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

Authors: Beidou Ding; Naiqiang Xiao; Shuxun Zhang; Yong Wang
Journal: Mathematical Problems in Engineering
 (2017)

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## 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:(1) quantitative relationship between steel Leeb hardness and carbon steel strength was studied on the basis of regression analysis of experimental data; (2) 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; (3) 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.


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## Body

## 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:(1) quantitative relationship between steel Leeb hardness and carbon steel strength was studied on the basis of regression analysis of experimental data; (2) 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; (3) 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 variabley(u) and m regression variables φiu have linear relationship, and then general form of multiple linear regression models is(1)yu=β1φ1u+β2φ2u+⋯+βmφmu.Suppose that the experiment is conducted for n times and n groups of measured values are obtained:(2)u1⋮y1⋮un-1yn-1unyn.Equation (3) can be obtained by substituting (2) into (1)(3)yi=β1φ1ui+β2φ2ui+⋯+βmφmui+εi,i=1,2,…,n.It is expressed in the form of matrix as(4)Y=y1y2⋮yn,X=φ1u1φ2u1…φmu1φ1u2φ2u2…φmu2⋮⋮⋮φ1unφ2un…φmun,β=β1β2⋮βn,ε=ε1ε2⋮εn.In (4), X is called regression variables matrix. Y is response variable matrix and Y~NnX·β,σ2I, while ε is a unobservable random error variable and ε~Nn0,σ2I, in which I is unit matrix. β is vector consisting of regression coefficients β1,β2,…,βm, and u is unknown constant vector.Then (4) can be expressed in the form of matrix as(5)Y=X·β+ε.


### 2.2. Least Squares Estimation of Regression Coefficients

The regression coefficientsβ can be estimated by the least square method. To an estimated values of β, if the least square of the random errors ε of the regression equation is minimum, the fitting effect of the regression equation is the best. Order the estimated value of β and record it as β^; then the appropriate β^ is to minimize the quadratic sum of random errors; namely,(6)min⁡βεT·ε=minβ⁡Y-X·βT·Y-X·β=Y-X·β^TY-X·β^≜Qβ^.It is written into form of components as below(7)Qβ1,β2,…,βm=∑i=1nyi-β1φ1ui-β2φ2ui-⋯-βmφmui2.Then(8)Qβ^1,β^2,…,β^m=minβiQβ1,β2,…,βm.Order∂Q/∂βj=0,j=1,2,…,m; then (9) can be obtained by taking necessary conditions of extreme values according to multivariate function; namely,(9)XT·X·β^=XT·Y.It can be proved that, for any given X, Y, when X is full rank, r(X)=r(XT·X)=m, to get minimum quadratic sum of random errors of (10); namely,(10)Qβ^=minβQβ,and the solution of normal equations system is β^=XT·X-1·XT·Y, 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 y is independent of u; namely, y(u)=β0, and the mean experiment value is y-=1/n∑i=1nyi, and the quadratic sum of total deviations is named as SST; namely,(11)SST=∑i=1nyi-y-2=∑i=1nyi-y^i+y^i-y-2=∑i=1nyi-y^i2+∑i=1ny^i-y-2+2∑i=1nyi-y^iy^i-y-=SSE+SSR,where the quadratic sum of residual is(12)SSE=∑i=1nyi-y^i2=Y-X·β^TY-X·β^=YT·Y-YTX·β^.The quadratic sum of regression is(13)SSR=∑i=1ny^i-y-2.Multiple correlation coefficientsR=SSR/SST are hereby defined and used to evaluate the fitting effect of the regression equation on the sample data. The greater the R 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 ofR 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 fT=n-1 and fR=m-1, respectively. Then the freedom degree of the residual quadratic sum is fE=fT-fR=n-m. Naturally, the mean-square values of the quadratic sum of residual and the quadratic sum of regression are defined as MSR=1/m-1SSR and MSE = 1/n-mSSE, respectively.It can be proved that wheny=β0, yi~N0,σ2; then EMSR=E1/m-1SSR=σ2,EMSE=E1/m-1SSE=σ2. It is indicated that MSE is unbiased estimation of σ2; namely,(14)SSEσ2~χ2n-m,SSRσ2~χ2m-1.In the meantime, SSR and SSE are mutually independent, and then statistical magnitude ofF is constructed as(15)F=MSRMSE=SSR/m-1SSE/n-m~FfR,fE=Fm-1,n-m.After a significance level is taken,Fαm-1,n-m is calculated and compared with Fm-1,n-m. When Fm-1,n-m<Fαm-1,n-m, it is believed that the model is significant; then y=β0 does not hold; namely, η has obvious functional relationship with u. At the moment Fm-1,n-m<Fαm-1,n-m; it is believed that the model is not significant; then yu=β0 holds; namely, y(u) does not have obvious functional relationship with u.


