MultiPhysicsVault/.raw/AFirstCourseInTheFiniteElementMethod/AFirstCourseInTheFiniteElementMethod_081.md

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$$
\hat {f} _ {3 y} ^ {(2)} = 2 2. 9 5 \mathrm{kip}, \quad \hat {m} _ {3} ^ {(2)} = - 2 1 7 1 \mathrm{k-in}.
$$

$$
F _ {1 x} = F _ {3 x} = 4 6. 8 \mathrm{kip}, \quad F _ {1 y} = 7 7. 1 \mathrm{kip}, \quad M _ {1} = 3 6 1 \mathrm{k-in}.
$$

$$
F _ {3 y} = 2 2. 9 5 \mathrm{kip}, \quad M _ {3} = 2 1 7 1 \mathrm{k-in}.
$$

5.6 $d_{2x} = -0.000269$ in., $d_{2y} = -0.0363$ in., $\phi_2 = -0.00347$ rad

$$
\hat {f} _ {1 x} ^ {(1)} = 4 6. 6 \text {   kip }, \quad \hat {f} _ {1 y} ^ {(1)} = 6. 0 7 \text {   kip }, \quad \hat {m} _ {1} ^ {(1)} = 4 9 1. 3 \text {   k - in }.
$$

$$
\hat {f} _ {2 x} ^ {(1)} = - 3 2. 4 \text {   kip }, \quad \hat {f} _ {2 y} ^ {(1)} = 8. 0 7 \text {   kip }, \quad \hat {m} _ {2} ^ {(1)} = - 8 3 1. 3 \text {   k - in }.
$$

$$
\hat {f} _ {2 x} ^ {(2)} = - \hat {f} _ {3 x} ^ {(2)} = - 0. 2 8 \text { kip }, \quad \hat {f} _ {2 y} ^ {(2)} = 5 8. 3 1 \text { kip }, \quad \hat {m} _ {2} ^ {(2)} = 1 1 2 3. 9 \text { k - in }.
$$

$$
\hat {f} _ {3 y} ^ {(2)} = 2 1. 6 9 \mathrm{kip}, \quad \hat {m} _ {3} ^ {(2)} = - 1 6 1 1. 8 \mathrm{k-in}.
$$

$$
\hat {f} _ {4 x} ^ {(3)} = - \hat {f} _ {2 x} ^ {(3)} = 5 0. 2 \text { kip }, \quad \hat {f} _ {4 y} ^ {(3)} = - \hat {f} _ {2 y} ^ {(3)} = - 1. 4 9 \text { kip }, \quad \hat {m} _ {4} ^ {(3)} = - 1 5 4. 2 \text { k - in }.
$$

$$
\hat {m} _ {2} ^ {(3)} = - 2 9 3. 2 \mathrm{k-in}.
$$

$$
F _ {1 x} = 2 8. 6 5 \mathrm{kip}, \quad F _ {1 y} = 3 7. 2 4 \mathrm{kip}, \quad M _ {1} = 4 9 1. 3 \mathrm{k} - \text { in }.
$$

$$
F _ {3 x} = 0. 2 8 \mathrm{kip}, \quad F _ {3 y} = 2 1. 6 9 \mathrm{kip}, \quad M _ {3} = - 1 6 1 1. 8 \mathrm{k} \text {-in}.
$$

$$
F _ {4 x} = - 2 8. 9 3 \mathrm{kip}, \quad F _ {4 y} = 4 1. 0 5 \mathrm{kip}, \quad M _ {4} = - 1 5 4. 2 \mathrm{k} \text {-in}.
$$

5.7 $d_{2x}=0.4308\times10^{-4}$ m, $d_{2y}=-0.9067\times10^{-4}$ m,

$$
\phi_ {2} = - 0. 1 4 0 3 \times 1 0 ^ {- 2} \mathrm{rad}
$$

$$
\hat {f} _ {1 x} ^ {(1)} = - \hat {f} _ {2 x} ^ {(1)} = 2 3. 8 \mathrm{kN}, \quad \hat {f} _ {1 y} ^ {(1)} = 1 7. 2 6 \mathrm{kN}, \quad \hat {m} _ {1} ^ {(1)} = 3 2. 7 7 \mathrm{kN} \cdot \mathrm{m}
$$

$$
\hat {f} _ {2 y} ^ {(1)} = 2 2. 7 4 \mathrm{kN}, \quad \hat {m} _ {2} ^ {(1)} = - 5 4. 6 4 \mathrm{kN} \cdot \mathrm{m}
$$

$$
\hat {f} _ {2 x} ^ {(2)} = - \hat {f} _ {3 x} ^ {(2)} = 1 1. 3 1 \mathrm{kN}, \quad \hat {f} _ {2 y} ^ {(2)} = 3 7. 1 9 \mathrm{kN}, \quad \hat {m} _ {2} ^ {(2)} = 6 5. 0 9 \mathrm{kN} \cdot \mathrm{m}
$$

$$
\hat {f} _ {3 v} ^ {(2)} = 4 2. 8 1 \mathrm{kN}, \quad \hat {m} _ {3} ^ {(2)} = - 8 7. 5 4 \mathrm{kN} \cdot \mathrm{m}
$$

$$
\hat {f} _ {2 x} ^ {(3)} = - \hat {f} _ {4 x} ^ {(3)} = 1 7. 5 5 \mathrm{kN}, \quad \hat {f} _ {2 y} ^ {(3)} = - \hat {f} _ {4 y} ^ {(3)} = 1. 4 0 \mathrm{kN}
$$

$$
\hat {m} _ {2} ^ {(3)} = - 1 0. 5 1 \mathrm{kN} \cdot \mathrm{m}, \quad \hat {m} _ {4} ^ {(3)} = - 5. 3 0 \mathrm{kN} \cdot \mathrm{m}
$$

$$
F _ {1 x} = - 1 7. 2 6 \mathrm{kN}, \quad F _ {1 y} = 2 3. 8 0 \mathrm{kN}, \quad M _ {1} = 3 2. 7 7 \mathrm{kN} \cdot \mathrm{m}
$$

$$
F _ {3 x} = - 1 1. 3 1 \mathrm{kN}, \quad F _ {3 y} = 4 2. 8 1 \mathrm{kN}, \quad M _ {3} = - 8 7. 5 4 \mathrm{kN} \cdot \mathrm{m}
$$

$$
F _ {4 x} = - 1 1. 4 2 \mathrm{kN}, \quad F _ {4 y} = 1 3. 4 0 \mathrm{kN}, \quad M _ {4} = - 5. 3 0 \mathrm{kN} \cdot \mathrm{m}
$$

