Chapter 10, Selection of Material and Shape Video Solutions, Materials Selection in Mechanical Design

Problem 1

Stiffness shape factors for tubes (Fig. E10.1). Evaluate the shape factor $\phi_{B}^{e}$ for stiffness-limited design in bending of a square box section of outer edgelength $h=100 \mathrm{~mm}$ and wall thickness $t=3 \mathrm{~mm}$. Is this box shape more efficient than tube made of the same material with diameter $2 r=100 \mathrm{~mm}$ and wall thickness $t=3.82 \mathrm{~mm}$ (giving it the same mass per unit length, $m / L$ )? Treat both as thin-walled shapes.
Use the expressions in Table $10.3$ of the text for the shape factor $\phi_{B}^{e}$.

Dheeraj Sharma

Numerade Educator

02:49

Problem 2

Strength shape factors for tubes (Fig. El0.1 again). Evaluate the shape factor $\phi_{B}^{f}$ for strength-limited design in bending of a square box section of outer edge-length $h=100 \mathrm{~mm}$ and wall thickness $t=3 \mathrm{~mm}$. Is this box shape more efficient than tube made of the same material with diameter $2 r=100 \mathrm{~mm}$ and wall thickness $t=3.82 \mathrm{~mm}$ (giving it the same mass per unit length, $m / L$ )? Treat both as thin-walled shapes.
Use the expressions given in Table $10.3$ of the text for the shape factors $\phi_{B}$.

Km Neeraj

Numerade Educator

02:45

Problem 3

Deriving shape factors for stiffness-limited design (Fig. E10.2). Derive the expression for the shape-efficiency factor $\phi_{B}^{e}$ for stiffness-limited design for a circular tube with outer radius $5 t$ and wall thickness $t$, loaded in bending (Fig. E10.2). Do not assume that the thin-wall approximation is valid.

Narayan Hari

Numerade Educator

02:09

Problem 4

Deriving shape factors for stiffness-limited design (Fig. E10.3). Derive the expression for the shape-efficiency factor $\phi_{B}^{e}$ for stiffness-limited design for a channel section of thickness $t$, overall flange width $5 t$ and overall depth $10 t$, bent about its major axis (the dash-dot line in Fig. E10.3). Do not assume that the thin-wall approximation is valid.

Nick Johnson

Numerade Educator

02:09

Problem 4

Nick Johnson

Numerade Educator

04:22

Problem 5

Deriving shape factors for stiffness-limited design (Fig. E10.4). Derive the expression for the shape-efficiency factor $\phi_{B}^{e}$ for stiffness-limited design for a square box section of wall thickness $t$, and height and width $h_{1}=10 t$ bent about its major axis (the dash-dot line in Fig. E10.4). Do not assume that the thin-wall approximation is valid.

Carson Merrill

Numerade Educator

05:04

Problem 6

Deriving shape factors for strength-limited design. Determine the value of the shape-efficiency factor $\phi_{B}$ for strength-limited design in bending using the dimensions shown on the diagrams
a. For the tube-section shown in Fig. E10.5(A)
b. For the box section shown in Fig. E10.5(B) and
c. For the channel section shown in Fig. E10.5(C).
The expression for $\phi_{B}^{f}$ for the first two sections can be read from Table 10.3. You will have to derive the expression for the third.

Eric Mockensturm

Numerade Educator

04:04

Problem 7

Deriving indices that include shape. A beam of length $L$, loaded in bending, must support a specified bending moment $M$ without failing and be as light as possible. Show that to minimize the mass of the beam per unit length, $m / L$, one should select a material and a section-shape to maximize the quantity
$$
M=\frac{\left(\phi_{B}^{f_{f}} \sigma_{f}^{2 / 3}\right.}{\rho}
$$
where $\sigma_{f}$ is the failure stress and $\rho$ the density of the material of the beam, and $\phi_{B}$ is the shapeefficiency factor for failure in bending.