## 2.1. Regression Model

It is assumed that response variabley(u) and m regression variables φiu have linear relationship, and then general form of multiple linear regression models is(1)yu=β1φ1u+β2φ2u+⋯+βmφmu.Suppose that the experiment is conducted for n times and n groups of measured values are obtained:(2)u1⋮y1⋮un-1yn-1unyn.Equation (3) can be obtained by substituting (2) into (1)(3)yi=β1φ1ui+β2φ2ui+⋯+βmφmui+εi,i=1,2,…,n.It is expressed in the form of matrix as(4)Y=y1y2⋮yn,X=φ1u1φ2u1…φmu1φ1u2φ2u2…φmu2⋮⋮⋮φ1unφ2un…φmun,β=β1β2⋮βn,ε=ε1ε2⋮εn.In (4), X is called regression variables matrix. Y is response variable matrix and Y~NnX·β,σ2I, while ε is a unobservable random error variable and ε~Nn0,σ2I, in which I is unit matrix. β is vector consisting of regression coefficients β1,β2,…,βm, and u is unknown constant vector.Then (4) can be expressed in the form of matrix as(5)Y=X·β+ε.


## 2.2. Least Squares Estimation of Regression Coefficients

The regression coefficientsβ can be estimated by the least square method. To an estimated values of β, if the least square of the random errors ε of the regression equation is minimum, the fitting effect of the regression equation is the best. Order the estimated value of β and record it as β^; then the appropriate β^ is to minimize the quadratic sum of random errors; namely,(6)min⁡βεT·ε=minβ⁡Y-X·βT·Y-X·β=Y-X·β^TY-X·β^≜Qβ^.It is written into form of components as below(7)Qβ1,β2,…,βm=∑i=1nyi-β1φ1ui-β2φ2ui-⋯-βmφmui2.Then(8)Qβ^1,β^2,…,β^m=minβiQβ1,β2,…,βm.Order∂Q/∂βj=0,j=1,2,…,m; then (9) can be obtained by taking necessary conditions of extreme values according to multivariate function; namely,(9)XT·X·β^=XT·Y.It can be proved that, for any given X, Y, when X is full rank, r(X)=r(XT·X)=m, to get minimum quadratic sum of random errors of (10); namely,(10)Qβ^=minβQβ,and the solution of normal equations system is β^=XT·X-1·XT·Y, 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 y is independent of u; namely, y(u)=β0, and the mean experiment value is y-=1/n∑i=1nyi, and the quadratic sum of total deviations is named as SST; namely,(11)SST=∑i=1nyi-y-2=∑i=1nyi-y^i+y^i-y-2=∑i=1nyi-y^i2+∑i=1ny^i-y-2+2∑i=1nyi-y^iy^i-y-=SSE+SSR,where the quadratic sum of residual is(12)SSE=∑i=1nyi-y^i2=Y-X·β^TY-X·β^=YT·Y-YTX·β^.The quadratic sum of regression is(13)SSR=∑i=1ny^i-y-2.Multiple correlation coefficientsR=SSR/SST are hereby defined and used to evaluate the fitting effect of the regression equation on the sample data. The greater the R 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 ofR 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 fT=n-1 and fR=m-1, respectively. Then the freedom degree of the residual quadratic sum is fE=fT-fR=n-m. Naturally, the mean-square values of the quadratic sum of residual and the quadratic sum of regression are defined as MSR=1/m-1SSR and MSE = 1/n-mSSE, respectively.It can be proved that wheny=β0, yi~N0,σ2; then EMSR=E1/m-1SSR=σ2,EMSE=E1/m-1SSE=σ2. It is indicated that MSE is unbiased estimation of σ2; namely,(14)SSEσ2~χ2n-m,SSRσ2~χ2m-1.In the meantime, SSR and SSE are mutually independent, and then statistical magnitude ofF is constructed as(15)F=MSRMSE=SSR/m-1SSE/n-m~FfR,fE=Fm-1,n-m.After a significance level is taken,Fαm-1,n-m is calculated and compared with Fm-1,n-m. When Fm-1,n-m<Fαm-1,n-m, it is believed that the model is significant; then y=β0 does not hold; namely, η has obvious functional relationship with u. At the moment Fm-1,n-m<Fαm-1,n-m; it is believed that the model is not significant; then yu=β0 holds; namely, y(u) does not have obvious functional relationship with u.


