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Study on Statistical Characteristic of Strength of Steel Plate Used for Penstocks of Hydropower Stations Jianguo Hou Shaojun Fu Xuwen An Yingming He College of Civil & Architectural Engrg., Wuhan Univ. of Hydr.&Elec. Engrg.,Wuhan 430072 China Contact

Abstract:

This paper collects more than 20,000 groups of testing data of steel plates' strength, such as steel grades Q235 and Q345, which are produced by representative steel mills in China during recent years, divided into groups in terms of grouping stipulations by thickness of steel plates in the criteria (GB 700-88, Carbon Structural Steels;GB/T 1591-94, High Strength Low Alloy Structural Steels), analyzes yield strength s_s and ultimate tensile strength s_b of these steel plates by utilizing statistical theory and come up with the relevant function of probability distribution, its type and the statistical parameters etc. . The statistic analytic result shows that mechanical properties of steel plates produced by Chinese steel mills are stable and dependable and can meet the requirements of steel plates prescribed in national standards GB 700-88 and GB/T 1591-94, and the result also has testified the strength values of steels which are stipulated in the present national steel standards, and can be settled for the requirements that a normal value of material strength should be confirmed on the basis of the rule that dividing value of probability distribution is equal to 0.05 and the assurance ratio must be not less than 95%,writed in Unified Standard of Reliability of Hydraulic Engineering Structures (GB 50199-94).For the steel with higher ratio of yield strength to ultimate tensile strength s_s /s_b, on the basis of referring to many home and overseas documents and some Chinese experts' suggestions, this paper also advises that restrictive ratio A(A= s_s /s_b) of steel plates may be increased from 0.67 prescribed in old code to 0.75 in order to make full use of high strength steel. In conclusion ,the forenamed research findings offer an important information for revising present code SD144-85 in terms of limit state design method based on probability, and it takes on an important practical significance and a referential worthiness in the field of steel productions and their quality control in China.

Keywords: penstocks; steel plate, strength, probability distribution, statistical parameters

1 Introduction

"Design Code of Penstocks of Hydropower Stations"(SD144-85) [1], Chinese Electric Power Industry Standard, is being edited in terms of the probabilistic limit state method of "Unified Standard for Reliability of Hydraulic Engineering Structures"(GB50199-94) [2]. Whereas the first work is to finish calibrating and analyzing the reliability of SD144-85 by using probabilistic method, make clear the general reliability level of designed penstocks by active code. Afterward, come up with a suggestion that be taking target value of reliability is always used as a design base in new code. Statistical parameters of basic variables influenced on reliability of penstocks are the groundwork of analyzing reliability of penstocks, therefore, strengths of steel plates are the absolutely necessarily basic data during analyzing the reliability of penstocks. This study is carried out to obtain the statistical characteristics of strengths of steel plates used for penstocks of hydropower stations.
The typic steel plates of penstocks of hydropower stations in China are carbon steel of steel grade Q235 and high strength low-alloy steel of steel grade Q345. Therefore, we collected more than 20,000 groups of strengths testing data of the steel grades Q235, Q345, which are produced by representative steel mills in China during recent years. On the basis of statistics and analysis, the probability distribution function, type of distribution and the statistical parameters of above mentioned steel plates' strength were determined. It provides an important scientific reference for analyzing the reliability of penstocks, and it takes on an important practical significance and a referential worthiness for studying preferably the reliability of penstocks and steel production and quality control in China.

2 Collecting and preprocessing testing data

2.1 Collecting testing data
There are more than 20,000 groups of strength testing data in this research, they are taken from the strength testing in the representative steel mills in China .The thickness of steel plates scope from 2.5mm to 150mm.The quantity of the specimens is very large. It is representative that each statistical parameter of yield strength and ultimate tensile strength are obtained from counting and analyzing the testing data. Table 1 lists the data from various steel mills.

2.2 Preprocessing
With the purpose of ensuring the accuracy and the dependability of the statistic analysis results, the experimental data must be checked and amended appropriately before statistic analysis, the abnormal data should be discarded and manipulated, and fill up the deficient data. Sometimes all data should be reordered by quantity and transformed properly, for instance, logarithmic transformation and exponential transformation. Therefore, according to the characteristics of steels used for penstocks of hydropower stations, this study preprocesses the collected test data of steel plates' strength of various steel grades produced by Chinese mills as table 1.