5.9 $d_{2x} = -4.95 \times 10^{-5} \mathrm{~m}, d_{2y} = -2.56 \times 10^{-5} \mathrm{~m}, \phi_2 = 2.66 \times 10^{-3} \mathrm{rad}$

$$
\hat {f} _ {1 x} ^ {(1)} = - \hat {f} _ {2 x} ^ {(1)} = 2 6. 9 \mathrm{kN}, \quad \hat {f} _ {1 y} ^ {(1)} = - \hat {f} _ {2 y} ^ {(1)} = - 4 2. 0 \mathrm{kN}
$$

$$
\hat {m} _ {1} ^ {(1)} = 5 5. 9 \mathrm{kN} \cdot \mathrm{m}, \quad \hat {m} _ {2} ^ {(1)} = 1 1 1. 7 \mathrm{kN} \cdot \mathrm{m}
$$

$$
\hat {f} _ {2 x} ^ {(2)} = - \hat {f} _ {3 x} ^ {(2)} = - 4 2. 0 \mathrm{kN}, \quad \hat {f} _ {2 y} ^ {(2)} = - \hat {f} _ {3 y} ^ {(2)} = 2 6. 9 \mathrm{kN}
$$

$$
M _ {1} = 5 5. 9 \mathrm{kN} \cdot \mathrm{m}, \quad M _ {3} = 4 4. 7 \mathrm{kN} \cdot \mathrm{m}
$$

5.10 $d_{2y} = -0.1423 \times 10^{-2} \, m,\quad \phi_{2} = -0.5917 \times 10^{-3} \, rad$

$$
\hat {f} _ {1 x} ^ {(1)} = 0, \hat {f} _ {1 y} ^ {(1)} = 1 0 \mathrm{kN}, \hat {m} _ {1} ^ {(1)} = 2 3. 3 \mathrm{kN} \cdot \mathrm{m}, \hat {f} _ {2 x} ^ {(1)} = 0,
$$

$$
\hat {f} _ {2 y} ^ {(1)} = - 1 0 \mathrm{kN}, \quad \hat {m} _ {2} ^ {(1)} = 6. 7 \mathrm{kN} \cdot \mathrm{m}
$$

5.11 $d_{2y} = -3.712 \times 10^{-5} \, m,\quad F_{1x} = 5440 \, N,\quad F_{1y} = 10000 \, N,\quad M_{1} = 112 \, N \cdot m$

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5.12 $d_{1x} = -0.2143 \, m,\quad d_{1y} = -0.250 \, m,\quad \phi_{1} = 0.0893 \, rad,\quad d_{2x} = -0.2143 \, m,$ $d_{2y} = -0.357 \times 10^{-4} \, m,\quad \phi_{2} = 0.0714 \, m$

5.13 $d_{2x} = 0.0559$ in., $d_{2y} = 0.00382$ in., $\phi_2 = -0.000150$ rad $d_{3x} = 0.0558$ in., $d_{3y} = -0.000133$ in., $\phi_3 = 0.000149$ rad $F_{1x} = -198$ lb, $F_{1y} = -4770$ lb, $M_1 = 27460$ lb·in. $F_{4x} = -4802$ lb, $F_{4y} = 4770$ lb, $M_4 = 27430$ lb·in.

5.14 $d_{2x}=0.0174$ in., $d_{2y}=-0.0481$ in., $\phi_{2}=-0.00165$ rad $\hat{f}_{1x}^{(1)}=19160\ \mathrm{lb},\quad\hat{f}_{1y}^{(1)}=-1385\ \mathrm{lb},\quad\hat{m}_{1}^{(1)}=-59050\ \mathrm{lb}\cdot\mathrm{in}.$ $\hat{f}_{2x}^{(1)}=-19160\ \mathrm{lb},\quad\hat{f}_{2y}^{(1)}=1385\ \mathrm{lb},\quad\hat{m}_{2}^{(1)}=-176,000\ \mathrm{lb}\cdot\mathrm{in}.$

5.15 $d_{2x} = -1.76 \times 10^{-2} \, m,\quad d_{2y} = -1.87 \times 10^{-5} \, m,\quad \phi_{2} = 5.00 \times 10^{-3} \, rad$ $d_{3x} = -1.76 \times 10^{-2} \, m,\quad \phi_{3} = -2.49 \times 10^{-3} \, rad$ $F_{1x} = 20.0 \, kN,\quad F_{1y} = 13.1 \, kN,\quad M_{1} = -57.4 \, kN \cdot m,\quad F_{3y} = -13.1 \, kN$

5.16 $d_{3y} = -2.83 \times 10^{-5} \, m,\quad d_{4x} = 1.0 \times 10^{-5} \, m,\quad d_{4y} = -2.83 \times 10^{-5} \, m$ 5.17 $d_{3y} = -0.397 \, in., \quad \phi_{3} = 0$

5.18 $d_{2x}=d_{2y}=-0.01\times10^{-3}\ m,\quad\phi_{2}=1.766\times10^{-4}\ rad$

5.19 $d_{1x}=0.702\ in.,\quad d_{1y}=0.00797\ in.,\quad \phi_{1}=-0.00446\ rad$ $\hat{f}_{3x}^{(1)}=-\hat{f}_{1x}^{(1)}=-19.93\ kip,\quad\hat{f}_{3y}^{(1)}=-\hat{f}_{1y}^{(1)}=18.1\ kip,\quad\hat{m}_{3}^{(1)}=1309\ k\cdot\ in.$ $\hat{m}_{1}^{(1)}=863\ k\cdot\ in.$

