Chapter 4.6, Problem 92P

(a)

To determine

Find the residual stress at $y=2\text{in}\text{.}$

Answer to Problem 92P

The residual stress is $\underset{\_}{{\sigma }_{\text{res}}=13.36\text{ksi}}$.

Explanation of Solution

Given information:

The yield stress for the beam is ${\sigma }_Y=42\text{ksi}$.

The Young’s modulus of steel is $E=29\times {10}^6\text{psi}$.

Calculation:

Show the cross-section of the beam as shown in Figure 1.

Refer to Figure 1.

Calculate the area of the cross section $\left(A\right)$ as shown below.

$A=bd$ (1)

Here, b is the width of the cross section and d is the depth of the cross section.

Calculate the area of the portion (1) $\left(A_1\right)$ as shown below.

Substitute $3\text{in}\text{.}$ for b and $1\text{in}\text{.}$ for d in Equation (1).

$\begin{array}{c}{A}_1=3\times 1\\ =3\text{in}{\text{.}}^2\end{array}$

Calculate the area of the portion (2) $\left(A_2\right)$ as shown below.

Substitute $1\text{in}\text{.}$ for b and $1\text{in}\text{.}$ for d in Equation (1).

$\begin{array}{c}{A}_2=1\times 1\\ =1\text{in}{\text{.}}^2\end{array}$

Calculate the moment of inertia $\left(I\right)$ as shown below.

$I=\frac{b{d}^3}{12}+A{h}^2$ (2)

Calculate the moment of inertia of portion (1) $\left(I_1\right)$ as shown below.

Substitute $3\text{in}\text{.}$ for b, $1\text{in}\text{.}$ for d, $3\text{in}{\text{.}}^2$ for A and $1.5\text{in}\text{.}$ for h in Equation (2).

$\begin{array}{c}{I}_1=\frac{3\times 1^3}{12}+3\times {1.5}^2\\ =7\text{in}{\text{.}}^4\end{array}$

Hence, $I_3=7\text{in}{\text{.}}^4$

Calculate the moment of inertia of portion (2) $\left(I_2\right)$ as shown below.

Substitute $1\text{in}\text{.}$ for b, $2\text{in}\text{.}$ for d, $2\text{in}{\text{.}}^2$ for A and $0$ for h in Equation (2).

$\begin{array}{c}{I}_2=\frac{1\times 2^3}{12}+0\\ =0.667\text{in}{\text{.}}^4\end{array}$

Calculate the total moment of inertia $\left(I\right)$ as shown below.

$I=I_1+I_2+I_3$

Substitute $7\text{in}{\text{.}}^4$ for $I_1$, $0.667\text{in}{\text{.}}^4$ for $I_2$, and $7\text{in}{\text{.}}^4$ for $I_3$.

$\begin{array}{c}I=7+0.667+7\\ =14.667\text{in}{\text{.}}^4\end{array}$

Calculate the centroid (c) as shown below.

$c=\frac{h}{2}$

Substitute $4\text{in}\text{.}$ for h.

$\begin{array}{c}c=\frac{4}{2}\\ =2\text{in}\text{.}\end{array}$

Sketch the stress acting on the cross-section of the beam as shown in Figure 2.

Refer Figure 2.

Calculate the area of the portion (2) $\left(A_2\right)$ as shown below.

$A_2=\frac{1}{2}bh$

Substitute $1\text{in}\text{.}$ for b and $1\text{in}\text{.}$ for d.

$\begin{array}{c}{A}_2=\frac{1}{2}\times 1\times 1\\ =0.5\text{in}{\text{.}}^2\end{array}$

Calculate the reaction applied to portion (1) $\left(R_1\right)$ as shown below.

$R_1={\sigma }_Y{A}_1$

Substitute $42\text{ksi}$ for ${\sigma }_Y$ and $3\text{in}{\text{.}}^2$ for $A_1$.

$\begin{array}{c}{R}_1=42\times 3\\ =126\text{kips}\end{array}$

Calculate the reaction applied to portion (2) $\left(R_2\right)$ as shown below.

$R_2={\sigma }_Y{A}_2$

Substitute $42\text{ksi}$ for ${\sigma }_Y$ and $0.5\text{in}{\text{.}}^2$ for $A_2$.

$\begin{array}{c}{R}_2=42\times 0.5\\ =21\text{kips}\end{array}$

Calculate the moment $\left(M\right)$ as shown below.

$M=2\left(R_1{y}_1+R_2{y}_2\right)$

Substitute $126\text{kips}$ for $R_1$, $\left(1+\frac{1}{2}\right)\text{in}\text{.}$ for $y_1$, $21\text{kips}$ for $R_2$, and $\left(\frac{2}{3}\times 1\right)\text{in}\text{.}$ for $y_2$.

$\begin{array}{c}M=2\left(126\times \left(1+\frac{1}{2}\right)+21\times \left(\frac{2}{3}\times 1\right)\right)\\ =406\text{kip}\cdot \text{in}\text{.}\end{array}$

Calculate the stress $\left({\sigma }^{\prime }\right)$ as shown below.

