Chapter 3.4, Problem 11E

Refer to Exercise 12 in Section 3.2. Assume that τ₀ = 50 ± 1 MPa, w = 1.2 ± 0.1 mm, and k = 0.29 ±0.05 mm^–1.

1. a. Estimate τ, and find the uncertainty in the estimate.
2. b. Which would provide the greatest reduction in the uncertainty in τ: reducing the uncertainty in τ₀ to 0.1 MPa, reducing the uncertainty in w to 0.01 mm, or reducing the uncertainty in k to 0.025 mm^–1?
3. c. A new, somewhat more expensive process would allow both τ₀ and w to be measured with negligible uncertainty. Is it worthwhile to implement the process? Explain.

To determine

Find the estimate of the maximum shear stress of a cracked concrete member.

Find the uncertainty in the maximum shear stress of a cracked concrete member.

Answer to Problem 11E

The estimate of the maximum shear stress of a cracked concrete member is $\underset{\_}{\tau =32.6\pm 3.4\text{MPa}}$.

The uncertainty in the maximum shear stress of a cracked concrete member is $\underset{\_}{{\sigma }_{\tau }=3.4\text{MPa}}$.

Explanation of Solution

Given info:

The maximum shear stress for a crack width of zero is measured to be ${\tau }_0=50\pm 1\text{MPa}$, the crack width is measured to be $w=1.2\pm 0.1\text{mm}$ and the constant estimated from experimental data is measured to be $k=0.29\pm 0.05{\text{mm}}^{-1}$.

Calculation:

The form of the measurements of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of constant estimated from experimental data is $k=0.29\pm 0.05{\text{mm}}^{-1}$.

Here, the measured value of the constant is $k=0.29{\text{mm}}^{-1}$ and the uncertainty in the constant is ${\sigma }_k=0.05{\text{mm}}^{-1}$.

The form of the measurements of maximum shear stress for a crack width of zero is ${\tau }_0=50\pm 1\text{MPa}$.

Here, the measured value of the zero crack width is ${\tau }_0=50\text{MPa}$ and the uncertainty in the zero crack width is ${\sigma }_{{\tau }_0}=1\text{MPa}$.

The form of the measurements of crack width is $w=1.2\pm 0.1\text{mm}$.

Here, the measured value of the crack width is $w=1.2\text{mm}$ and the uncertainty in the crack width is ${\sigma }_w=0.1\text{mm}$.

Measured value of maximum shear stress of a cracked concrete member:

The formula for maximum shear stress of a cracked concrete member is $\tau ={\tau }_0\left(1-kw\right)$.

Here, ${\tau }_0=50\text{MPa}$,$w=1.2\text{mm}$ and $k=0.29{\text{mm}}^{-1}$.

The measured value of maximum shear stress of a cracked concrete member is obtained as follows:

$\begin{array}{c}\tau ={\tau }_0\left(1-kw\right)\\ =50\left(1-k\times 1.2\right)\\ =50-60k\\ =50-60\times 0.29\end{array}$

$=32.6$

Thus, the measured value of maximum shear stress of a cracked concrete member is $\underset{\_}{\tau =35.5\text{MPa}}$.

Uncertainty:

The uncertainty of a process is determined by the standard deviation of the measurements. In other words it can be said that, measure of variability of a process is known as uncertainty of the process.

Therefore, it can be said that uncertainty is simply $\left(\sigma \right)$.

Standard deviation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

The general formula for standard deviation is,

$s=\sqrt{\frac{{\displaystyle \sum x_i{}^2-\frac{{\left({\displaystyle \sum x_i}\right)}^2}{n}}}{n-1}}$.

From the properties of uncertainties for functions of one measurement it is known that,

• If $X_1,X_2,\mathrm{...},X_n$ are independent measurements with uncertainties ${\sigma }_{X_1},{\sigma }_{X_2},\mathrm{...},{\sigma }_{X_n}$ and if $U=U\left(X_1,X_2,\mathrm{..},X_n\right)$ is a function of $X_1,X_2,\mathrm{...},X_n$ then the uncertainty in the variable U is ${\sigma }_U=\sqrt{{\left(\frac{\partial U}{\partial X_1}\right)}^2{\sigma }_{X_1}^2+{\left(\frac{\partial U}{\partial X_2}\right)}^2{\sigma }_{X_2}^2+\mathrm{....}+{\left(\frac{\partial U}{\partial X_n}\right)}^2{\sigma }_{X_n}^2}$.