5.20 $d_{3x}=1.24$ in., $d_{3y}=0.00203$ in., $\phi_{3}=-0.000556$ rad $\hat{f}_{1x}^{(1)}=-2.76\ \text{kip},\quad\hat{f}_{1y}^{(1)}=1.79\ \text{kip},\quad\hat{m}_{1}^{(1)}=0,\quad\hat{f}_{2x}^{(1)}=2.76\ \text{kip},\quad\hat{f}_{2y}^{(1)}=-1.79\ \text{kip},$ $\hat{m}_{2}^{(1)}=322\ k\cdot\text{in}.$

5.21 Use a W16 × 31 for all sections

5.22 $\sigma_{bending\ max} = 11924$ psi

5.23 $d_{5x}=0.0204$ in., $d_{5y}=0.00122$ in., $\phi_{5}=0.000207$ rad

5.24 $d_{5x} = 2.82$ in., $d_{5y} = 0.00266$ in., $\phi_5 = -0.00139$ rad

5.25 a. $d_{2y} = -2.12 \times 10^{-3}$ in. b. $d_{3y} = -6.07 \times 10^{-2}$ in.

5.26 $d_{2x}=0.596\times10^{-5}$ in., $d_{2y}=-0.332\times10^{-2}$ in., $\phi_{2}=-0.100\times10^{-3}$ rad $F_{1x}=130\ lb,\quad F_{1y}=10360\ lb,\quad F_{4x}=-130\ lb,\quad F_{4y}=10360\ lb$

5.27 $d_{3y} = -0.0153$ in., $f_{1x}^{(1)} = 30$ kN, $f_{1y}^{(1)} = -6.67$ kN, $m_{1}^{(1)} = 0$

5.28 $d_{2x} = 5.70\mathrm{mm},\quad d_{2y} = -0.0244\mathrm{mm},\quad \phi_{2} = 0.00523\mathrm{rad}$

5.29 $d_{3y} = -1.83$ in., $d_{4y} = -1.22$ in.

5.30 $d_{3y}=6.67\ in.,\quad d_{4y}=-6.67\ in.,\quad\phi_{3}=-\phi_{4}=-3.20\ rad$ $F_{1x}=11.69\ kN,\quad F_{1y}=30\ kN,\quad M_{1}=-1810\ kN\cdot m$ $F_{6x}=-11.69\ kN,\quad F_{6y}=30\ kN,\quad M_{6}=1810\ kN\cdot m$

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5.31 $d_{2y} = -1.58 \times 10^{-2}$ in.   
5.32 $d_{2x} = 4.30\mathrm{mm},\quad \phi_{2} = -0.241\times 10^{-3}\mathrm{rad}$

$$
F _ {1 x} = - 8 3 3 9 \mathrm{N}, \quad F _ {1 y} = - 4 9 9 5 \mathrm{N}, \quad M _ {1} = 2 6, 7 0 0 \mathrm{N} \cdot \mathrm{m},
$$

$$
F _ {4 x} = - 6 6 6 1 \mathrm{N}, \quad F _ {4 y} = 4 9 9 5 \mathrm{N}, \quad M _ {4} = 2 3, 3 3 0 \mathrm{N} \cdot \mathrm{m}
$$

5.33 $d_{7x} = 0.0264\mathrm{m},\quad d_{7y} = 0.463\times 10^{-4}\mathrm{m},\quad \phi_{7} = 0.171\times 10^{-2}\mathrm{rad}$

$$
\hat {f} _ {1 x} ^ {(1)} = - 2 1. 1 \mathrm{N}, \quad \hat {f} _ {1 y} ^ {(1)} = 3 0. 4 \mathrm{N}, \quad \hat {m} _ {1} ^ {(1)} = 7 4. 9 5 \mathrm{N} \cdot \mathrm{m}
$$

$$
\hat {f} _ {3 x} ^ {(1)} = 2 1. 1 \mathrm{N}, \quad \hat {f} _ {3 y} ^ {(1)} = - 3 0. 4 \mathrm{N}, \quad \hat {m} _ {3} ^ {(1)} = 4 6. 6 5 \mathrm{N} \cdot \mathrm{m}
$$

5.35 $d_{9x}=0.0174\ m,\quad\hat{f}_{1x}^{(1)}=-22.6\ kN,\quad\hat{f}_{1y}^{(1)}=16.0\ kN,\quad\hat{m}_{1}^{(1)}=53.6\ kN\cdot m$

$$
\hat {f} _ {3 x} ^ {(1)} = 2 2. 6 \mathrm{kN}, \quad \hat {f} _ {3 y} ^ {(1)} = - 1 6. 0 \mathrm{kN}, \quad \hat {m} _ {3} ^ {(1)} = 4 2. 4 \mathrm{kN} \cdot \mathrm{m}
$$

5.36 $d_{6y} = -2.80 \times 10^{-7} \, m,\quad d_{7y} = -4.87 \times 10^{-7} \, m$   
5.37 $d_{5\mathrm{y}} = -1.29\times 10^{-2}\mathrm{m}$   
5.38 $d_{2x}=1.43\times10^{-1}$ m   
5.39 Truss: $d_{7x}=0.0260\ m,\quad d_{7y}=0.00566\ m,$

$$
\text { Frame: } d _ {7 x} = 0. 0 1 8 0 \mathrm{m}, \quad d _ {7 y} = 0. 0 0 4 2 4 \mathrm{m}
$$

$$
\text { Truss,   element   1:   } \hat {f} _ {1 x} = - 4 9, 7 3 0 \text {   N,   } \hat {f} _ {1 y} = 0
$$

$$
\text { Frame,   element   1: } \hat {f} _ {1 x} = - 4 3, 0 6 0 \mathrm{N}, \quad \hat {f} _ {1 y} = 2 2 6 7 0 \mathrm{N}
$$

5.40 $d_{max} = -0.0105 \, m$ at midspan

$$
\mathrm{M} _ {\max} = 1. 5 6 8 \times 1 0 ^ {6} \mathrm{N} - \mathrm{m} \quad \text { at   C }
$$