## 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:(16)HL=1000×VRVA.In the equation, HL is Leeb hardness value;VR is impact speed of the bulb; VA 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.Figure 1
Test specimen.Figure 2
Test scheme.During experiment process of Leeb hardness measuring of carbon steel plates, Leeb hardness tester EQLIOTIP3UNITD was used to measure Leeb hardness values of carbon steel plates (seen in Figure3(a)) and standard test block of D type pressure head of Leeb hardness tester was used to calibrate Leeb hardness values (seen in Figure 3(b)).Figure 3
(a)
Leeb hardness gauge
(b)
Steel hardness test


### 3.3. Experimental Process and Results

It is required that specimens of the steel plates should have enough mass and rigidity in experimental standard and will not go through displacement or springing during impact process. Mass range of specimens participating in the experiment was 2 kg <m < 5 kg; in order to meet experimental requirements, specimens should be coupled so as to reduce energy loss under impact and avoid affecting calibration Leeb hardness values. Combined with field operating conditions, impact device adopted in this paper turned upward (seen in Figure 3(b)). Before specimens were measured, standard test block was tested. Leeb hardness value of the standard test block was 538 HLD, and measuring results were as shown in Table 1 for the convenience of correction of experimentally measured data.Table 1
Testing data of standard test block of Leeb hardness tester.
Calibration value (HLD)
Testing data (HLD)
Mean value (HLD)
538
540
538
538
537
539
538.4It can be known from Table1 that error between measured value and calibration value of standard test block is smaller than 12 HLD which meets national standard requirements. According to experimental standard requirements, each group of specimens should be measured for nine times, distance between indentations of every two points should be greater than 3 mm, and the distance from each point to specimen edge was greater than 5 mm. Two maximum values and two minimum values were excluded among nine measured values which met conditions to solve average value of five measured values, and then this average value was modified, which was the final Leeb hardness value. Table 2 is some partial testing results of the specimens measured by Leeb hardness.Table 2
Results of Leeb hardness test (partial data).
Number
Testing data
Modified value
FK00001
393
392
399
397
394
395.0
FK00001
410
412
412
407
409
410.0
FK00002
410
412
412
412
412
411.6
FK00002
426
425
424
428
426
425.8
FK00004
416
416
411
414
409
413.2
FK00004
443
442
444
442
442
442.6
FK00005
381
384
384
388
384
384.2
FK00005
424
422
423
424
432
425.0
FK00006
434
431
433
434
431
432.6
FK00006
443
444
445
445
445
444.4


### 3.4. Inference of Regression Equation of Carbon Steel Strength Based on Leeb Hardness

In order to infer the correlation between hardness and yield strength of carbon steel, the data of hardness and yield strength were regressed by the least square method and statistical method for the 162 test data. The correlation coefficientr2 and the F value of regression equation fitting were analyzed. At the same time, through residual analysis the optimal regression model was determined and the large residual outliers were removed from the experimental data. Changes of HLD-Re linear regression model statistics were shown in Figure 4. From Figure 4, the correlation coefficient r2 and theF value increased with the optimization number, while the residual value decreased. After five optimization times, the correlation coefficient r2 was increased from 0.6 to 0.697;F value increased from 191.8 to 301.9, while the residual value σ2 decreased from 1070.2 to 620.5. It was shown that this statistic reached the optimal value and the regression model better met the original data.Figure 4
Changes of HLD-Re linear regression model statistics.
(a)
Changes of correlation coefficientr2
(b)
Changes ofF value
(c)
Changes of residual valueσ2Experimental data of all steel plants were put together, linear regression analysis method was used to analyze experimental data, corresponding relation between Leeb hardness and strength of steel was obtained, and summary sheet of data regression analysis was as shown in Table3. The quantitative relationships between Leeb hardness with yield strength Re and ultimate tensile strength Rm of steel plants in Jiangsu were obtained through regression analysis method. It was found that, based on portable Leeb hardness tester, incorporating NDT technology of inferring steel strength with Leeb hardness method into the standard was feasible.Table 3
Regression analysis summary between HLD and yield strengthRe and ultimate tensile strength Rm.
Function expression
Correlation coefficientr2
Significant levelα
F value
Test number
R
e
=
-
477.1893
+
2.0090
H
L
D
0.6974
0.05
301.8938
133
R
m
=
-
405.7281
+
2.164
5
H
L
D
0.6000
0.05
210.7
141


### 3.5. Comparisons Analysis of the Regression Equation

Figure5 was the comparisons analysis of the regression equation among Duan et al.’s [3] paper and specifications in DGJ32/TJ 116-2011 [1]. DGJ32/TJ 116-2011 has an upper and lower limit of tensile strength in the same hardness value, so the converted tensile strength corresponding to the measured Leeb hardness was in a range, and it was not much practical effectiveness. From Figure 5, the values of Leeb hardness test were in the range 366~454 HLD and the most tensile strength test values were within the certain range. The inferred tensile strength according to Duan et al.’s [3] paper was close to the upper limit of converted tensile strength according to the specification in DGJ32/TJ116-2011. The formula based on multiple linear regressions method fitted the experimental data well. Furthermore, the inferred tensile strength was in the middle of the upper and lower bounds and could be used to estimate the steel strength in different engineering practice.Figure 5
Comparisons analysis of the regression equation among Duan et al.’s [3] paper and specifications in DGJ32/TJ 116-2011 [1].


