Advanced mechanics of materials

Arthur P. Boresi, Richard J. Schmidt, Omar M. Sidebottom

Chapter 9 Curved Beams - all with Video Answers

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Chapter Questions

Problem 1

The frame shown in Fig. E9.1 has a rectangular cross section with a thickness of $10 \mathrm{~mm}$ and depth of $40 \mathrm{~mm}$. The load $P$ is located $120 \mathrm{~mm}$ from the centroid of section $B C$. The frame is made of steel having a yield stress of $Y=430 \mathrm{MPa}$. The frame has been designed using a factor of safety of $S F=$ 1.75 against initiation of yielding. Determine the maximum allowable magnitude of $P$, if the radius of curvature at section $B C$ is $R=40 \mathrm{~mm}$.

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Problem 2

Solve Problem 9.1 for the condition that $R=35 \mathrm{~mm}$.

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06:20

Problem 3

The curved beam in Fig. P9.3 has a circular cross section $50 \mathrm{~mm}$ in diameter. The inside diameter of the curved beam is $40 \mathrm{~mm}$. Determine the stress at $B$ for $P=20 \mathrm{kN}$.

Vipender Yadav

Numerade Educator

Problem 4

Let the crane hook in Fig. E9.2 have a trapezoidal cross section as shown in row (c) of Table 9.2 with (see Fig. P9.4) $a=45 \mathrm{~mm}, c=80 \mathrm{~mm}, b_1=$ $25 \mathrm{~mm}$, and $b_2=10 \mathrm{~mm}$. Determine the maximum load to be carried by the hook if the working stress limit is $150 \mathrm{MPa}$.

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Problem 5

A curved beam is built up by welding together rectangular and elliptical cross-section curved beams; the cross section is shown in Fig. P9.5. The center of curvature is located $20 \mathrm{~mm}$ from $B$. The curved beam is subjected to a positive bending moment $M_x$. Determine the stresses at points $B$ and $C$ in terms of $M_x$.

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Problem 6

A commercial crane hook has the cross-sectional dimensions shown in Fig. P9.6 at the critical section that is subjected to an axial load $P=$ $100 \mathrm{kN}$. Determine the circumferential stresses at the inner and outer radii for this load. Assume that area $A_1$ is half of an ellipse [see row $(j)$ in Table 9.2] and area $A_3$ is enclosed by a circular arc.

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Problem 7

A crane hook has the cross-sectional dimensions shown in Fig, P9.7 at the critical section that is subjected to an axial load $P=90.0 \mathrm{kN}$. Determine the circumferential stresses at the inner and outer radii for this load. Note that $A_1$ and $A_3$ are enclosed by circular arcs.

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03:08

Problem 8

The curved beam in Fig. P9.8 has a triangular cross section with the dimensions shown. If $P=40 \mathrm{kN}$, determine the circumferential stresses at $B$ and $C$.

Surendra Kumar

Numerade Educator

Problem 9

A curved beam with a rectangular cross section strikes a $90^{\circ}$ arc and is loaded and supported as shown in Fig. P9.9. The thickness of the beam is $50 \mathrm{~mm}$. Determine the hoop stress $\sigma_{\mathrm{pg}}$ along line $A-A$ at the inside and outside radii and at the centroid of the beam.

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Problem 10

For the curved beam in Problem 9.5, determine the radial stress in terms of the moment $M_x$ if the thickness of the web at the weld is $10 \mathrm{~mm}$.

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Problem 11

In Fig. P9.11 is shown a cast iron frame with a U-shaped cross section. The ultimate tensile strength of the case iron is $\sigma_y=320 \mathrm{MPa}$.
(a) Determine the maximum value of $P$ based on a factor of safety $S F=$ 4.00 which is based on the ultimate strength.
(b) Neglecting the effect of stress concentrations at the fillet at the junction of the web and flange, determine the maximum radial stress when this load is applied.
(c) Is the maximum radial stress less than the maximum circumferential stress?

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Problem 12

A T-section curved beam has the cross section shown in Fig. P9.12. The center of curvature lies $40 \mathrm{~mm}$ from the flange. If the curved beam is subjected to a positive bending moment $M_x=2.50 \mathrm{kN} \cdot \mathrm{m}$, determine the stresses at the inner and outer radii. Use Bleich's correction factors. What is the maximum shear stress in the curved beam?

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Problem 13

Determine the radial stress at the junction of the web and the flange for the curved beam in Problem 9.12. Neglect stress concentrations. Use the Bleich correction.