5.41 $d_{max} = 0.0524 \, m$

$$
\mathrm{M} _ {\max} = 6. 2 2 \times 1 0 ^ {4} \mathrm{N} - \mathrm{m}
$$

5.45 Tapered beam $n = 3$

$$
\text { one   element: } \quad d _ {1 y} = - 0. 2 2 2 \times 1 0 ^ {- 1} \text {   in. }
$$

$$
\text { two   elements: } \quad d _ {1 y} = - 0. 1 8 9 \times 1 0 ^ {- 1} \text { in. }
$$

$$
\text { four   elements: } d _ {1 y} = - 0. 1 8 1 \times 1 0 ^ {- 1} \text { in. }
$$

$$
\text { eight   elements:   } d _ {1 y} = - 0. 1 7 9 \times 1 0 ^ {- 1} \text {   in. }
$$

5.46 $\underline{K}=15\frac{GJ_{0}}{L}\begin{bmatrix}1&-1\\ -1&1\end{bmatrix}$

5.48 $d_{2y} = -0.214$ in.   
5.49 $d_{2y} = -0.729$ in.   
5.51 $d_{1y} = -0.690 \times 10^{-2} \mathrm{~m}$   
5.52 $d_{5y} = -0.1776$ in.   
5.53 $d_{4y} = -1.026$ in.   
5.55 $d_{3y} = -2.54 \times 10^{-3} \mathrm{~m}$   
5.57 $d_{5y} = -2.22 \times 10^{-2} \mathrm{~m}$

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5.58 $d_{2y}=0.491$ in., $d_{3z}=0.837$ in.   
5.59 $d_{7z} = -0.251$ in.

# Chapter 6

6.1 Use Eq. (6.2.10) in Eq. (6.2.18) to show $N_{i} + N_{j} + N_{m} = 1$ .

6.3 a. $k = 4.0 \times 10^{6}$ $\begin{bmatrix} 2.5 & 1.25 & -2.0 & -1.5 & -0.5 & 0.25 \\ & 4.375 & -1.0 & -0.75 & -0.25 & -3.625 \\ & & 4.0 & 0 & -2.0 & 1.0 \\ & & & 1.5 & 1.5 & -0.75 \\ & & & & 2.5 & -1.25 \\ & & & & & 4.375 \end{bmatrix}$ lb/in.

Symmetry

$\mathbf{b}.\underline{k} = 13.33\times 10^{6}\left[ \begin{array}{cccccc}1.54 & 0.75 & -1.0 & -0.45 & -0.54 & -0.3\\ & 1.815 & -0.3 & -0.375 & -0.45 & -1.44\\ & & 1.0 & 0 & 0 & 0.3\\ & & & 0.375 & 0.45 & 0\\ & & & & 0.54 & 0\\ & & & & & 1.44 \end{array} \right]$ lb/in. Symmetry

6.4 a. $\sigma_{x} = 19.2\mathrm{ksi}$ , $\sigma_{y} = 4.8\mathrm{ksi}$ , $\tau_{xy} = -15.0\mathrm{ksi}$

$$
\sigma_ {1} = 2 8. 6 \mathrm{ksi}, \quad \sigma_ {2} = - 4. 6 4 \mathrm{ksi}, \quad \theta_ {p} = - 3 2. 2 ^ {\circ}
$$

b. $\sigma_{x} = 32.0\mathrm{ksi}$ , $\sigma_{y} = 8.0\mathrm{ksi}$ , $\tau_{xy} = -25.0\mathrm{ksi}$

$$
\sigma_ {1} = 4 7. 7 \mathrm{ksi}, \quad \sigma_ {2} = - 7. 7 3 \mathrm{ksi}, \quad \theta_ {p} = - 3 2. 2 ^ {\circ}
$$

6.6 a. $\underline{k}=2.074\times10^{5}\left[\begin{array}{cccccc}8437.5 & 1687.5 & -7762.5 & -337.5 & -675 & -1350 \\ 1687.5 & 3937.5 & 337.5 & -2137.5 & -2025 & -1800 \\ -7762.5 & 337.5 & 8437.5 & -1687.5 & -675 & 1350 \\ -337.5 & -2137.5 & -1687.5 & 3937.5 & 2025 & -1800 \\ -675 & -2025 & -675 & 2025 & 1350 & 0 \\ -1350 & -1800 & 1350 & -1800 & 0 & 3600\end{array}\right]N/m$

$\mathbf{b}.\quad \underline{k} = 4.48\times 10^{7}\left[ \begin{array}{cccccc}25.0 & 0 & -12.5 & 6.25 & -12.5 & -6.25\\ & 9.375 & 9.375 & -4.6875 & -9.375 & -4.6875\\ & & 15.625 & -7.8125 & -3.125 & -1.5625\\ & & & 27.343 & 1.5625 & -3.125\\ & & & & 15.625 & 7.8125\\ \text{Symmetry} \end{array} \right]\mathrm{N / m}$

6.7 a. $\sigma_{x} = -5.289$ GPa, $\sigma_{y} = -0.156$ GPa, $\tau_{xy} = 0.233$ GPa

$$
\sigma_ {1} = - 0. 1 4 5 9 \mathrm{GPa}, \quad \sigma_ {2} = - 5. 3 0 \mathrm{GPa}, \quad \theta_ {p} = - 2. 5 9 ^ {\circ}
$$

b. $\sigma_{x} = 0$ $\sigma_{y} = 42.0\mathrm{MPa}$ $\tau_{xy} = 33.6\mathrm{MPa}$

$$
\sigma_ {1} = 6 0. 6 \mathrm{MPa}, \quad \sigma_ {2} = - 1 8. 6 \mathrm{MPa}, \quad \theta_ {p} = - 2 9 ^ {\circ}
$$