## 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:(16)HL=1000×VRVA.In the equation, HL is Leeb hardness value;VR is impact speed of the bulb; VA 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.Figure 1
Test specimen.Figure 2
Test scheme.During experiment process of Leeb hardness measuring of carbon steel plates, Leeb hardness tester EQLIOTIP3UNITD was used to measure Leeb hardness values of carbon steel plates (seen in Figure3(a)) and standard test block of D type pressure head of Leeb hardness tester was used to calibrate Leeb hardness values (seen in Figure 3(b)).Figure 3
(a)
Leeb hardness gauge
(b)
Steel hardness test


## 3.3. Experimental Process and Results

It is required that specimens of the steel plates should have enough mass and rigidity in experimental standard and will not go through displacement or springing during impact process. Mass range of specimens participating in the experiment was 2 kg <m < 5 kg; in order to meet experimental requirements, specimens should be coupled so as to reduce energy loss under impact and avoid affecting calibration Leeb hardness values. Combined with field operating conditions, impact device adopted in this paper turned upward (seen in Figure 3(b)). Before specimens were measured, standard test block was tested. Leeb hardness value of the standard test block was 538 HLD, and measuring results were as shown in Table 1 for the convenience of correction of experimentally measured data.Table 1
Testing data of standard test block of Leeb hardness tester.
Calibration value (HLD)
Testing data (HLD)
Mean value (HLD)
538
540
538
538
537
539
538.4It can be known from Table1 that error between measured value and calibration value of standard test block is smaller than 12 HLD which meets national standard requirements. According to experimental standard requirements, each group of specimens should be measured for nine times, distance between indentations of every two points should be greater than 3 mm, and the distance from each point to specimen edge was greater than 5 mm. Two maximum values and two minimum values were excluded among nine measured values which met conditions to solve average value of five measured values, and then this average value was modified, which was the final Leeb hardness value. Table 2 is some partial testing results of the specimens measured by Leeb hardness.Table 2
Results of Leeb hardness test (partial data).
Number
Testing data
Modified value
FK00001
393
392
399
397
394
395.0
FK00001
410
412
412
407
409
410.0
FK00002
410
412
412
412
412
411.6
FK00002
426
425
424
428
426
425.8
FK00004
416
416
411
414
409
413.2
FK00004
443
442
444
442
442
442.6
FK00005
381
384
384
388
384
384.2
FK00005
424
422
423
424
432
425.0
FK00006
434
431
433
434
431
432.6
FK00006
443
444
445
445
445
444.4


## 3.4. Inference of Regression Equation of Carbon Steel Strength Based on Leeb Hardness

In order to infer the correlation between hardness and yield strength of carbon steel, the data of hardness and yield strength were regressed by the least square method and statistical method for the 162 test data. The correlation coefficientr2 and the F value of regression equation fitting were analyzed. At the same time, through residual analysis the optimal regression model was determined and the large residual outliers were removed from the experimental data. Changes of HLD-Re linear regression model statistics were shown in Figure 4. From Figure 4, the correlation coefficient r2 and theF value increased with the optimization number, while the residual value decreased. After five optimization times, the correlation coefficient r2 was increased from 0.6 to 0.697;F value increased from 191.8 to 301.9, while the residual value σ2 decreased from 1070.2 to 620.5. It was shown that this statistic reached the optimal value and the regression model better met the original data.Figure 4
Changes of HLD-Re linear regression model statistics.
(a)
Changes of correlation coefficientr2
(b)
Changes ofF value
(c)
Changes of residual valueσ2Experimental data of all steel plants were put together, linear regression analysis method was used to analyze experimental data, corresponding relation between Leeb hardness and strength of steel was obtained, and summary sheet of data regression analysis was as shown in Table3. The quantitative relationships between Leeb hardness with yield strength Re and ultimate tensile strength Rm of steel plants in Jiangsu were obtained through regression analysis method. It was found that, based on portable Leeb hardness tester, incorporating NDT technology of inferring steel strength with Leeb hardness method into the standard was feasible.Table 3
Regression analysis summary between HLD and yield strengthRe and ultimate tensile strength Rm.
Function expression
Correlation coefficientr2
Significant levelα
F value
Test number
R
e
=
-
477.1893
+
2.0090
H
L
D
0.6974
0.05
301.8938
133
R
m
=
-
405.7281
+
2.164
5
H
L
D
0.6000
0.05
210.7
141


## 3.5. Comparisons Analysis of the Regression Equation

Figure5 was the comparisons analysis of the regression equation among Duan et al.’s [3] paper and specifications in DGJ32/TJ 116-2011 [1]. DGJ32/TJ 116-2011 has an upper and lower limit of tensile strength in the same hardness value, so the converted tensile strength corresponding to the measured Leeb hardness was in a range, and it was not much practical effectiveness. From Figure 5, the values of Leeb hardness test were in the range 366~454 HLD and the most tensile strength test values were within the certain range. The inferred tensile strength according to Duan et al.’s [3] paper was close to the upper limit of converted tensile strength according to the specification in DGJ32/TJ116-2011. The formula based on multiple linear regressions method fitted the experimental data well. Furthermore, the inferred tensile strength was in the middle of the upper and lower bounds and could be used to estimate the steel strength in different engineering practice.Figure 5
Comparisons analysis of the regression equation among Duan et al.’s [3] paper and specifications in DGJ32/TJ 116-2011 [1].