Steel grade Manufacturer	Structural carbon steel of steel grade Q235 (groups)	Low-alloy structural steel of steel grade Q345 (groups)	Sum (groups)
Wuhan steel mill	4222	1282	5504
Anshan steel mill	1283	829	2112
Chongqing steel mill	3570	2840	6410
Wuyang steel mill	5045	1015	6060
Sum	14120	5966	20086
Table 1 :Test data of steel plates of various steel grades produced by Chinese mills

2.2.1 Grouping according to thickness of steel plates
At the sight of the production of steel plate, the probability of the defect of steel plate increases with its thickness. If the designers take steel plate as the perfect homogeneous material while designing and accepting penstocks, there should be some hidden damage for penstocks of hydropower stations in their safety operations. Hence, when analyzing the reliability of penstocks, we should consider the mutation of strength caused by choosing different thickness of steel plates. For the sake of the comparable of statistic analysis results to national standard steel plates, the collected more than 20,000 groups of testing data of steel plates' strength is divided into groups in terms of grouping stipulations by thickness t of steel plate in the standards (GB 700-88, Structural Carbon Steels [3]; GB/T 1591-94, High Strength Low Alloy Structural Steels [4] ). Table 2 lists the grouping result.

Grouping scope Steel grade	(2.5~16mm)	(16~40mm) (16~35mm)*	(40~60mm) (35~50mm)*	(60~100mm) (50~100mm)*	(100~150mm)
No. Of specimens of Q235 ni	3991	7380	1861	718	170
No. Of specimens of Q345 ni	2632	2230	646	396	36
Table 2 : Grouping scope of steel plate thickness(t)and the number of specimens (ni)

The star * denotes grouping scope of steel plates of steel grade Q345.

2.2.2 Checking and amending test data of strength


1. Dealing with the abnormal data
   During the experiment or observation, sometimes a few numerical data in the sample may be overlarge or over small, deviate far from normal data, these data usually are referred to as abnormal data or outlying observation. Distinctly, we should first recognize the abnormal data and then cull them in order to ensure the precision of the analyzing results before analyzing and computing further.
   The methods dealing with abnormal data usually are 3s rule, Crubbs law and Chauvenet law etc.. As far as the introduction of the reference [5], the recognition effect of Crubbs law is best. Hence, this paper takes the obvious level to a = 5.0% and deals with the abnormal data by Crubbs law.
2. Filling up the deficient data
   A few deficient data were found through analyzing the collected testing data. Hence, for the sake of reflecting the statistic law of testing data of strength and ensuring the accuracy and the reliability of the statistic analysis results, we filled up the deficient data appropriately.
3. Transforming the data
   The testing data have to be transformed properly in order to compute and analyze conveniently. It is clear that the normal distribution is more sophisticated than others in theory, and it is very convenient to practice. Furthermore, transforming the data to the form of the normal distribution properly can enlarge the application fields of the random simulation by the computers and enhance the practical effects of the random simulation by the computers better.
   Having been studied and analyzed through the experiment and the theory, the resistance of structure is often subject to normal distribution and logarithmic normal distribution. Because strength of steel plates is the major component constituted the structural resistance of penstocks, all testing data X(x₁,x₂,...,x_n)were converted by following transformations.
   
   1. logarithmic transformation: Y=lnX ;
   2. exponential transformation: Y=e^X .

3 Counting and analysis

3.1 Computing statistical parameters
In accordance with statistical theory, the digital feature of random variables is given by the following equations:

(1)

(2)

or

(3)

For the sake of convenience of making use of programs to compute the statistical parameters and S² accurately, the transformations of equations(1) to (3) are given by the following formulas:

(4)

(5)

where

(6)

(7)

Initial value

Mean of specimens

(8)

Mean square

(9)

Variation coefficient

(10)

Tables 3 and 4 list the computing results.

Grouping scopes (mm) Statistical parameters	2.5~16	16~40	40~60	60~100	100~150	AVE.
AVE. of yield strength s (N/mm2)	303.41	270.02	247.28	230.30	224.82	273.76
AVE. of ultimate tensile strength b (N/mm2)	456.87	446.45	442.33	437.20	431.76	448.16
Standard deviation of yield strength Ss (N/mm2)	22.69	25.07	24.78	22.70	21.38	32.53
Standard deviation of ultimate tensile strength S b (N/mm2)	21.73	20.02	22.26	21.61	19.30	21.75
Variation coefficient of yield strength ds	0.07	0.09	0.10	0.10	0.10	0.12
Variation coefficient of ultimate tensile strength db	0.05	0.04	0.05	0.05	0.04	0.05
The Number of specimens	3924	7371	1861	718	170	14044
Table 3 : Statistical parameters of s s and s b of steel plates of steel grade Q235

