Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4393279, 11 pages

https://doi.org/10.1155/2017/4393279

## Research on NDT Technology in Inference of Steel Member Strength Based on Macro/Micro Model

^{1}State Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China^{2}Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering, China University of Mining and Technology, Xuzhou 221116, China^{3}National Space Frame and Steel Structures Quality Supervision and Inspection Center, Xuzhou 221006, China

Correspondence should be addressed to Beidou Ding

Received 10 February 2017; Revised 5 June 2017; Accepted 11 June 2017; Published 16 July 2017

Academic Editor: Marek Lefik

Copyright © 2017 Beidou Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In consideration of correlations among hardness, chemical composition, grain size, and strength of carbon steel, a new nondestructive testing technology (NDT) of inferring the carbon steel strength was explored. First, the hardness test, chemical composition analysis, and metallographic analysis of 162 low-carbon steel samples were conducted. Second, the following works were carried out: quantitative relationship between steel Leeb hardness and carbon steel strength was studied on the basis of regression analysis of experimental data; influences of chemical composition and grain size on tension properties of carbon steel were analyzed on the basis of stepwise regression analysis, and quantitative relationship between conventional compositions and grain size with steel strength was obtained; according to the macro and/or micro factors such as hardness, chemical compositions, and grain size of carbon steel, the fitting formula of steel strength was established based on MLR (multiple linear regressions) method. The above relationships and fitting formula based on MLR method could be used to estimate the steel strength with no damage to the structure in engineering practice.

#### 1. Introduction

The hardness and strength are two macroscopic performance indexes of steel material. The tensile strength is usually the main parameter evaluating the bearing capacity of structure or components. As detection of steel hardness is simple and rapid and will not damage structure or component, establishment of mathematical model with “macro” index such as steel hardness and strength is convenient for evaluation of material strength of steel structure in service. In addition, internal crystal structure and chemical composition are “micro” constitution of carbon steel. In fact, besides carbon composition, carbon steel also contains a small quantity of Mn, Si, S, P, O, N, and other compositions.

Metal harness has correlation with strength as expressed in Jiangsu engineering construction standard DGJ32/TJ 116-2011 [1] and international ISO standard ISO/TR10108 [2]. In recent years, many researchers [3–6] have used mathematical statistical analysis method to establish correlations between steel tensile strength with Brinell hardness and Rockwell hardness, and regression analysis method is used to solve their expressions. Over the past decades, steel mills have obtained many achievements in researches on quantitative relationship between mechanical properties and chemical composition of rebar. Furthermore, some regression models were formed according to their own production practices and used to guide design of chemical composition and formulation of internal control standard [7, 8]. A neural network with feed-forward topology and back propagation algorithm was used to predict the effects of chemical composition and tensile test parameters on hardness of heat affected zone (HAZ) in X70 pipeline steels [9]. The existences of minor compositions also affect the carbon steel quality and performance greatly. Influences of chemical composition of carbon steel and internal lattice size “micro” model on strength needed to be studied and integrated into the regression model. However, there is a lack of researches on significance of minor chemical compositions to the mechanical properties of the steel. With development of technologies like mathematical method, physical metallurgy, and rolling, relationship between microstructure of hot rolling product and mechanical properties as well as mathematical model evolving from microstructure has obtained rapid development. Researches have carried out a large quantity of work in studying quantitative relationship between grain size and mechanical properties of steel, where the most important is Hall-Petch formula with the most extensive application [10–12]. All in all, at present, many model researches consider influences of single factors among carbon steel hardness, microstructure, and chemical composition on carbon steel strength, respectively. However, there are few model researches on inference of carbon steel strength by combining micro factors, chemical composition and grain size of carbon steel, and macro factor, hardness [13–16].

The main objective of the present paper is to develop MLR (multiple linear regressions) method for inferring the carbon steel strength with its hardness, chemical compositions, and grain size. In order to acquire the correlations among steel hardness, chemical composition, and strength, hardness test and metallographic analysis of carbon steel were carried out in this paper. MLR method was used to propose a macro/micro mathematical model based on experiment data of hardness, grain size, and chemical compositions of carbon steel. The models could be used to infer carbon steel strength to verify its effectiveness. This paper is arranged as follows. First, regression analysis method is introduced which consists of the establishment of the regression model, least squares estimation of regression coefficients, and the significance test of linear regression model. Second, the hardness test, chemical composition analysis, and metallographic analysis of 162 low-carbon steel samples were conducted. The following works were completed: quantitative relationship between steel Leeb hardness and carbon steel strength was studied on the basis of regression analysis of experimental data; influences of chemical composition and grain size on mechanical properties of carbon steel were analyzed on the basis of stepwise regression analysis, and quantitative relationship between conventional compositions and grain size with steel strength was obtained; according to the macro and/or micro factors such as hardness, chemical compositions, and grain size of carbon steel, the fitting formula of steel strength was established based on MLR (multiple linear regressions) method. The above relationships and fitting formula could be used to estimate the carbon steel strength in different NDT engineering practice.