## 4. Experimental Study of Inference of Carbon Steel Strength Based on Micro Model

Besides carbon composition, carbon steel also contains compositions like Mn, Si, S, P, O, and N. The existence of minor compositions will affect quality and performance of carbon steel greatly. Through test of chemical compositions in the steel specimens, their influences on steel performance were analyzed.


### 4.1. Test of Spectrograph Determination of Chemical Composition

Spark source atomic emission spectrometer takes spark discharge as excitation light source to sputter and excite the sample. Then, the qualitative and quantitative analysis of atoms in the tested sample can be conducted by detecting emission of composition spectrum happening in this process. As five major conventional compositions, C, Si, Mn, S, and P, have great influences on mechanical properties of steel, this experiment concentrated on detecting contents of C, Si, Mn, S, and P compositions. The experimental samples were 162 groups of carbon steel plates mentioned in previous section. For the convenience of experimental operation, steel plates used in this experiment were small steel plates cut from 162 groups of large steel plates.In research process of this experiment, operations were as below:(1) 162 groups of steel plates (shown in Figure 6(a)) were processed, and for the convenience of experimental operations, steel plates were processed into small steel plates being about 10 cm × 5 cm. (2) Surface treatment of small steel plates was conducted: surface dust and external pollutants and greasy dirt were removed to avoid affecting measurement of chemical composition. (3) Bruker direct-reading spectrometer (shown in Figure 6(b)) was calibrated, excitation light source was used to excite specimen surface, and contents of various compositions were obtained through quantitative spectral measurement. (4) Measuring channels of chemical compositions were opened, contents of five major conventional compositions of 162 groups of steel plates were measured, and partial measuring results were as seen in Table 4.Table 4
Percentages of regular compositions of test specimens (partial data).
Number
C%
Si%
Mn%
P%
S%
FK00001
0.136
0.125
0.337
0.026
0.023
FK00001
0.117
0.22
0.287
0.018
0.016
FK00002
0.146
0.331
1.474
0.02
0.0033
FK00002
0.118
0.338
1.412
0.025
0.024
FK00004
0.124
0.282
1.254
0.019
0.0093
FK00004
0.145
0.384
1.299
0.015
0.013
FK00005
0.112
0.159
0.423
0.021
0.018
FK00005
0.13
0.348
1.325
0.016
0.014Figure 6
(a)
Sample for determination of chemical compositions
(b)
Bruker direct-reading spectrometer


### 4.2. Inferring Strength of Steel Based on Stepwise Regression

Stepwise function in statistical toolkit was used to build multiple linear regression models of chemical composition and strength, and analysis results were as seen in Table5. The greater the regression sums of squares, the better the results of regression equation. It can be seen from Table 5 that Mn and S have great influences on yield strength Re of carbon steel, while for its ultimate tensile strength Rm, besides Mn, S and P also have great influences. Hence, the above compositions were imported into regression equation.Table 5
Partial regression squares on the relationship between the determination of chemical composition and strength.
C
Si
Mn
S
P
Yield strength(Re)
0.157
0.024
1.428
0.561
0.115
Tension strength(Rem)
0.00129
0.088
0.719
0.037
0.154After variablesMn, S, and/or P, namely, compositions were introduced, coefficient of determination and root-mean-square (RMS) error decreased. Meanwhile the corresponding regression model of yield strength Re and ultimate tensile strength Rm of carbon steel could be defined as(17)Re=334.914+59.1846Mn-1492.38S,Rm=421.399+36.1576Mn-1492.38S+8.70659P.


