures in Japan are designed using the allowable stress method, which calls for assessment of structural safety based on material safety factors and design loads. Factors of uncertainty (those that cannot be evaluated or predicted deterministically due to the randomness of the phenomena or lack of data) for material strength and loads considered at the design stage are consolidated into the overall assessment. Thus, this method cannot provide a reasonable and quantitative assessment of structural safety. Moreover, this method does not include recognition or quantitative assessment of the failure mode. It is essential that a reliability assessment for structures is introduced during the maintenance process to remove allowable stress from the safety index and to facilitate discussion of the structure’s safety in terms of failures.
The structural reliability theory, developed in the aeronautical engineering field, has been applied to evaluate the quantitative safety of structures in civil and mechanical engineering.^{3,4} However, it is sometimes difficult to evaluate the failure probabilities of civil engineering structures because:
 Real structures generally consist of multiple members, and the loaddisplacement relationships are nonlinear; and
 Complete information on load and resistance is rarely available because natural hazards (such as earthquakes, wind, and waves) are potentially random or stochastic phenomena.
To overcome these limitations, we developed a procedure to evaluate the structural reliability of radial gates. The procedure involves five steps:
 1) Specifying failure modes for target structures under the considered loads, on the basis of past failure examples and results of nonlinear structural analysis. When multiple failure modes are identified, the failure probability is calculated by considering the correlation between each mode.
 2) Selecting parameters, such as corrosion, to represent aging deterioration of the gate and setting a range for each parameter. Then, a histogram representing a relationship between the target parameter and its frequency is generated, based on statistical data. When actual data is not available, empirical judgments from civil engineering experts are used. Finally, the probabilistic method is applied in order to estimate the probability density function (PDF) of each parameter.
 3) Performing a finite element analysis of the structure to obtain response values of the target elements or sections, which are unit boundaries for computational purposes. This analysis is performed for all the various combinations of the parameters.
 4) Modeling the PDF of the load variables and resistance variables, which together make up the performance function of the structure. Load variables are modeled considering the occurrence probability of deterioration parameters. Resistance variables are stochastically modeled based on the statistical data.
 5) Applying a structural reliability evaluation method (such as first order second moment or advanced first order second moment)4 and a first order Gausssecond moment method to compute the failure probability and a safety index
This method considers corrosion and trunnion pin friction on radial gates as deterioration parameters and employs precise structural analysis methods. For the sake of simplicity, we assume that the two deterioration parameters are independent of each other. A more realistic probability distribution of response values is established by considering the joint probability for each parameter combination. The structural failure probability of the target gate is evaluated without conducting heavy calculations, such as Monte Carlo Simulation. This characterizes the procedure’s potential for evaluating structural reliability more efficiently and precisely.
Testing the method numerically
To verify this method, we conducted several tests using the finite element method. Electric power companies in Japan used the analytical model generated as a design reference in Japan.^{5} This model was selected to facilitate meaningful discussion among facility managers on safety through comparison between the design standards (basic case) and results of investigations of existing structures.
We performed a threedimensional structural analysis on a simulated gate using Abaqus software from Dassault Systemes.
The finite element analysis was conducted as follows:
 Selfweight analysis. The selfweight of each part was calculated based on the steel volume and unit weight.
 Stress analysis under design water level load. A static stress analysis was conducted under design water level conditions.
 Stress analysis under hoisting load. A static hoisting load was applied to the cable attachment and an analysis was conducted on stress due to trunnion pin friction.
The water levels and seismic intensities shown in Table 1 were considered to be external forces. Two load combinations were set for this analysis. For design floods, the water level is 12.6 meters. Seismic intensity is not considered. For earthquakes, the flood surcharge water level (the maximum rise of the water level above the reservoir top water level during a design flood) is 12.4 meters.
Recently, Japan has experienced several large magnitude earthquakes, and strong ground motion of more than 0.5 gravity has been measured at dam sites. We considered three cases for earthquakes (seismic intensity of 0.12 gravity, 0.15 gravity, and 0.20 gravity), in accordance with the dam design guidelines in Japan.
Two aging factors were considered in the analysis corrosion of the steel members and variations in trunnion pin friction coefficients. Using data from 71 gates and 62 penstocks,^{6,7} we established the relationship between years in operation and mean corrosion. With regard to friction coefficients, there is little data available. Thus, we evaluated the friction coefficients using data gathered in a survey we performed of 30 civil engineering experts.
We estimated the PDF of each load variable using the LevenbergMarquardt method, a technique for nonlinear least square problems.^{8} In this analysis, we assumed that there is no direct correlation between the corrosion and friction coefficients.
To model resistance as a random variable, we adopted data on yield strength of penstocks manufactured in Japan in the 1950s.^{6,7} Using the data, we assumed that the resistance variable had a normal distribution, with a mean value of 281 megapascals (MPa) and a standard deviation 19.6 MPa.
Based on past failure accidents and nonlinear structural analyses, buckling of the strut arms near the trunnion is considered to be the gate’s limit state.
While several methods have been developed to calculate structural failure probability,^{3,4} we chose to adopt the Gaussian firstorder approximation method.^{9} We defined the performance function of the radial gate as:

Equation 1:
 Z = R – S

where:
 Z is the performance function of the gate;
 R is the resistance variable; and
 S is the load variable.
The safety index by the firstorder Gaussian approximation method is defined as:

Equation 2:
 b = mz / sz

where:
 b is the safety index;
 mz is the mean value of the performance function (Z); and
 sz is the standard deviation of Z.
The failure probability can be obtained from the following equation:

Equation 3:
 Pf = P(Z < 0) = 1 – F (mz / sz)

where:
 Pf is the failure probability;
 P is the probability;
 Z is the performance function;
 F is the cumulative probability distribution function of standard normal distribution;
 mz is the mean value of Z; and
 sz is the standard deviation of Z.
Equations 1 through 3 illustrate the following issues:
 The safety index (b) is the difference between the mean value of the performance function (Z) and the failure criterion;
 The safety index is invariant even though the performance function Z is represented by another numerical formula; and
 If the value of the safety index is small, structural reliability is low.
Results
Figure 1 shows the results of finite element analysis for the flood water level load. The left image does not include aging factors while the right image includes the largest deterioration values. The largest stress occurs on the strut arms near the main girder in the left image and near the trunnion in the right image. Interestingly, trunnion pin friction affected the stress level considerably more than did corrosion. The analyses also revealed that the state under earthquake conditions (which factors in design seismic intensity and surcharge water level) poses a higher risk than the state under the design flood level.
Figure 2 shows the probability density distributions of four load variables and a resistance variable obtained from the results of the numerical simulation. Peaks are at around 100 MPa for the load variables and near 275 MPa for the resistance variable.
Table 1 shows the results of the structural reliability evaluation. The safety index (b) for the design flood level is 3.94 and the failure probability (Pf) is 4.056◊105. Even under the worst earthquake conditions, the safety index is 3.16 and the failure probability is 7.960◊104. Thus, the structural reliability of the example is well within the reliability level of civil engineering structures.^{10,11}

null
Conclusions
Information on structural reliability is essential when evaluating current and future risk because of the uncertainties in maintenance and operation. We believe our reliability evaluation method can be an efficient tool in establishing an effective riskbased maintenance program for deteriorated radial gates. s
Drs. Nakajima and Yamamoto may be reached at Central Research Institute of Electric Power Industry, 1646 Abiko, Abikoshi, Chibaken 2701194 Japan; (81) 471821181; Email: [email protected] or [email protected].
Notes
1 Kawamura, K., and S. Nakano, “Summary of Investigation Reporton Wachi Dam Gate Accident,” Civil Engineering Journal, Volume 10, No. 9, 1968, pages 2632 (in Japanese).
2 Forensic Report Excerpt Spillway Tainter Gate 3 Failure, Folsom Dam, California, USA, U.S. Department of the Interior, Bureau of Reclamation, Mid Pacific Region, 1996.
3 Freudenthal, A.M., “Safety, Reliability and Structural Design,” Journal of Structural Division, Volume 87, No. 3, 1961, pages 116.
4 Shueller, G.I., “Structural Reliability – Recent Advances,” Proceedings of the International Conference on Structural Safety and Reliability, Springer Netherlands, pages 135, 1997.
5 Technical Guideline for Hydraulic Gate & Penstock: Design Reference of Hydraulic Gate, Japan Hydraulic Gate and Penstock Association, Tokyo, Japan, 1986 (in Japanese).
6 Numazaki, Y., “Investigation on Corrosion and Resistance of Existing Penstocks,” Electric Power Civil Engineering, Volume 54, No. 151, November 1977, pages 3740 (in Japanese).
7 Taguchi, Y., T. Murase, and K. Andoh, “Diagnostic Technique on Penstock Healthiness Investigation on Corrosion of Penstock Support Base Using Special Ultrasonic Waves,” Hydraulic Gate and Penstock, No. 211, June 2002, pages 5459 (in Japanese).
8 Nocedal, J., and S.J. Wright, Numerical Optimization, Springer Series in Operations Research, SpringerVerlag, pages 262266, 1999.
9 Hasofer, A.M., and N.C. Lind, “Exact Invariant SecondMoment Code Format,” Journal of Engineering Mechanics Division, Volume 100, No. EM1, February 1974, pages 111121.
10 Nagao, T., “Reliability Based Design Way for Caisson Typed Breakwaters,” Journal of Structural Mechanics and Earthquake Engineering, Volume 689, No. I57, October 2001, pages 173182 (in Japanese).
11 Barker, M., et al, “Spillway Gate Reliability and Handling of Risk for Radial and Drum Gates” Proceedings of NZCOLD/ANCOLD 2003 Conference on Dams, Australian National Committee on Large Dams, Tatura, Victoria, Australia, 2003.
Masato Nakajima, PhD, a research engineer in the Earthquake Engineering Department at the Central Research Institute of Electric Power Industry (CRIEPI) in Japan, is responsible for probabilistic safety assessment of dams based on reliability theory. Kosuke Yamamoto, PhD, a senior research engineer in the Structural Engineering Department at CRIEPI, is project coordinator for maintenance technology of hydropower civil engineering facilities.
This article has been evaluated and edited in accordance with reviews conducted by two or more professionals who have relevant expertise. These peer reviewers judge manuscripts for technical accuracy, usefulness, and overall importance within the hydroelectric industry.