Angela Guo

Numerade Educator

01:16

Problem 8

Calculating shape factors from stiffness data. The elastic shape factor measures the gain in stiffness by shaping, relative to a solid square section of the same cross-section area and thus mass per unit length. Shape factors can be determined by experiment by measuring the stiffness and mass of a structure and using these to calculate $\phi_{B}^{e}$ by inverting Eq. (10.19) of the text and solving for $\phi_{B}^{e}$. Apply this approach to calculate the shape factor $\phi_{B}^{e}$ from the following experimental data, measured on an aluminium alloy beam (Fig. E10.6) loaded in 3-point bending (for which $C_{1}=48$ - see Appendix B, Section B3) by using the data shown in the table.
$$
\begin{array}{|l|l|}
\hline \text { Attribute } & \text { Value } \\
\hline \text { Bending stiffness } S_{\mathrm{B}} & 7.2 \times 10^{5} \mathrm{~N} / \mathrm{m} \\
\hline \text { Mass/unit length } \mathrm{m} / \mathrm{L} & 1 \mathrm{~kg} / \mathrm{m} \\
\hline \text { Beam length } L & 1 \mathrm{~m} \\
\hline \text { Beam material } & 6061 \text { aluminium alloy } \\
\hline \text { Material density } \rho & 2670 \mathrm{~kg} / \mathrm{m}^{3} \\
\hline \text { Material modulus } E & 69 \mathrm{GPa} \\
\hline
\end{array}
$$

Manik Pulyani

Numerade Educator

03:11

Problem 9

Calculating shape factors from stiffness data. A steel truss bridge shown in Fig. E10.7 has a span $L$ and is simply supported at both ends. It weighs $m$
tonnes. As a rule of thumb, bridges are designed with a stiffness $S_{B}$ such that the central deflection $\delta$ of a span under its self-weight is less than $1 / 300$ of the length $L$ (thus $S_{B} \geq 300 \mathrm{mg} / L$ where $g$ is the acceleration due to gravity, $9.81 \mathrm{~m} / \mathrm{s}^{2}$ ). Use this information to calculate a minimum value for the shape factor $\phi_{B}^{e}$ of the three steel truss bridge spans listed in the table. Take the density $\rho$ of steel to be $7900 \mathrm{~kg} / \mathrm{m}^{3}$ and its modulus $E$ to be $205 \mathrm{GPa}$. The constant $C_{1}=384 / 5=76.8$ for uniformly distributed load (Appendix B, Section B3).

Chai Santi

Numerade Educator

07:36

Problem 10

Deriving indices for bending. A beam, loaded in bending, must support a specified bending moment $M^{*}$ without failing and be as light as possible. Section shape is a variable, and 'failure' here means the first onset of plasticity. Derive the material index. The table summarizes the requirements.

Vipender Yadav

Numerade Educator

01:08

Problem 11

Deriving indices for torsion (Fig. E10.8). A shaft of length $L$, loaded in torsion, must support a specified torque $T^{*}$ without failing and be as cheap as possible. Section shape is a variable and 'failure' again means the first onset of plasticity. Derive the material index. The table summarizes the requirements.

Hast Aggarwal

Numerade Educator

05:37

Problem 12

Material and shape to suppress buckling (Fig. E10.9). The figure shows a concept for a lightweight display stand. The stalk must support a mass $m$ of $100 \mathrm{~kg}$, to be placed on its upper surface at a height $h$, without failing by elastic buckling. It is to be made of stock tubing and must be as light as possible. Use the methods of this chapter to derive a material index for the tubular material of the stalk of the stand that includes the shape of the section, described by the shape factor
$$
\phi_{B}^{e}=\frac{12 I}{A^{2}}
$$
where $I$ is the second moment of area and $A$ is the section area of the stalk. The table summarizes the requirements.