## 4.1. Test of Spectrograph Determination of Chemical Composition

Spark source atomic emission spectrometer takes spark discharge as excitation light source to sputter and excite the sample. Then, the qualitative and quantitative analysis of atoms in the tested sample can be conducted by detecting emission of composition spectrum happening in this process. As five major conventional compositions, C, Si, Mn, S, and P, have great influences on mechanical properties of steel, this experiment concentrated on detecting contents of C, Si, Mn, S, and P compositions. The experimental samples were 162 groups of carbon steel plates mentioned in previous section. For the convenience of experimental operation, steel plates used in this experiment were small steel plates cut from 162 groups of large steel plates.In research process of this experiment, operations were as below:(1) 162 groups of steel plates (shown in Figure 6(a)) were processed, and for the convenience of experimental operations, steel plates were processed into small steel plates being about 10 cm × 5 cm. (2) Surface treatment of small steel plates was conducted: surface dust and external pollutants and greasy dirt were removed to avoid affecting measurement of chemical composition. (3) Bruker direct-reading spectrometer (shown in Figure 6(b)) was calibrated, excitation light source was used to excite specimen surface, and contents of various compositions were obtained through quantitative spectral measurement. (4) Measuring channels of chemical compositions were opened, contents of five major conventional compositions of 162 groups of steel plates were measured, and partial measuring results were as seen in Table 4.Table 4
Percentages of regular compositions of test specimens (partial data).
Number
C%
Si%
Mn%
P%
S%
FK00001
0.136
0.125
0.337
0.026
0.023
FK00001
0.117
0.22
0.287
0.018
0.016
FK00002
0.146
0.331
1.474
0.02
0.0033
FK00002
0.118
0.338
1.412
0.025
0.024
FK00004
0.124
0.282
1.254
0.019
0.0093
FK00004
0.145
0.384
1.299
0.015
0.013
FK00005
0.112
0.159
0.423
0.021
0.018
FK00005
0.13
0.348
1.325
0.016
0.014Figure 6
(a)
Sample for determination of chemical compositions
(b)
Bruker direct-reading spectrometer


## 4.2. Inferring Strength of Steel Based on Stepwise Regression

Stepwise function in statistical toolkit was used to build multiple linear regression models of chemical composition and strength, and analysis results were as seen in Table5. The greater the regression sums of squares, the better the results of regression equation. It can be seen from Table 5 that Mn and S have great influences on yield strength Re of carbon steel, while for its ultimate tensile strength Rm, besides Mn, S and P also have great influences. Hence, the above compositions were imported into regression equation.Table 5
Partial regression squares on the relationship between the determination of chemical composition and strength.
C
Si
Mn
S
P
Yield strength(Re)
0.157
0.024
1.428
0.561
0.115
Tension strength(Rem)
0.00129
0.088
0.719
0.037
0.154After variablesMn, S, and/or P, namely, compositions were introduced, coefficient of determination and root-mean-square (RMS) error decreased. Meanwhile the corresponding regression model of yield strength Re and ultimate tensile strength Rm of carbon steel could be defined as(17)Re=334.914+59.1846Mn-1492.38S,Rm=421.399+36.1576Mn-1492.38S+8.70659P.


## 5. Experimental Studies of Microstructures of Carbon Steel Plates

### 5.1. Experimental Study of Microstructures

Metallographic microstructure test of specimens was conducted, their internal organization structures were analyzed, grain size d was calculated, and influence of grain size on material strength was analyzed. The main experimental processes were as below:(1) cut out specimens: according to experimental requirements, grinding wheel slicer was used for cutting and cooling measures should be taken during cutting process to reduce structural change of specimens caused by heating (as shown in Figure 7(a)). (2) Polish the cut specimens: this procedure was conducted step by step from rough grinding to fine grinding, and damage of surface layer should be reduced during grinding process of metallographic specimens as much as possible. (3) Remove deformed layer left in grinding procedure and make the deformed layer generated by polishing not to affect observation of microstructure. (4) Observe chemical corrosion of specimens. During the operation, the polished specimens were corroded in 4% nitric acid alcohol solution, and angle of microscopic plane was different from that of original grinding plane after corrosion. Under irradiation of vertical light rays, light rays entering objective lens through reflection were different, and grains with different shades could be seen (as shown in Figure 8). (5) DMI5000 metalloscope was used to observe the prepared metallographic samples and take photos through computer and image acquisition software (seen in Figure 7(b)).Figure 7
(a) Test specimen; (b) DMI5000 Leica metallographic microscope.
(a)
(b)Figure 8
Metallurgical structure of test specimen (×100).


### 5.2. Experimental Analysis

According to dislocation piling up theory of single-crystal and polycrystalline materials, relationship between material tension strength and grain size was concluded, namely, Hall-Petch formula [14]:(18)σ=σ0+kd-1/2.Experimental data of ten groups of specimens extracted in this paper were as shown in Table6, mathematical regression analysis was implemented, and it is obtained that yield strength and tensile strength satisfied positively related mathematical linear relationships with grain size d-1/2. This was because after metal grains were refined, ratio of grain boundaries increased, and grain boundary’s effect on hindering dislocation piling up was enhanced, and as dislocation piled up nearby the grain boundary, carbon steel strength increased as grains were refined. Actually as grain size decreased, grain boundaries increased, and then more energy would be consumed when the specimen was damaged, and thus it could be seen that carbon steel strength had linear relationship with grain size d-1/2. It was concluded that the relationship between the tensile strength (yield strength) and the grain size of low-carbon steel complied the Hall-Petch formula with different coefficients, as shown in Figure 9.Table 6
Grain size andRe of steel plate.
Number
Yield strength (Mpa)
Tensile strength (Mpa)
Grain size (mm)
d
-
1
/
2
1
303
445
0.201632
2.227
2
309
450
0.128744
2.787
3
314
454
0.0945
3.253
4
321
460
0.065578
3.905
5
328
467
0.048155
4.557
6
331
469
0.042741
4.837
7
345
482
0.026508
6.142
8
353
489
0.021083
6.887
9
358
493
0.018496
7.353
10
365
499
0.015621
8.001Figure 9
Relationship between grain size and steel strength.