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6.9 a. $\sigma_{x} = -15.0\mathrm{ksi}$ , $\sigma_{y} = -45.0\mathrm{ksi}$ , $\tau_{xy} = -18.0\mathrm{ksi}$

$$
\sigma_ {1} = - 6. 5 7 \mathrm{ksi}, \quad \sigma_ {2} = - 5 3. 4 \mathrm{ksi}, \quad \theta_ {p} = - 2 5. 1 ^ {\circ}
$$

b. $\sigma_{x} = -15.0$ ksi, $\sigma_{y} = -45$ ksi, $\tau_{xy} = -21.0$ ksi

$$
\sigma_ {1} = - 4. 1 9 \mathrm{ksi}, \quad \sigma_ {2} = - 5 5. 8 \mathrm{ksi}, \quad \theta_ {p} = - 2 7. 2 ^ {\circ}
$$

c. $\sigma_{x} = -30$ ksi, $\sigma_{y} = -90$ ksi, $\tau_{xy} = -21$ ksi

$$
\sigma_ {1} = - 2 3. 3 8 \mathrm{ksi}, \quad \sigma_ {2} = - 9 6. 6 \mathrm{ksi}, \quad \theta_ {p} = - 1 7. 4 7 ^ {\circ}
$$

f. $\sigma_{x} = -22.5\mathrm{ksi}$ , $\sigma_{y} = -67.5\mathrm{ksi}$ , $\tau_{xy} = -21.0\mathrm{ksi}$

$$
\sigma_ {1} = - 1 4. 2 \mathrm{ksi}, \quad \sigma_ {2} = - 7 5. 8 \mathrm{ksi}, \quad \theta_ {p} = - 2 1. 5 ^ {\circ}
$$

6.10 a. $\sigma_{x} = -52.5\mathrm{MPa}$ , $\sigma_{y} = -32.8\mathrm{MPa}$ , $\tau_{xy} = -5.38\mathrm{MPa}$

$$
\sigma_ {1} = - 3 1. 4 \mathrm{MPa}, \quad \sigma_ {2} = - 5 3. 9 \mathrm{MPa}, \quad \theta_ {p} = - 1 4. 3 ^ {\circ}
$$

b. $\sigma_{x} = -31.4\mathrm{MPa}$ , $\sigma_{y} = -13.5\mathrm{MPa}$ , $\tau_{xy} = 5.38\mathrm{MPa}$

$$
\sigma_ {1} = - 1 2. 0 \mathrm{MPa}, \quad \sigma_ {2} = - 3 2. 9 \mathrm{MPa}, \quad \theta_ {p} = - 1 5. 5 ^ {\circ}
$$

c. $\sigma_{x} = -27.6\mathrm{MPa}$ , $\sigma_{y} = -19.5\mathrm{MPa}$ , $\tau_{xy} = 4.04\mathrm{MPa}$

$$
\sigma_ {1} = - 1 7. 9 \mathrm{MPa}, \quad \sigma_ {2} = - 2 9. 3 \mathrm{MPa}, \quad \theta_ {p} = - 2 2. 5 ^ {\circ}
$$

d. $\sigma_{x} = -31.6$ MPa, $\sigma_{y} = -28.9$ MPa, $\tau_{xy} = -6.73$ MPa

$$
\sigma_ {1} = - 2 3. 0 \mathrm{MPa}, \quad \sigma_ {2} = - 3 8. 0 \mathrm{MPa}, \quad \theta_ {p} = 3 9 ^ {\circ}
$$

6.11 a. $f_{s1x} = 0, \quad f_{s1y} = 0, \quad f_{s2x} = p_0Lt / 6, \quad f_{s2y} = 0$

$$
f _ {s 3 x} = p _ {0} L t / 3, \quad f _ {s 3 y} = 0
$$

b. $f_{s1x}=0,\quad f_{s2x}=p_{0}Lt/12,\quad f_{s3x}=p_{0}Lt/4$

6.12 b. $f_{s1y} = f_{s2y} = p_{0}Lt/\pi$

6.13 $d_{3x}=0.5\times10^{-3}$ in., $d_{3y}=-0.275\times10^{-2}$ in.

$$
d _ {4 x} = - 0. 6 0 9 \times 1 0 ^ {- 3} \text {   in. }, \quad d _ {4 y} = - 0. 2 9 3 \times 1 0 ^ {- 2} \text {   in. }
$$

$$
\sigma_ {x} ^ {(1)} = 8 2 4 \mathrm{psi}, \quad \sigma_ {y} ^ {(1)} = 2 4 7 \mathrm{psi}, \quad \tau_ {x y} ^ {(1)} = - 1 5 8 7 \mathrm{psi}
$$

$$
\sigma_ {1} ^ {(1)} = 2 1 4 9 \mathrm{psi}, \quad \sigma_ {2} ^ {(1)} = - 1 0 7 7 \mathrm{psi}, \quad \theta_ {p} ^ {(1)} = - 4 0 ^ {\circ}
$$

$$
\sigma_ {x} ^ {(2)} = - 8 2 6 \text {   psi }, \quad \sigma_ {y} ^ {(2)} = 2 9 2 \text {   psi }, \quad \tau_ {x y} ^ {(2)} = - 4 1 1 \text {   psi }
$$

$$
\sigma_ {1} ^ {(2)} = 4 2 6 \mathrm{psi}, \quad \sigma_ {2} ^ {(2)} = - 9 6 0 \mathrm{psi}, \quad \theta_ {p} ^ {(2)} = 1 8. 1 5 ^ {\circ}
$$

6.14 a. $d_{2x}=0.281\times10^{-4}$ m, $d_{2y}=-0.330\times10^{-4}$ m

$$
d _ {5 x} = 0. 1 1 5 \times 1 0 ^ {- 4} \mathrm{m}, \quad d _ {5 y} = - 0. 1 0 3 \times 1 0 ^ {- 4} \mathrm{m}
$$

$$
\sigma_ {x} ^ {(2)} = 1 6. 4 \mathrm{MPa}, \quad \sigma_ {y} ^ {(2)} = 1 5. 2 \mathrm{MPa}
$$

$$
\tau_ {x y} ^ {(2)} = - 6. 9 9 \mathrm{MPa}, \quad \sigma_ {1} ^ {(2)} = 2 2. 8 \mathrm{MPa}
$$

$$
\sigma_ {2} ^ {(2)} = 8. 8 0 \mathrm{MPa}, \quad \theta_ {p} ^ {(2)} = - 4 2. 7 ^ {\circ}
$$

$$
\sigma_ {x} ^ {(1)} = 1 0. 6 \mathrm{MPa}, \quad \sigma_ {y} ^ {(1)} = 3. 1 8 \mathrm{MPa}
$$

$$
\tau_ {x y} ^ {(1)} = - 3. 3 4 \mathrm{MPa}, \quad \sigma_ {1} ^ {(1)} = 1 1. 9 \mathrm{MPa}
$$

$$
\sigma_ {2} ^ {(1)} = 1. 9 0 \mathrm{MPa}, \quad \theta_ {p} ^ {(1)} = - 2 1. 0 ^ {\circ}
$$