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07:24

Problem 14

A load $P=12.0 \mathrm{kN}$ is applied to the clamp shown in Fig. P9.14. Determine the circumferential stresses at points $B$ and $C$, assuming that the curved beam formula is valid at that section.

Chai Santi

Numerade Educator

Problem 15

Determine the radial stress at the junction of the web and inner flange of the curved beam portion of the clamp in Problem 9.14. Neglect stress concentrations.

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Problem 16

The curved beam in Fig. P9.16 is made of a steel $(E=200 \mathrm{GPa})$ that has a yield point stress $Y=420 \mathrm{MPa}$. Determine the magnitude of the bending moment $M_x=M_Y$ required to initiate yielding in the curved beam, the angle change of the free end, and the horizontal and vertical components of the deflection of the free end.

Victor Salazar

Numerade Educator

Problem 17

Determine the deflection of the curved beam in Problem 9.3 at the point of load application. The curved beam is made of an aluminum alloy for which $E=72.0 \mathrm{GPa}$ and $G=27.1 \mathrm{GPa}$. Let $k=1.3$.

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02:29

Problem 18

The triangular cross-section curved beam in Problem 9.8 is made of steel $(E=200 \mathrm{GPa}, G=77.5 \mathrm{GPa}$ ). Determine the separation of the points application of the load. Let $k=1.5$.

Chai Santi

Numerade Educator

Problem 19

Determine the deflection across the center of curvature of the cast iron curved beam in Problem 9.11 for $P=126 \mathrm{kN} . E=102.0 \mathrm{GPa}$ and $G=$ $42.5 \mathrm{GPa}$. Let $k=1.0$ with the area in shear equal to the product of the web thickness and the depth.

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02:36

Problem 20

The ring in Fig, P9.20 has an inside diameter of $100 \mathrm{~mm}$, an outside diameter of $180 \mathrm{~mm}$, and a circular cross section. The ring is made of a steel having a yield stress $Y=520 \mathrm{MPa}$. Determine the maximum allowable magnitude of $P$ if the ring has been designed with a factor of safety $S F=$ 1.75 against initiation of yielding.

Naman Kumar

Numerade Educator

01:41

Problem 21

If $E=200 \mathrm{GPa}$ and $G=77.5 \mathrm{GPa}$ for the steel in Problem 9.20, determine the deflection of the ring for a load $P=60 \mathrm{kN}$. Let $k=1.3$.

Chai Santi

Numerade Educator

Problem 22

An aluminum alloy ring has a mean diameter of $600 \mathrm{~mm}$ and a rectangular cross section with $200 \mathrm{~mm}$ thickness and a depth of $300 \mathrm{~mm}$ (radial direction). The ring is loaded by diametrically opposite radial loads $P=4.00 \mathrm{MN}$. Determine the maximum tensile and compressive circumferential stresses in the ring.

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Problem 23

If $E=72.0 \mathrm{GPa}$ and $G=27.1 \mathrm{GPa}$ for the aluminum alloy ring in Problem 9.22, determine the separation of the points of application of the loads. Let $k=1.5$.

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Problem 24

The link in Fig. P9.24 has a circular cross section and is made of a steel having a yield point stress of $Y=250 \mathrm{MPa}$. Determine the magnitude of $P$ that will initiate yield in the link.

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02:37

Problem 25

Let the curved beam in Fig, 9.10 have a rectangular cross section with depth $h$ and width $b$. Show that the ratio of the bending moment for fully plastic load $P_P$ to the fully plastic moment for pure bending $M_P=Y b h^2 / 4$ is given by the relation
$$
\frac{M}{M_P}=\frac{4 D}{h} \sqrt{1+\frac{4 D^2}{h^2}}-\frac{8 D^2}{h^2}
$$

Chai Santi

Numerade Educator

Problem 26

Let the curved beam in Problem 9.1 be made of a steel that has a flat top stress-strain diagram at the yield point stress $Y=430 \mathrm{MPa}$. From the answer to Problem 9.1, the load that initiates yielding is equal to $P_Y=$ $S F(P)=6.05 \mathrm{kN}$. Since $D=3 h$, assume $M=M_P$ and calculate $P_P$. Determine the ratio $P_P / P_Y$.

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03:04

Problem 27

Let the steel in the curved beam in Example 9.5 have a flat top at the yield point stress $Y=280 \mathrm{MPa}$. Determine the fully plastic moment for the curved beam. Note that the original cross section must be used. The distortion of the cross section increases the fully plastic moment for a positive moment.

Surendra Kumar

Numerade Educator