## 5.1. Experimental Study of Microstructures

Metallographic microstructure test of specimens was conducted, their internal organization structures were analyzed, grain size d was calculated, and influence of grain size on material strength was analyzed. The main experimental processes were as below:(1) cut out specimens: according to experimental requirements, grinding wheel slicer was used for cutting and cooling measures should be taken during cutting process to reduce structural change of specimens caused by heating (as shown in Figure 7(a)). (2) Polish the cut specimens: this procedure was conducted step by step from rough grinding to fine grinding, and damage of surface layer should be reduced during grinding process of metallographic specimens as much as possible. (3) Remove deformed layer left in grinding procedure and make the deformed layer generated by polishing not to affect observation of microstructure. (4) Observe chemical corrosion of specimens. During the operation, the polished specimens were corroded in 4% nitric acid alcohol solution, and angle of microscopic plane was different from that of original grinding plane after corrosion. Under irradiation of vertical light rays, light rays entering objective lens through reflection were different, and grains with different shades could be seen (as shown in Figure 8). (5) DMI5000 metalloscope was used to observe the prepared metallographic samples and take photos through computer and image acquisition software (seen in Figure 7(b)).Figure 7
(a) Test specimen; (b) DMI5000 Leica metallographic microscope.
(a)
(b)Figure 8
Metallurgical structure of test specimen (×100).


## 5.2. Experimental Analysis

According to dislocation piling up theory of single-crystal and polycrystalline materials, relationship between material tension strength and grain size was concluded, namely, Hall-Petch formula [14]:(18)σ=σ0+kd-1/2.Experimental data of ten groups of specimens extracted in this paper were as shown in Table6, mathematical regression analysis was implemented, and it is obtained that yield strength and tensile strength satisfied positively related mathematical linear relationships with grain size d-1/2. This was because after metal grains were refined, ratio of grain boundaries increased, and grain boundary’s effect on hindering dislocation piling up was enhanced, and as dislocation piled up nearby the grain boundary, carbon steel strength increased as grains were refined. Actually as grain size decreased, grain boundaries increased, and then more energy would be consumed when the specimen was damaged, and thus it could be seen that carbon steel strength had linear relationship with grain size d-1/2. It was concluded that the relationship between the tensile strength (yield strength) and the grain size of low-carbon steel complied the Hall-Petch formula with different coefficients, as shown in Figure 9.Table 6
Grain size andRe of steel plate.
Number
Yield strength (Mpa)
Tensile strength (Mpa)
Grain size (mm)
d
-
1
/
2
1
303
445
0.201632
2.227
2
309
450
0.128744
2.787
3
314
454
0.0945
3.253
4
321
460
0.065578
3.905
5
328
467
0.048155
4.557
6
331
469
0.042741
4.837
7
345
482
0.026508
6.142
8
353
489
0.021083
6.887
9
358
493
0.018496
7.353
10
365
499
0.015621
8.001Figure 9
Relationship between grain size and steel strength.