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$\mathbf b . ~ d _ { 1 x } = - d _ { 2 x } = - 0 . 1 6 5 \times 1 0 ^ { - 5 } \mathrm { m } , ~ d _ { 1 y } = d _ { 2 y } = - 0 . 1 2 5 \times 1 0 ^ { - 4 } \mathrm { m }$

$$
d _ {5 x} = 0. 2 7 4 \times 1 0 ^ {- 1 2} \mathrm{m}, d _ {5 y} = - 0. 1 6 3 \times 1 0 ^ {- 4} \mathrm{m}
$$

$$
\sigma_ {x} ^ {(1)} = 5. 9 9 \times 1 0 ^ {5} \mathrm{N} / \mathrm{m} ^ {2}, \quad \sigma_ {y} ^ {(1)} = - 3. 7 8 \times 1 0 ^ {6} \mathrm{N} / \mathrm{m} ^ {2}
$$

$$
\tau_ {x y} ^ {(1)} = 4. 0 5 \times 1 0 ^ {- 1} \mathrm{N} / \mathrm{m} ^ {2}, \quad \sigma_ {1} ^ {(1)} = 5. 9 9 \times 1 0 ^ {5} \mathrm{N} / \mathrm{m} ^ {2}
$$

$$
\sigma_ {2} ^ {(1)} = - 3. 7 8 \times 1 0 ^ {6} \mathrm{N} / \mathrm{m} ^ {2}, \quad \theta_ {p} ^ {(1)} = 0 ^ {\circ}, \quad \sigma_ {x} ^ {(3)} = 5. 6 4 \times 1 0 ^ {6} \mathrm{N} / \mathrm{m} ^ {2}
$$

$$
\sigma_ {y} ^ {(3)} = 1. 8 8 \times 1 0 ^ {7} \mathrm{N} / \mathrm{m} ^ {2}, \quad \tau_ {x y} ^ {(3)} = - 1. 1 1 \times 1 0 ^ {- 1} \mathrm{N} / \mathrm{m} ^ {2}
$$

$$
\sigma_ {1} ^ {(3)} = 1. 8 8 \times 1 0 ^ {7} \mathrm{N} / \mathrm{m} ^ {2}, \quad \sigma_ {2} ^ {(3)} = 5. 6 4 \times 1 0 ^ {6} \mathrm{N} / \mathrm{m} ^ {2}, \quad \theta_ {p} ^ {(3)} = - 9 0 ^ {\circ}
$$

6.15 All $f _ { b x } \mathbf { \bar { s } }$ are equal to 0.

a. $f _ { b 1 y } = f _ { b 2 y } = f _ { b 3 y } = f _ { b 4 y } = - 1 0 . 2 8 \ \mathrm { N } , \quad f _ { b 5 y } = - 2 0 . 5 6 \ \mathrm { N }$

b. $f _ { b 1 y } = f _ { b 2 y } = f _ { b 3 y } = f _ { b 4 y } = - 8 . 0 3 \ \mathrm { N } , \quad f _ { b 5 y } = - 1 6 . 0 6 \ \mathrm { N }$

6.18 b. Yes, c. Yes g. No

6.20 a. $n _ { b } = 8 ,$ b. $n _ { b } = 1 2$

# Chapter 7

7.9 $d _ { 2 x } = d _ { 3 x } = 0 . 6 4 7 \times 1 0 ^ { - 3 } \ \mathrm { i n . , } \quad d _ { 2 y } = 0 . 6 6 6 \times 1 0 ^ { - 4 } \ \mathrm { i n . }$

$d _ { 3 y } = - 0 . 6 6 6 \times 1 0 ^ { - 4 } \mathrm { i t }$ n., skew effect

7.10 Stress approaches 2.5 psi near edge of whole for model of 70 nodes, 54 elements.

7.11 At depth 4 in. equal to width, stress approaches uniform $\sigma _ { y } = - 1 0 0 0 \ \mathrm { p s i }$ i.

7.12 $\sigma _ { 1 } = 8 8 3 6$ psi at top and bottom of hole

7.13 $\sigma _ { 1 } = 3 7 2$ psi at fillet

7.14 For refined mesh at re-entrant corner, $\sigma _ { 1 } = 2 0 1 6 0 \mathrm { p s i }$ .

7.15 $\sigma _ { V M } = 9 3 . 7 $ psi at load

7.17 For the model with 12 in. $\times \textstyle { \frac { 1 } { 2 } }$ in. size elements, finite element solution yields free-end deflection of -0:499 in.; exact solution is -1:15 in. (See Table 7–1 in text for other results.)

7.19 $\sigma _ { 1 } = 3 ~ \mathrm { k N } / \mathrm { m } ^ { 2 }$ (round hole model)

$\sigma _ { 1 } = 3 . 5 1 \mathrm { ~ k N } / \mathrm { m } ^ { 2 }$ (square hole with corner radius)

7.21 $\sigma _ { V M } = 8 . 1 \ \mathrm { M P a }$

7.22 $\mathbf { a . \ } \sigma _ { 1 } = 5 8 7 0 0 \ \mathrm { p s i }$

7.23 s1 ¼ 19 MPa at hole

7.25 Largest von Mises stress 35–45 MPa at inside edge at junction of narrow to larger section of wrench

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7.27 Largest principal stress $\sigma_{1}=1005$ MPa at narrowest width of member (70-element, 94-node model)   
7.35 For a 1 cm thick wrench, $\sigma_{VM} = 502$ MPa