Grouping scopes (mm) Statistical parameter	2.5~16	16~ 35	35~ 50	50~ 100	100~ 150	AVE.
AVE. of yield strength s (N/mm2)	381.74	355.56	333.13	319.69	303.29	362.01
AVE. of ultimate tensile strength b (N/mm2)	553.08	539.20	527.15	527.83	513.94	543.13
Standard deviation of yield strength S s (N/mm2)	25.22	27.91	25.06	28.34	25.30	33.45
Standard deviation of ultimate tensile strength S b (N/mm2)	28.10	31.05	27.32	28.20	27.38	30.45
Variation coefficient of yield strength ds	0.07	0.08	0.08	0.09	0.08	0.09
Variation coefficient of ultimate tensile strength db	0.05	0.06	0.05	0.05	0.05	0.06
Number of specimens	2632	2230	646	396	36	5940
Table 4 : Statistical parameters of s s and s b of steel plates of steel grade Q345

Parameters of normal distribution density function are given by

(11)

Parameters of logarithmic normal distribution density function are given by

(12)

3.2 Computing empirical distribution and fitting verification
3.2.1 Computing empirical distribution
Given that the size of a sample is N and the observation data are X{x₁,x₂,...,x_n }, X may be regarded as an entirety, so the empirical distribution can be obtained by

(13)

Where

K_x-the number of the observation data less than x.
The empirical distribution converges to the ideal distribution F(x) and the probability is equal to 1 with the increasing of the size of the sample, i.e. As far as this, the empirical distribution will determined as follows:

1. Computing the minimum and the maximum of the positional characteristic parameters of the testing data respectively, that is a=min{x₁,x₂,...,x_n }, b=max{x₁,x₂,...,x_n };
2. The quantity of grouping k is worked out by the following empirical formulas, i.e. k = 1.87(n-1)^2/5 , and then total zone [a, b] is subdivided to k non-cross subintervals,
3. Calculating the empirical frequency is the frequency count;
4. Drawing the histogram of s_s, seeing Figs. 1 to 2.

3.2.2 Fitting verification by c² method
The testing data of the steel plates' strength is worked out preliminary to be satisfied with the following distributions through Figs.1 to 2.

The normal distribution

(14)

and the logarithmic normal distribution

(15)

The following steps would adopt the c² fitting method to verify the distributions. The obvious level was taken by the stipulation of ISO, i.e.a   = 5%.

Fig 1: Yield strength histogram of steel grade Q235

Fig 2: Yield strength histogram of steel grade Q345

1. Postulation H₀:The distribution function of total sample X is F₀(x), i.e. equations(14) and (15) ;
2. According to the postulation F(x)=F₀(x), the probability can be calculated by equation(14) or equation(15), then p_i= F(a_i)-F(a_i-1) is obtained ,i=1,2,...,k;
3. Calculating the real frequency v_i and the ideal frequency np_i, i=1,2,...,k,if np_i <5, these zone should be combined with the near zone to make this case included into the new big zone in order to satisfy the requirement of np_i ³5;
4. The statistic value of c² can be calculated by the following equation

(16)

5. Looking up the c² distribution table of the critical value to determine the value of c²_0.05(k-1), then compared to x² to discriminate whether the postulation is right. Given that c² > c²_0.05(k-1), refusing H₀; Given that c²£ c²_0.05(k-1) , accepting H₀.
   The calculation result shows that the yield strength of steel grades Q235 and Q345 are satisfied to normal distribution, and the ultimate tensile strength of steel grades Q235 and Q345 are satisfied to logarithmic normal distribution at the obvious levela   = 5%.

4 Determination of steel plates' strength in analysis of reliability of penstocks

While the structural reliability is analyzed, the uncertainty of mechanical properties of materials (e.g. strength, elastic modulus) is mainly consisted of the variability of the material performance in structure caused by material quality, craft, loading, environments, dimension etc.. GB 50199-94[2] specifies that, for a variety of structural material, the national typical productive level should be taken the weighted average value and the variability coefficient of the material performance f of normal test sample provided by all steel mills in all over China during the same period. Moreover, the distinction of the material performance between the actual structure and normal rest sample, and the disparity between the practical conditions and the test conditions should also be considered. These disparities can be reflected by transformation factors or functions, or determined by the engineering empirical methods in accordance with the results of the corresponding compared test. So the uncertainty of material performance in structure can be expressed by

(17)

Where K₀-the random variable reflected the distinction between the performance of the structural material and the test sample material(i.e. f_c and f_s);
K_f-the random variable reflected the uncertainty of the performance of material;
f_k-the normal value of the performance of material specified by the specification;
w₀-the influencing coefficient or function which considers material defects, construction quality, effect of size, loading velocity, test methods and time effect etc. .
Thereby the average value (m _KM) of K_M and the variability coefficient (d _KM)are given by

(18)

Where
µ_f , µ_K0 , µ_Kr - these are the average of strength f_s of the test samples and the average of the random variable K₀ and K_f respectively;
d_f , d_K0 - these are the variability coefficient of strength f_s of the test samples and the variation coefficient of the random variable K₀ respectively.


