Summary 

Using the Weibull's statistic the way that can be used to represent the characteristic tensions of a material by means of an only parameter is described. This representation is achieved in particular with the establishment of the occurrence probability of some characteristic tension. The quality level determination called "quality control" in the classic methodology, through sampling procedures is of singular importance when is wanted to declare the acceptance or rejection of materials as the concrete and the construction steel. Then knowing the probability of occurrence of certain characteristic tension we can determine, by means of a simulation process the number of samples that is necessary to test to obtain a specific deviation with a previously fixed precision. The results obtained for the concrete and the steel allow to define this way a variable plan of statical sampling, according to the preset precision.
 

Introduction

To carry out an effective control of a resistant material (in this case construction steel and cement) it is necessary to take a number of samples of a group that in statistic –and with practical purposes– it is supposed homogeneous and with an infinite number of samples. 

If we suppose that the material is characterized by the main value of its properties and its dispersion and that follows a Weibull's statistic, in this work we determine the number of samples that are necessary to test in each case to determine the specific deviation (quotient of the dispersion  regarding the main value ), with a preset error.
 

Outline of the problem

The Weibull's statistic [1,2] for the case of traction or uniaxial compression , is given by the next formula 
 

(1)

where the cumulative probability of the  variable is ,  the volume of the sample,  the unit of volume,  the fracture's specific risk function,  the ultimate tension below which there is not fracture,  and , manufacture parameters. 

We suppose that the expression (1) is valid for compression or traction of fragile materials, like this way for flexotraction. This, although is not exact, is closely fulfilled. In the traction of a ductile material case as the construction steel, the formula (1) is good, all this with characteristic parameters of each material and way of testing it. 

The calculation of  and  is the habitual one in statistic supposing : 
 

		(2)
is the Euler's function.

As we can see, the specific dispersion is 
 

(3)

Therefore  does only depend on  and defines the problem. Now, the problem is: Which is the necessary number of tests to determine  with a preset error? This question is answered in the next section.
 

Obtaining of the number of tests

According to the above-mentioned, we can make , , , what means that is a group of tests where the volume and the material stays constant, as well as the form of manufacturing it. So the cumulative probability that takes place  is given by 
 

(4)

By means of the formula (3) we can determine when we know approximately , that characterizes the test. With this value is determined. Then we can simulate many tests with the formula (4), and by means of a aleatory numbers group  is determined the  collection. With this group  is determined  for different numbers of simulated tests and the necessary number is resolved to get closer to the  value which we start. 

In the case of normalized mortars it has been determined the break tensions to the flexotracción and the compression being obtained the following values of m, being kept in mind that the values group around two families; m, m₁, m₂: 
 

For the 3, 6, and 9 mm of diameter SAE 1020 and SAE 1045 steels it has been obtained for the fluence tension, the following values of : 
 

The quantity of necessary tests was determined by means of simulation to determine  with an error of 5% and 10% (Chart III). 
 

	m
	5	2500	250
	9.3	780	150
	12.3	350	93
	13.5	260	86
	16.7	210	84
	18.9	185	84
	21	165	80
	21.7	176	77
	23.5	174	75
	24	170	80
	25	160	76
	27	150	74
	50	95	50

Chart III Values of the number of tests to obtain  with an approach of 5%
and 10%, where  is the dispersion of 

In the above-mentioned it has been supposed that N has a normal distribution, so it is verified 
 

(5)

Therefore the probability that N is bigger than a given N it is 
 

(6)

With a change of variables we have 
 

		(7)

If we make  we have 
 

That is to say that if we take , the made error can be of 15%. 
 

Conclusions

To determine if a homogeneous departure of a given material, cement, steel, etc. fulfills a preset condition of quality (that we will suppose when the specific dispersion  of a property has a same or smaller value to one given) we proceed like next. 

First is carried out a series of tests, 20 or 30 and  is determined, then with this value and the formulas (3) m is obtained. With this m, by means of a simulation on (4) an  is obtained for each number N of simulations. Varying N the number of simulations is obtained for which  is smaller or similar to the one obtained previously. 

The number of tests is determined this way to carry out for an effective control. However it is necessary to keep in mind that it is very important to use much common sense to choose the homogeneous sampling. Even with much care very strong apartments take place as the one for the SAE-1045 steel (Chart II) where a value appears m = 5 that changes the whole control scene. 

All this is designed for a probability of error of 15%. 

Is important to mention that the normal control according to the standards in use are merely symbolic values. Only in non–destructive tests the required conditions of the rigorous statistic can be taken.
 

Acknowledgements

The authors would like to express their thanks to the Fondo Nacional de Desarrollo Científico y Tecnológico, FONDECYT, for Grant N° 1961105 project. 
 

Bibliography
 

1. Weibull, W., "To Statical theory of the strength of materials", Engineer Vatescamp Akad., Händl, 151 (1939) 1 - 45. 
2. Kittl, P. and Díaz, G., "Weibull's fractures statistic or probabilistic strength of materials: State of the Art", Head. Mech., 24 (1988) 99 - 207.