#### 2. Regression Analysis Methods

Regression analysis method is a commonly used method in statistical analysis and mathematical modeling of relationships between random variables. As most random variables are randomly obtained through the experiment, the validity and significance of random variables in this model need further verification with statistical experiment. For regression analysis, it is necessary to establish a mathematical model, namely, common functional relationship. In the function model, its independent variables are regression variables and its dependent variables are called response variables. If there is only one variable in the model, it is then called single regression model, and if there are multiple variables, then it is called multiple regression model.

##### 2.1. Regression Model

It is assumed that response variable and regression variables have linear relationship, and then general form of multiple linear regression models isSuppose that the experiment is conducted for times and groups of measured values are obtained:Equation (3) can be obtained by substituting (2) into (1)It is expressed in the form of matrix as

In (4), is called regression variables matrix. is response variable matrix and , while is a unobservable random error variable and , in which is unit matrix. is vector consisting of regression coefficients , and is unknown constant vector.

Then (4) can be expressed in the form of matrix as

##### 2.2. Least Squares Estimation of Regression Coefficients

The regression coefficients can be estimated by the least square method. To an estimated values of , if the least square of the random errors of the regression equation is minimum, the fitting effect of the regression equation is the best. Order the estimated value of and record it as ; then the appropriate is to minimize the quadratic sum of random errors; namely,

It is written into form of components as below

Then

Order ; then (9) can be obtained by taking necessary conditions of extreme values according to multivariate function; namely,It can be proved that, for any given , , when is full rank, , to get minimum quadratic sum of random errors of (10); namely,and the solution of normal equations system is , namely, estimated value of regression coefficients.

##### 2.3. Significance Test of Linear Regression Model

This process mainly checks whether the model certainly has close relationship with regression variables, namely, whether it accords with (1). It is assumed that is independent of ; namely, , and the mean experiment value is , and the quadratic sum of total deviations is named as SST; namely,where the quadratic sum of residual is

The quadratic sum of regression is

Multiple correlation coefficients are hereby defined and used to evaluate the fitting effect of the regression equation on the sample data. The greater the value is, the closer the relationship between regression variables and response variables is and vice versa.

Hence, it is necessary to build statistical magnitude to determine value of in this paper. Firstly, the freedom degree should be confirmed. The freedom degree of the total deviations quadratic sum and the regression quadratic sum are and , respectively. Then the freedom degree of the residual quadratic sum is . Naturally, the mean-square values of the quadratic sum of residual and the quadratic sum of regression are defined as and MSE = SSE, respectively.

It can be proved that when , ; then It is indicated that MSE is unbiased estimation of ; namely,

In the meantime, SSR and SSE are mutually independent, and then statistical magnitude of is constructed as

After a significance level is taken, is calculated and compared with . When , it is believed that the model is significant; then does not hold; namely, has obvious functional relationship with . At the moment ; it is believed that the model is not significant; then holds; namely, does not have obvious functional relationship with .

#### 3. Experimental Study of Inference of Carbon Steel Strength Based on Leeb Hardness

To obtain the correlation between hardness and strength, the experiments of steel hardness and tensile strength were conducted, a comparative analysis of experimental data was implemented, and corresponding mathematical model was obtained on the basis of mathematical regression method.

##### 3.1. Experimental Principles

Leeb hardness tester is manufactured according to principles of elastic impact and it is used to measure hardness of steel materials. When the impact device of the hardness tester is used to release the impact body made of tungsten carbide or diamond bulb from a fixed position, the specimen surface of the sample is impacted. The impact speed and rebound speed of the bulb on specimen surface are measured, and its Leeb hardness value is expressed with ratio of rebound speed to impact speed of the bulb. And the calculation formula of Leeb hardness is as below:

In the equation, HL is Leeb hardness value; is impact speed of the bulb; is rebound speed of the bulb. Leeb hardness tester can be configured with six kinds of impact heads, that is, D, DC, D + 15, G, E, and C type pressure heads, respectively. Except that E type plunger chip is made of diamond, other forms are made of tungsten carbide.

##### 3.2. Experimental Scheme

Steel plates used in the experiment came from 162 groups of steel plates from different steel plants in Jiangsu Province of China, including 82 Q235 steel plates and 80 Q345 steel plates, totally 162 experimental steel plates. Their thicknesses were, respectively, 6 mm, 8 mm, 10 mm, 12 mm, 14 mm, 30 mm, and so forth, their masses satisfied 2 kg <* m* < 5 kg, and material sampling was as seen in Figure 1. The experimental aim was to find empirical formula between Leeb hardness and tensile strength of structure steel. Based on the experimental aim, D type impact head was used for measurement in this experiment, and experimental scheme was as seen in Figure 2.