Measurement of Residual Stresses

Abstract

The hole-drilling method is the preferred method to determine residual stresses in components made of coarse-grained aluminium alloys. In the literature there are very few data available on the uncertainty of measurement for the hole-drilling method. Especially about uncertainty of the mentioned case you cannot find any data. In order to be able to assess the measured values with regard to their significance, it is essential to know the uncertainty; especially if you want to use those readings for calculations. By using the hole-drilling method you normally have only one measurement result for the principal strains respectively for the principal stresses as well as for the direction of the principal stresses. With the help of a special strain gage rosette with eight single grids and several measured objects there are 240 results for the residual stress in every hole-drilling depth. By means of an analysis of variance the values were broken down into the following distribution parts: two distributions due to the variance of the measured objects and to the variance of the measurement points and one distribution which is a result for the measurement uncertainty. This uncertainty depends on the depth of the drilled hole. At the surface it is about 20 % to 30 % and from a depth of about 0.4 mm it is nearly constant at about 13 %.

Appendix: Performing of an Analysis of Variance

y _{i j ν} are single measurment values of strains for each cylinder head (i=1..5) and for each measurement point (j=1..3). To estimate an uncertainty you have to perform the following steps. The result you get, is one point on the “V”-curve in Fig. 6. To get the whole curve, you have to do the same calculation for every depth and every single grid.

Table 1 Measurement values and calculated sums for hole depth 0 and one single grid

Calculation of Sums

$$\begin{array}{@{}rcl@{}} Y_{ij.} & = & \sum\limits_{v=1}^{n} y_{ij\nu} = \sum\limits_{v=1}^{5} y_{ij\nu} \end{array} $$

(4)

$$\begin{array}{@{}rcl@{}} & = & -0.024 + 0.203 +0.199 + 0.213 + 0.188 = 0.778 \\ Yi.. & = & \sum\limits_{j=1}^{s} Yij. = \sum\limits_{j=1}^{3} Yij. \end{array} $$

(5)

$$\begin{array}{@{}rcl@{}} & = & 0.778 -1.291 +1.599 = 1.086 \\ Y... & = & \sum\limits_{i=1}^{r} Yi.. = \sum\limits_{i=1}^{5} Yi..\\ & = & 1.086 + 1.959 + 2.083 - 0.370 + 0.770 = 5.529 \end{array} $$

(6)

Sums of Squares

$$\begin{array}{@{}rcl@{}} N & = & r \cdot s \cdot n = 5 \cdot 3 \cdot 5 = 75 \end{array} $$

(7)

Between cylinder heads:

$$\begin{array}{@{}rcl@{}} SQ_{A} & = & \sum\limits_{i=1}^{r} \frac{Yi..^{2}}{s \cdot n} - \frac{Y...^{2}}{N} \\ &=& \frac{1.180 + 3.839 + 4.339 + 0.137 + 0.594}{3 \cdot 5} \\ &&- \frac{5.529^{2}}{3 \cdot 5 \cdot 5}\\ & = & \frac{10.089}{15} - \frac{30.569}{75} = 0.6726 - 0.4075 = 0.265 \end{array} $$

(8)

Between measurement points:

$$\begin{array}{@{}rcl@{}} SQ_{B} & = & \sum\limits_{i=1}^{r} \sum\limits_{j=1}^{s} \frac{Yij.^{2}}{n} - \sum\limits_{i=1}^{r} \frac{Yi..^{2}}{s \cdot n} = \frac{65.525}{5} - \frac{10.088}{15} \\ &=& 12.433 \end{array} $$

(9)

Error sum of squares:

$$\begin{array}{@{}rcl@{}} SQ_{R} & = & \sum\limits_{i=1}^{r} \sum\limits_{j=1}^{s} \sum\limits_{v=1}^{n} y_{ijv}^{2} - \sum\limits_{i=1}^{r} \sum\limits_{j=1}^{s} \frac{Yij.^{2}}{n}\\ & = & 13.920 - \frac{65.525}{5} = 0.815 \end{array} $$

(10)

Total sum of squares:

$$\begin{array}{@{}rcl@{}} SQ_{Tot} & = & SQ_{A} + SQ_{B} + SQ_{R} \end{array} $$

(11)

Degrees of Freedom

$$\begin{array}{@{}rcl@{}} f_{A} & = & r - 1 = 5 - 1 = 4 \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} f_{B} & = & r (s - 1) = 5(3 - 1) = 5 \cdot 2 = 10 \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} f_{R} & = & N - r \cdot s = 75 - 15 = 60 \end{array} $$

(14)

Mean Sums of Squares

$$\begin{array}{@{}rcl@{}} MQ_{A} & = & \frac{SQ_{A}}{f_{A}} = \frac{0.265}{4} = 0.06625 \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} MQ_{B} & = & \frac{SQ_{B}}{f_{B}} = \frac{12.433}{10} = 1.243 \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} MQ_{R} & = & \frac{SQ_{R}}{f_{R}} = \frac{0.815}{60} = 0.014 \end{array} $$

(17)

Variance Estimate

The coefficient of variance V is defined by the following equation and describes the measurement uncertainty.

$$\begin{array}{@{}rcl@{}} MQ_{A} & = & s \cdot n \cdot \hat{\sigma}_{a}^{2} + n \cdot \hat{\sigma}_{b}^{2} + \hat{\sigma}_{e}^{2} \end{array} $$

(18)

$$\begin{array}{@{}rcl@{}} MQ_{B} & = & n \cdot \hat{\sigma}_{b}^{2} + \hat{\sigma}_{e}^{2} \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} MQ_{R} & = & \hat{\sigma}_{e}^{2} \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \hat{\sigma}_{a}^{2} & = & \frac{1}{s \cdot n} \left( MQ_{A} - MQ_{B}\right) \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} \hat{\sigma}_{b}^{2} & = & \frac{1}{n} \left( MQ_{B} - MQ_{R}\right) \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} \hat{\sigma}_{e}^{2} & = & MQ_{R} = 0.014 \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} V & = & \frac{\hat{\sigma}_{e}}{\overline{X}} \cdot 100~\% \end{array} $$

(24)