## 6. Regression Analysis of Experimental Data in Inferring Carbon Steel Strength Based on Macro/Micro Model

With the combination of previous contents and comprehensive consideration, the macro/micro factors, which are steel hardness, five major conventional chemical compositions, and grain size, were used to infer strength formula of carbon steel with a comprehensive MLR analysis. During comprehensive analysis process, SPSS mathematical statistics software based on least square method was used to conduct MLR analysis. The residual analysis showed that residual errors complied with normal distribution, and it was indicated that the model met assumptions of least square method. On the condition that the macro-micro model met assumptions of normal distribution, coefficients of yield strength and tensile strength models of steel were estimated; estimated values of coefficient matrixes were, respectively, seen in Tables7 and 8.Table 7
Coefficient estimates of yield strength model.
Yield strength model
Coefficientβ estimates
95% confidence interval ofβ
Lower limit
Upper limit
Constant
−314.302
−443.432
−185.173
HLD
1.526
1.212
1.840
d
-
1
/
2
3.318
0.720
5.917
C percentage (%)
−174.066
−334.770
−13.363
Si percentage (%)
−2.324
−88.397
83.749
Mn percentage (%)
37.646
19.459
55.832
P percentage (%)
609.908
−244.132
1463.948
S percentage (%)
−217.509
−962.534
527.516Table 8
Coefficient estimates of tensile strength model.
Tensile strength model
Coefficientβ estimates
95% confidence interval ofβ
Lower limit
Upper limit
Constant
−227.221
−349.639
−104.803
HLD
1.469
1.171
1.767
d
-
1
/
2
1.571
−0.893
4.035
C percentage (%)
171.233
18.882
323.583
Si percentage (%)
62.335
−19.265
143.934
Mn percentage (%)
54.287
37.046
71.529
P percentage (%)
1296.782
487.133
2106.431
S percentage (%)
−213.827
−920.127
492.474According to estimated coefficient values of independent values and constant terms in Table7, yield strength was used to express mathematical regression models of factors as (19)Re=-314.302-174.066C-2.324Si+37.646Mn+609.908P-217.509S+1.526HLD+3.318d-1/2.According to estimated coefficient values of independent variables and constant terms in Table8, the mathematical regression models of yield strength Rm were expressed as (20)Rm=-227.221+171.233C+62.335Si+54.287Mn+1296.782P-213.827S+1.469HLD+1.571d-1/2.The regression analysis of yield strengthRe based on macro/micro model was also shown in Table 9, and the regression analysis of tensile strength Rm based on macro/micro model was shown in Table 10.Table 9
Regression analysis of yield strengthRe based on macro/micro model.
Regression formula (5)
R
e
=
-
314.302
-
174.066
C
-
2.324
S
i
+
37.646
M
n
+
609.908
P
-
217.509
S
+
1.526
H
L
D
+
3.318
d
-
1
/
2
Regression model statistics
r
2
=
0.686
F
=
45.952
RMSE = 28.059Table 10
Regression analysis of tensile strengthRm based on macro/micro model.
Regression formula (6)
R
m
=
-
227.221
+
171.233
C
+
62.335
S
i
+
54.287
M
n
+
1296.782
P
-
213.827
S
+
1.469
[
H
L
D
]
+
1.571
d
-
1
/
2
Regression model statistics
r
2
=
0.763
F
=
67.778
RMSE = 26.601It can be found that the value of fitting degreer2 and statistical magnitude F of yield strength were 0.686 and 45.952, respectively, while those of tensile strength were 0.763 and 67.778. Thus it can be known that when the micro factors such as conventional chemical compositions and grain size were incorporated into the mathematical model, the updated macro-micro model was improved.


## 7. Conclusions

Quantitative relationships between mechanical properties of steel with hardness and chemical composition were mainly studied in this paper, and a new NDT technology of steel strength inferring was explored. During research process, steel plates which were commonly used in steel structure were mainly used in this paper and they came from 86 steel plants of Jiangsu Province, steel plate thicknesses included 6 mm, 7 mm, 8 mm, 10 mm, 12 mm, 14 mm, 18 mm, 20 mm, and 30 mm, hardness value test and detection of chemical composition as well as metallographic test of these steel plates were conducted, and the following conclusions were drawn:(1)
Through analysis of experimental data from different perspectives, it is known that Leeb strength had quantitative relationships with yield strength and tensile strength of steel, and then the fitting formulas between Leeb hardness with yield strength and tensile strength were obtained (see Table11).(2)
Influences of chemical compositions on mechanical properties of steel were analyzed on the basis of stepwise regression analysis method, and it is obtained that Mn, S, or P compositions had great influences on yield strength and tensile strength. Stepwise regression analysis was conducted, stepwise test was made, and then fitting formulas (3) and (4) about quantitative relationships between conventional compositions and mechanical properties of steel were obtained.(21)Re=334.914+59.1846Mn-1492.38S,Rp=421.399+36.1576Mn-1492.38S+8.70659P.(3)
Metallographic microstructures of steel had different morphologies and the relationship between the tensile strength (yield strength) and the grain size of low-carbon steel complied with the Hall-Petch formula with different coefficients.(4)
Based on the built conversion methods of steel hardness, chemical compositions, and strength, metallographic theory was introduced, mathematical macro/micro models between strength with hardness, grain size, and chemical compositions were built, and corresponding fitting formulas were obtained (see Table12).Table 11
Regression formulas (1) and (2)
Regression model statistics
r
2
F
R
e
=
-
477.1893
+
2.0090
H
L
D
0.6974
301.8938
R
m
=
-
405.7281
H
L
D
+
2.1645
0.6000
210.7000Table 12
Regression formula (5)
R
e
=
-
314.302
-
174.066
C
-
2.324
S
i
+
37.646
M
n
+
609.908
P
-
217.509
S
+
1.526
H
L
D
+
3.318
d
-
1
/
2
Regression model statistics
r2=0.686
F=45.952
RMSE = 28.059
Regression formula (6)
R
m
=
-
227.221
+
171.233
C
+
62.335
S
i
+
54.287
M
n
+
1296.782
P
-
213.827
S
+
1.469
[
H
L
D
]
+
1.571
d
-
1
/
2
Regression model statistics
r2=0.763
F=67.778
RMSE = 26.601



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*Source: 4393279-2017-07-16.xml*