When the test is being proceeded, the more quickly loading, the higher value of strength (i.e. yield strength) showed on the experimental equipment. Hence, the loading velocity is often clearly specified in all kinds of material test criteria. In fact, the loading velocity of the test when leaving the factory is different for the different factory. Even if the test is carried out according to the stipulation of the criteria, the loading speed of test is faster than the speed at which the structure is subject to the static loads. In addition that the disparity between top and low yield point, and the mistake of measuring the thickness of steel plate are different between facts and tests. Considering the influences of factors above mentioned, and then suppose that the strength of steel plate is subject to triangle distribution in a scope 1.0 to 0.85, the statistical parameter K₀ of distinction between a structure and a sample can be obtained from statistic analysis. µ_K0 and d_K0 are equal to 0.92 and 0.035 respectively for steel plates of Q235, µ_K0 and d_K0 are equal to 0.95 and 0.022 respectively for Q345 steel plates of Q345.

In terms of the statistical parameters of steel plates strength (i.e.s_s,s_b) listed in tables 3 and 4, and the strength indexes specified for different thickness of steel plates in current steel standards and the statistical parameter of K₀, the Statistical parameter of the strength s_s of steel plates of steel grades Q235 and Q345 can be acquired to analyze reliability of penstocks by equation (18). The solution lists in table5 [6].

Steel grade	m f	d f
Q235	273.76	0.12	0.92	0.035	1.145	0.125
Q345	362.01	0.09	0.95	0.022	1.109	0.093
Table 5 : Statistical parameterss adopted in analyzing reliability of penstocks for steel plates of steel grades Q235 and Q345

5 Conclusions

We can make the following conclusions in terms of the analysis results from more than 20,000 groups testing data of steel plates.
1. After the verification of hypothesis, the yield strength of steel grades Q235 and Q345 are approximately satisfied to normal distribution, and the ultimate tensile strength of steel grades Q235 and Q345 are approximately satisfied to logarithmic normal distribution.
2. The average and the standard deviation and the variation coefficient about yield strength and ultimate tensile strength of steel grades Q235 and Q345 are obtained from analyzing the collected more than 20,000 groups of testing data, referring to tables 3 to 5. These achievements can be used to analyze the reliability of penstocks when revising the code SD144-85 [1]. It also has the practical significance and the referential merit to the production of steel and the quality control in China.
3. If s _s and s _b specified in the current national standard of steel plates are compared with the average of the measured values of s _s and s _b in terms of the collected testing data of steel plates produced by a few representative Chinese steel mills, it can be discovered that the values about s _s and s _b specified by current standards is basically equal to the average of the measured s _s and s _b minus 1.645 times of standard deviation. That shows that it is practicable to take the strength indexes s _s specified in the current standards as a basis of the determination of the strength normal value of steel plates of steel grades Q235 and Q345, and it is basically satisfied to the requirements, which are that the dividing value is equal to 0.05 and the assurance probability is not less than 0.95 in GB 50199-94 [2]. This also indicates that the mechanical properties of steel plates produced by Chinese steel mills in recent years are stable and reliable.
4. If tensile strength limit of steel plates is regarded as the basic parameter, the analysis of the reliability of penstocks according to the stipulation of the current code SD144-85s_s =0.67s _b whens_s /s _b > 0.67 shows the reliable index of tensile strength limit is usually higher than that of yield strength. Therefore, with relation to the high strength steel with higher ratio of yield strength to ultimate tensile strength (s_s /s_b), according to the analyzing results and referring to many home and overseas documents and some Chinese experts' suggestions, the restrictive ratio A(A=s_s /s_b) of yield strength to tensile strength limit of steel plates may be increased from 0.67 prescribed in old code to 0.75 in order to make full use of high strength steel and acquire a certain technical-economy benefits. It is preliminarily analyzed that steels can be saved some 10% or so in the structural designs if designers will adopt A=0.75, the structural reliability b will decrease a little, but it is still satisfied to the degree of acceptable value.

References

1. Design Code of Penstocks in Hydropower Stations(SD 144-85).Beijing:China Water and Power Press,1985 (in Chinese)
2. Unified Design Standard for Reliability of Hydraulic Engineering Structures (GB 50199-94).Beijing:China Planning Press,1994(in Chinese)
3. Carbon Structural Steels(GB 700-88).Beijing:China Standards Press,1988(in Chinese)
4. High Strength Low Alloy Structural Steels(GB/T 1591-94).Beijing:China Standards Press,1994(in Chinese)
5. Crubbs F. E., Procedures for Detecting Outlying Observations in Samples.Technometrics, 1969,1~21
6. Hou,J.G.,He,Y.M.,An,X.W.,and Fu.S.J.,Calibrating the Reliability and Determining Partial Factors for Penstocks of Hydraupower Stations.Wuhan Univ. of Hydr. & Engrg.,1999(in Chinese)

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