# Chapter 8

8.2 $\varepsilon_{x} = \frac{1}{3b} (-u_{1} + u_{2} + 4u_{4} - 4u_{5}),\varepsilon_{y} = \frac{1}{3h} (-v_{1} + v_{3} + 4v_{4} - 4v_{6})$

$$
\gamma_ {x y} = \frac {1}{3 h} \left(- u _ {1} + u _ {3} + 4 u _ {4} - 4 u _ {6}\right) + \frac {1}{3 b} \left(- v _ {1} + v _ {3} + 4 v _ {4} - 4 v _ {6}\right)
$$

$$
\sigma_ {x} = \frac {E}{1 - \nu^ {2}} (\varepsilon_ {x} + \nu \varepsilon_ {y}), \quad \sigma_ {y} = \frac {E}{1 - \nu^ {2}} (\varepsilon_ {y} + \nu \varepsilon_ {x}), \quad \tau_ {x y} = G \gamma_ {x y}
$$

8.3 $f_{s1x} = f_{s3x} = \frac{-pth}{6}, \quad f_{s5x} = \frac{-2pth}{3}$   
8.4 $f_{s1x} = 0,\quad f_{s3x} = \frac{-p_0\mathrm{th}}{6},\quad f_{s5x} = \frac{-p_0th}{3}$   
8.5 a. $\varepsilon_{x} = -5\times 10^{-5}y + 2.5\times 10^{-4},\quad \varepsilon_{y} = -1.67\times 10^{-4}x + 3.33\times 10^{-5},$

$$
\gamma_ {x y} = - 5 \times 1 0 ^ {- 5} x - 1. 1 1 \times 1 0 ^ {- 4} y + 4. 1 7 \times 1 0 ^ {- 4}
$$

$$
\sigma_ {x} = 3 2 9 0 \mathrm{psi}, \quad \sigma_ {y} = - 4 8 5 0 \mathrm{psi}, \quad \tau_ {x y} = 1 5 4 0 \mathrm{psi}
$$

b. $\varepsilon_{x} = -5\times 10^{-5}y + 1.67\times 10^{-4},\quad \varepsilon_{y} = -1.67\times 10^{-4}x + 5\times 10^{-5}$

$$
\gamma_ {x y} = - 5 \times 1 0 ^ {- 5} x - 4. 1 7 \times 1 0 ^ {- 5} y + 2. 0 8 \times 1 0 ^ {- 4}
$$

$$
\sigma_ {x} = 9 2 8 \text { psi }, \quad \sigma_ {y} = - 8 2 9 0 \text { psi }, \quad \tau_ {x y} = 6 3 2 \text { psi }
$$

8.6 $\varepsilon_{x} = 2.54\times 10^{-3}$

$$
\varepsilon_ {y} = - 7. 6 2 \times 1 0 ^ {- 3}
$$

$$
\gamma_ {x y} = - 7. 0 4 \times 1 0 ^ {- 3}
$$

8.7 $N_{1} = 1 - \frac{x}{20} +\frac{x^{2}}{1800},\quad N_{2} = \frac{-x + y}{60} +\frac{x^{2} + y^{2}}{1800} -\frac{xy}{900}$

$$
N _ {3} = \frac {- y}{6 0} + \frac {y ^ {2}}{1 8 0 0}, \quad N _ {4} = \frac {x y}{9 0 0} - \frac {y ^ {2}}{9 0 0}, \quad N _ {5} = \frac {y}{1 5} - \frac {x y}{9 0 0}, \text { etc. }
$$

# Chapter 9

9.1 a. $\underline{K}=25.132\times10^{6}\left[\begin{array}{cccccc}5 & 1 & 0 & -1 & 1 & 0 \\ 1 & 4 & -2 & -1 & -2 & -3 \\ 0 & -2 & 8 & 0 & 4 & 2 \\ -1 & -1 & 0 & 1 & 1 & 0 \\ 1 & -2 & 4 & 1 & 4 & 1 \\ 0 & -3 & 2 & 0 & 1 & 3\end{array}\right]$ lb/in.

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$$
\mathbf {b}. \underline {{K}} = 5 0. 2 6 5 \times 1 0 ^ {6} \left[ \begin{array}{c c c c c c} 2. 7 5 & 0 & - 2. 2 5 & 0. 5 & 0. 2 5 & - 0. 5 \\ 0 & 1 & 1 & - 1 & - 1 & 0 \\ - 2. 2 5 & 1 & 5. 7 5 & - 2. 5 & 0. 2 5 & 1. 5 \\ 0. 5 & - 1 & - 2. 5 & 4 & 0. 5 & - 3 \\ 0. 2 5 & - 1 & 0. 2 5 & 0. 5 & 1. 7 5 & 0. 5 \\ - 0. 5 & 0 & 1. 5 & - 3 & 0. 5 & 3 \end{array} \right] \mathrm{lb/in}.
$$

$$
f _ {s 2 r} = \frac {2 \pi b p _ {0} h}{6}, \quad f _ {s 3 r} = \frac {2 \pi b p _ {0} h}{3} \tag {9.2}
$$

$$
\begin{array}{l l} \text {9.3} & f _ {b 1 r} = f _ {b 2 r} = f _ {b 3 r} = 0. 3 8 2 \mathrm{lb} \\ & f _ {b 1 z} = f _ {b 2 z} = f _ {b 3 z} = - 6. 3 2 \mathrm{lb} \end{array}
$$

$$
\begin{array}{l l} \textbf {9 . 4} & \mathbf {a}. \sigma_ {r} = 8 0 0 0 \mathrm{psi}, \sigma_ {z} = 0, \sigma_ {\theta} = 8 0 0 0 \mathrm{psi}, \tau_ {r z} = 1 2 0 0 \mathrm{psi} \\ & \mathbf {b}. \sigma_ {r} = 5 8 3 0 \mathrm{psi}, \sigma_ {z} = - 3 7 7 0 \mathrm{psi}, \sigma_ {\theta} = 3 0 9 0 \mathrm{psi}, \tau_ {r z} = 4 0 0 \mathrm{psi} \end{array}
$$

$$
\mathbf {9 . 6} \quad \mathbf {a .} \underline {{k}} = 7. 0 3 7 \left[ \begin{array}{c c c c c c} 3 1 2 5 & 6 2 5 & 0 & - 6 2 5 & 6 2 5 & 0 \\ & 2 5 0 0 & - 1 2 5 0 & - 6 2 5 & - 1 2 5 0 & - 1 8 7 5 \\ & & 5 0 0 0 & 0 & 2 5 0 0 & 1 2 5 0 \\ & & & 6 2 5 & 6 2 5 & 0 \\ & & & & 2 5 0 0 & 6 2 5 \\ \text {Symmetry} & & & & & 1 8 7 5 \end{array} \right] \mathrm{kN/mm}
$$