Eric Mockensturm

Numerade Educator

04:20

Problem 13

Selection of material and shape to suppress buckling. Cylindrical tubing is available from stock in the following materials and sizes. Use this information and the material index to identify the best stock material for the stalk of the stand shown in Fig. E10.8.
$$
\begin{array}{|l|l|l|l|}
\hline \text { Material } & {}{||}{\text { Modulus } E(\mathrm{GPa}) \text { Tube Radius } r \text { Wall Thickness/Tube Radius, tir }} \\
\hline \text { Aluminium alloys } & 69 & 25 \mathrm{~mm} & 0.07 \text { to } 0.25 \\
\hline \text { Steel } & 210 & 30 \mathrm{~mm} & 0.045 \text { to } 0.1 \\
\hline \text { Copper alloys } & 120 & 20 \mathrm{~mm} & 0.075 \text { to } 0.1 \\
\hline \text { Polycarbonate (PC) } & 3.0 & 20 \mathrm{~mm} & 0.15 \text { to } 0.3 \\
\hline \text { Various woods } & 7-12 & 40 \mathrm{~mm} & \text { Solid circular sections only } \\
\hline
\end{array}
$$

Ameer Said

Numerade Educator

02:40

Problem 14

Microscopic shape: tube arrays (Fig. E10.10). Calculate the gain in bending efficiency, $\psi_{B}^{e}$, when a solid is formed into small, thin-walled tubes of radius $r$ and wall thickness $t$ that are then assembled and bonded into a large array, part of which is shown in the figure. Let the solid of which the tubes are made have modulus $E_{\text {s and density }} \rho_{s}$. Express the result in terms of $r$ and $t$.

Averell Hause

Carnegie Mellon University

01:16

Problem 16

Use of the four segment chart for stiffness-limited design. Use the 4segment chart for stiffness-limited design of Fig. $10.9$ of the text to compare the mass per unit length, $m / L$, of a section with $E I=10^{5} \mathrm{Nm}^{2}$ made from
a. structural steel with a shape factor $\phi_{B}^{e}$ of 20 , modulus $E=210 \mathrm{GPa}$ and density $\rho=7900 \mathrm{~kg} / \mathrm{m}^{3}$
b. carbon fiber reinforced plastic (CFRP) with a shape factor $\phi_{B}^{e}$ of 10 , modulus $E=70 \mathrm{GPa}$ and density $\rho=1600 \mathrm{~kg} / \mathrm{m}^{3}$, and

Manik Pulyani

Numerade Educator

02:03

Problem 17

Numerical comparison for stiffness-limited design. Show, by direct calculation, that the conclusion of Exercise $10.16$ - that the CFRP beam with $E I=10^{5} \mathrm{Nm}^{2}$ and $\phi_{B}^{e}=10$ weighs less than the steel beam with the same $\phi_{B}^{e}=20-$ is consistent with the idea that to minimize mass for a given stiffness one should maximize $\sqrt{E^{*}} / \rho^{*}$ with $E^{*}=E / \phi_{B}^{e}$ and $\rho^{*}=\rho / \phi_{B}^{e}$,

Chai Santi

Numerade Educator

05:47

Problem 18

Use of the four segment chart for strength. Use the 4-segment chart for strength-limited design of Fig. $10.12$ of the text to compare the mass per unit length, $m / L$, of a section with $Z \sigma_{f}=10^{4} \mathrm{Nm}$ (where $Z$ is the section modulus) made from
a. 6061 grade aluminium alloy with a shape factor $\phi_{B}^{f}$ of 3 , strength $\sigma_{f}=200 \mathrm{MPa}$ and density $\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}$, and
b. titanium alloy with a shape factor $\phi_{B}^{f}$ of 10 , strength $\sigma_{f}=480 \mathrm{MPa}$ and density $\rho=4420 \mathrm{~kg} / \mathrm{m}^{3}$.

Satpal Satpal

Numerade Educator

01:20

Problem 19

Numerical comparison for stiffness-limited design. Show, by direct calculation, that the conclusion of Exercise E10.18 - that the titanium alloy beam with $Z \sigma_{f}=10^{4} \mathrm{Nm}$ and $\phi_{B}^{f}$ of 10 is much lighter than the 6061 aluminium alloy beam of the same strength and $\phi_{B}^{f}-$ is consistent with the idea that to minimize mass for a given stiffness one should maximize $\frac{\left(\phi_{1} \sigma_{0}\right)^{2 / 3}}{p}$.

Surendra Kumar

Numerade Educator