${\sigma }^{\prime }=\frac{Mc}{I}$

Substitute $406\text{kip}\cdot \text{in}\text{.}$ for M, $2\text{in}\text{.}$ for c, and $14.667\text{in}{\text{.}}^4$ for I.

$\begin{array}{c}{\sigma }^{\prime }=\frac{406\times 2}{14.667}\\ =55.362\text{ksi}\end{array}$

Calculate the stress $\left({\sigma }^{″}\right)$ as shown below.

${\sigma }^{″}=\frac{My}{I}$

Substitute $406\text{kip}\cdot \text{in}\text{.}$ for M, $1\text{in}\text{.}$ for y, and $14.667\text{in}{\text{.}}^4$ for I.

$\begin{array}{c}{\sigma }^{″}=\frac{406\times 1}{14.667}\\ =27.681\text{ksi}\end{array}$

Calculate the residual stress at $y=c$ as shown below.

${\sigma }_{\text{res}}={\sigma }^{\prime }-{\sigma }_Y$

Substitute $55.362\text{ksi}$ for ${\sigma }^{\prime }$ and $42\text{ksi}$ for ${\sigma }_Y$.

$\begin{array}{c}{\sigma }_{\text{res}}=55.362-42\\ =13.362\text{ksi}\end{array}$

Calculate the residual stress at $y=y_Y$ as shown below.

${\sigma }_{\text{res}}={\sigma }^{″}-{\sigma }_Y$

Substitute $27.681\text{ksi}$ for ${\sigma }^{\prime }$ and $42\text{ksi}$ for ${\sigma }_Y$.

$\begin{array}{c}{\sigma }_{\text{res}}=27.681-42\\ =-14.319\text{ksi}\end{array}$

Sketch the stress distribution as shown in Figure 3.

Hence, the residual stress is $\underset{\_}{{\sigma }_{\text{res}}=13.36\text{ksi}}$.

(b)

To determine

Find the point where the residual stress is zero.

Answer to Problem 92P

The point where the residual stress is zero is $\underset{\_}{y_0=-1.517\text{in}\text{., 0, }1.517\text{in}\text{. }}$.

Explanation of Solution

Given information:

The yield stress for the beam is ${\sigma }_Y=42\text{ksi}$.

The Young’s modulus for steel is $E=29\times {10}^6\text{psi}$.

Calculation:

Consider that the residual stress is ${\sigma }_{\text{res}}=0$.

Calculate the yield stress $\left({\sigma }^{\prime }\right)$ as shown below.

${\sigma }^{\prime }=\frac{M{y}_0}{I}$

Calculate the point where the residual stress is zero as shown below.

${\sigma }_{\text{res}}={\sigma }^{\prime }-{\sigma }_Y$

Substitute $\frac{M{y}_0}{I}$ for ${\sigma }^{\prime }$ and $0$ for ${\sigma }_{\text{res}}$.

$\begin{array}{l}0=\frac{M{y}_0}{I}-{\sigma }_Y\\ {\sigma }_Y=\frac{M{y}_0}{I}\\ y_0=\frac{{\sigma }_{Y}I}{M}\end{array}$

Substitute $42\text{ksi}$ for ${\sigma }_Y$, $406\text{kip}\cdot \text{in}\text{.}$ for M, and $14.667\text{in}{\text{.}}^{{}^4}$ for I.

$\begin{array}{c}{y}_0=\frac{42\times 14.667}{406}\\ =1.517\text{in}\text{.}\end{array}$

Therefore, the point where the residual stress is zero is $\underset{\_}{y_0=-1.517\text{in}\text{., 0, }1.517\text{in}\text{. }}$.

(c)

To determine

Find the radius of curvature corresponding to the permanent deformation of the bar.

Answer to Problem 92P

The radius of curvature is $\underset{\_}{\rho =168.8\text{ft}}$.

Explanation of Solution

Given information:

The yield stress for the beam is ${\sigma }_Y=42\text{ksi}$.

The Young’s modulus of steel is $E=29\times {10}^6\text{psi}$.

Calculation:

Refer to part (a).

The residual stress ${\sigma }_{\text{res}}=-14.32\text{ksi}$.

Calculate the radius of curvature $\left(\rho \right)$ as shown below.

${\sigma }^{\prime }=\frac{M{y}_0}{I}$

Calculate the point where the residual stress is zero as shown below.

$\rho =-\frac{Ey}{\sigma }$

Substitute $29\times {10}^6\text{psi}$ for E, $-14.32\text{ksi}$ for $\sigma$ and $1\text{in}\text{.}$ for $y$.

$\begin{array}{c}\rho =-\frac{29\times {10}^6\text{psi}\times \frac{1\text{ksi}}{1,000\text{psi}}\times 1\text{in}\text{.}}{-14.32\text{ksi}}\\ =2,025.14\text{in}\text{.}\times \frac{1\text{ft}}{12\text{in}\text{.}}\\ =168.8\text{ft}\end{array}$

Therefore, the radius of curvature is $\underset{\_}{\rho =168.8\text{ft}}$.