Here, zero width, crack width and experimental constant are not constants. The maximum shear stress of a cracked concrete member is a function of zero width, crack width and experimental constant.

The uncertainty in the maximum shear stress of a cracked concrete member is,

$\begin{array}{c}{\sigma }_{\tau ={\tau }_0\left(1-kw\right)}=\sqrt{{\left(\frac{\partial M}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial M}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial M}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(1-kw\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-{\tau }_{0}w\right)}^2\times {\sigma }_k^2+{\left(-{\tau }_{0}k\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(1\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0.1\right)}^2}\end{array}$

               $=3.4$

Thus, the uncertainty in the maximum shear stress of a cracked concrete member is $\underset{\_}{{\sigma }_{\tau }=3.4\text{MPa}}$.

Estimate of the maximum shear stress of a cracked concrete member:

The estimate of the measurement of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The estimate of maximum shear stress of a cracked concrete member is,

$\begin{array}{c}\tau =\text{Measured value of }\tau \pm {\sigma }_{\tau }\\ =32.6\pm 3.4\text{MPa}\end{array}$

Thus, the estimate of the maximum shear stress of a cracked concrete member is $\underset{\_}{\tau =32.6\pm 3.4\text{MPa}}$.

To determine

Find the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa.

Find the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm.

Find the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$.

Compare the obtained two uncertainties.

Answer to Problem 11E

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa is $\underset{\_}{{\sigma }_{\tau }=3.3}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm is $\underset{\_}{{\sigma }_{\tau }=3.1}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$ is $\underset{\_}{{\sigma }_{\tau }=2.2}$.

The uncertainty in maximum shear stress of a cracked concrete member is less when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$.

Explanation of Solution

Calculation:

From part (a), the uncertainty in the zero crack width is ${\sigma }_{{\tau }_0}=1\text{MPa}$, uncertainty in the constant is ${\sigma }_k=0.05{\text{mm}}^{-1}$ and the uncertainty in the crack width is ${\sigma }_w=0.1\text{mm}$.

Uncertainty:

Here, zero width, crack width and experimental constant are not constants. The maximum shear stress of a cracked concrete member is a function of zero width, crack width and experimental constant.

The uncertainty in the maximum shear stress of a cracked concrete member is,

$\begin{array}{c}{\sigma }_{\tau ={\tau }_0\left(1-kw\right)}=\sqrt{{\left(\frac{\partial M}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial M}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial M}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(1-kw\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-{\tau }_{0}w\right)}^2\times {\sigma }_k^2+{\left(-{\tau }_{0}k\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(1\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0.1\right)}^2}\end{array}$

Uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa:

Here, ${\sigma }_{{\tau }_0}=0.1\text{MPa}$, ${\sigma }_k=0.05{\text{mm}}^{-1}$ and ${\sigma }_w=0.1\text{mm}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa is,

$\begin{array}{c}{\sigma }_{\tau }=\sqrt{{\left(0.652\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-60\right)}^2\times {\sigma }_k^2+{\left(-14.5\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(0.1\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0.1\right)}^2}\\ =3.3\end{array}$

Thus, the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa is $\underset{\_}{{\sigma }_{\tau }=3.3}$.

Uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm:

Here, ${\sigma }_{{\tau }_0}=1\text{MPa}$, ${\sigma }_k=0.05{\text{mm}}^{-1}$ and ${\sigma }_w=0.01\text{mm}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm is,

$\begin{array}{c}{\sigma }_{\tau }=\sqrt{{\left(0.652\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-60\right)}^2\times {\sigma }_k^2+{\left(-14.5\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(1\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0.01\right)}^2}\\ =3.1\end{array}$

Thus, the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm is $\underset{\_}{{\sigma }_{\tau }=3.1}$.

Uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$:

Here, ${\sigma }_{{\tau }_0}=1\text{MPa}$, ${\sigma }_k=0.025{\text{mm}}^{-1}$ and ${\sigma }_w=0.1\text{mm}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$  is,

$\begin{array}{c}{\sigma }_{\tau }=\sqrt{{\left(0.652\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-60\right)}^2\times {\sigma }_k^2+{\left(-14.5\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(1\right)}^2+{\left(-60\right)}^2\times {\left(0.025\right)}^2+{\left(-14.5\right)}^2\times {\left(0.1\right)}^2}\\ =2.2\end{array}$

Thus, the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$ is $\underset{\_}{{\sigma }_{\tau }=2.2}$.

Comparison:

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in ${\tau }_0$ is reduced to 0.1 MPa is $\underset{\_}{{\sigma }_{\tau }=3.3}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $w$ is reduced to 0.01 mm is $\underset{\_}{{\sigma }_{\tau }=3.1}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$ is $\underset{\_}{{\sigma }_{\tau }=2.2}$.

Here, $2.2<3.1<3.2$.

Thus, the uncertainty in maximum shear stress of a cracked concrete member is less when the uncertainty in $k$ is reduced to $0.025{\text{mm}}^{\text{-1}}$.

To determine

Check whether it is worth full to reduce the uncertainty in w and ${\tau }_0$ to “0” in order to reduce the uncertainty in maximum shear stress of a cracked concrete member.

Answer to Problem 11E

No, it is not worth full to reduce the uncertainty in w and ${\tau }_0$ to “0” in order to reduce the uncertainty in maximum shear stress of a cracked concrete member.

Explanation of Solution

Calculation:

From part (a), the uncertainty in the constant is ${\sigma }_k=0.05{\text{mm}}^{-1}$.

Uncertainty:

Here, zero width, crack width and experimental constant are not constants. The maximum shear stress of a cracked concrete member is a function of zero width, crack width and experimental constant.

The uncertainty in the maximum shear stress of a cracked concrete member is,

$\begin{array}{c}{\sigma }_{\tau ={\tau }_0\left(1-kw\right)}=\sqrt{{\left(\frac{\partial M}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial M}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial M}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial {\tau }_0}\right)}^2{\sigma }_{{\tau }_0}^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial k}\right)}^2{\sigma }_k^2+{\left(\frac{\partial \left({\tau }_0\left(1-kw\right)\right)}{\partial w}\right)}^2{\sigma }_w^2}\\ =\sqrt{{\left(1-kw\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-{\tau }_{0}w\right)}^2\times {\sigma }_k^2+{\left(-{\tau }_{0}k\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(1\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0.1\right)}^2}\end{array}$

Uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in w and ${\tau }_0$  is reduced to 0:

Here, ${\sigma }_{{\tau }_0}=0$, ${\sigma }_k=0.05{\text{mm}}^{-1}$ and ${\sigma }_w=0$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in w and ${\tau }_0$  is reduced to 0is,

$\begin{array}{c}{\sigma }_{\tau }=\sqrt{{\left(0.652\right)}^2\times {\sigma }_{{\tau }_0}^2+{\left(-60\right)}^2\times {\sigma }_k^2+{\left(-14.5\right)}^2\times {\sigma }_w^2}\\ =\sqrt{{\left(0.652\right)}^2\times {\left(0\right)}^2+{\left(-60\right)}^2\times {\left(0.05\right)}^2+{\left(-14.5\right)}^2\times {\left(0\right)}^2}\\ =3\end{array}$

Thus, the uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in w and ${\tau }_0$  is reduced to 0is $\underset{\_}{{\sigma }_{\tau }=3}$.

Comparison:

The uncertainty in the maximum shear stress of a cracked concrete member is $\underset{\_}{{\sigma }_{\tau }=3.4\text{MPa}}$.

The uncertainty in the maximum shear stress of a cracked concrete member when the uncertainty in w and ${\tau }_0$  is reduced to 0 is $\underset{\_}{{\sigma }_{\tau }=3}$.

Here, it can be seen that there is not much difference in ${\sigma }_{\tau }$ with the consideration of ${\sigma }_{{\tau }_0}=0$ and ${\sigma }_w=0$.

That is, there is not much difference between 3.0 and 3.4.

Therefore, the uncertainty in maximum shear stress of a cracked concrete member is not reduced much by estimating w and ${\tau }_0$ with negligible uncertainty.

Thus, it is not worth full to reduce the uncertainty in w and ${\tau }_0$ to “0” in order to reduce the uncertainty in maximum shear stress of a cracked concrete member.