$$
\mathbf {b}. \underline {{k}} = 1 1. 7 3 \left[ \begin{array}{c c c c c c} 2 4 7 5 & 0 & - 2 0 2 5 & 4 5 0 & 2 2 5 & - 4 5 0 \\ & 9 0 0 & 9 0 0 & - 9 0 0 & - 9 0 0 & 0 \\ & & 5 1 7 5 & - 2 2 5 0 & 2 2 5 & 1 3 5 0 \\ & & & 3 6 0 0 & 4 5 0 & - 2 7 0 0 \\ & & & & 1 5 7 5 & 4 5 0 \\ \text { Symmetry } & & & & & 2 7 0 0 \end{array} \right] \mathrm{kN/mm}
$$

$$
\begin{array}{l} \textbf {9 . 7} \quad \mathbf {a}. \sigma_ {r} = - 8 4 \mathrm{MPa}, \quad \sigma_ {z} = - 8 4 \mathrm{MPa}, \quad \sigma_ {\theta} = 2 5 2 \mathrm{MPa}, \quad \tau_ {r z} = - 1 0 1 \mathrm{MPa} \\ \mathbf {b}. \sigma_ {r} = - 1 0 3 \mathrm{MPa}, \quad \sigma_ {z} = - 1 0 3 \mathrm{MPa}, \quad \sigma_ {\theta} = 1 1 2 \mathrm{MPa}, \quad \tau_ {r z} = - 7 3 \mathrm{MPa} \end{array}
$$

9.14 Using 0.5 in. radii in corners, $\sigma _ { 1 } = 7 5 9 0$ psi at inside corner

9.18 $\sigma _ { 1 } = 4 6 2 1$ psi outer edge of hole, along axis of symmetry

9.19 $\sigma _ { \theta } = 2 2 , 7 1 1$ psi, $\sigma _ { r } = - 4 9 8 4$ psi, $u _ { r } = 0 . 0 3 7$ in.

9.20 s1 ¼ 64:1 MPa, u ¼ 0:0782 m top and bottom center of plates

9.24 $\sigma _ { V M } = 5 2 2 1$ psi at fillet, $\sigma _ { V M } = 1 6 3 7 . 5$ psi at groove

# Chapter 10

10.2 $\mathbf { a } , s = - { \frac { 1 } { 5 } } ;$ ; b. $N _ { 1 } = 0 . 4 , \quad N _ { 2 } = 0 . 6$

10.3 $\mathbf { a } , s = 0 ,$ b. $N _ { 1 } = 0 . 5 , \quad N _ { 2 } = 0 . 5$

10.5 ${ \bf a } . \ s = - 0 . 5 ,$ b. $N _ { 1 } = 0 . 3 7 5 .$ $N _ { 2 } = - 0 . 1 2 5$ ; $N _ { 3 } = 0 . 7 5$

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10.8 $d_{2x} = 4.859 \times 10^{-4}\mathrm{m}$ (right end), $d_{3x} = 2.793 \times 10^{-4}\mathrm{m}$ (center)

10.10 $\varepsilon_{x} = 0.0009375$ in./in., $\varepsilon_{y} = -0.00125$ in./in., $\gamma_{xy} = -0.000625$ rad

$$
\sigma_ {x} = 1 8. 5 \mathrm{ksi}, \quad \sigma_ {y} = - 3 1. 9 \mathrm{ksi}, \quad \tau_ {x y} = - 7. 2 1 \mathrm{ksi}
$$

10.15 a. $f_{s3t} = 500 \, \mathrm{lb}$ , $f_{s4t} = 500 \, \mathrm{lb}$ , b. $f_{s1t} = 83.33 \, \mathrm{lb}$ , $f_{s4t} = 41.67 \, \mathrm{lb}$

10.16 a. 1.917, b. 0.667, c. 0.400, d. 2.87, f. 0

# Chapter 11

11.1 a.

$$
\underline {{B}} = \frac {1}{8} \left[ \begin{array}{c c c c c c c c c c c c} 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 4 \\ 0 & 0 & 0 & 4 & 0 & 0 & 0 & 4 & 0 & - 4 & - 4 & 0 \\ 0 & 4 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & - 4 & - 4 \\ 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & - 4 & 0 & - 4 \end{array} \right]
$$

11.3 $\sigma_{x} = 77.9\mathrm{ksi}$ , $\sigma_{y} = 8.65\mathrm{ksi}$ , $\sigma_{z} = -49.0\mathrm{ksi}$

$$
\tau_ {x y} = 1 1. 5 \mathrm{ksi}, \quad \tau_ {y z} = - 2 3. 1 \mathrm{ksi}, \quad \tau_ {z x} = 5. 7 7 \mathrm{ksi}
$$

11.6 a. $\underline{B} = \frac{1}{18750}$

$$
\times \left[ \begin{array}{c c c c c c c c c c c c} - 6 2 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 2 5 & 0 & 0 \\ 0 & - 3 7 5 & 0 & 0 & 7 5 0 & 0 & 0 & 0 & 0 & 0 & - 3 7 5 & 0 \\ 0 & 0 & - 3 7 5 & 0 & 0 & 0 & 0 & 0 & 7 5 0 & 0 & 0 & - 3 7 5 \\ - 3 7 5 & - 6 2 5 & 0 & 7 5 0 & 0 & 0 & 0 & 0 & 0 & - 3 7 5 & 6 2 5 & 0 \\ 0 & - 3 7 5 & - 3 7 5 & 0 & 0 & 7 5 0 & 0 & 7 5 0 & 0 & 0 & - 3 7 5 & - 3 7 5 \\ - 3 7 5 & 0 & - 6 2 5 & 0 & 0 & 0 & 7 5 0 & 0 & 0 & - 3 7 5 & 0 & 6 2 5 \end{array} \right]
$$

11.7 $\sigma_{x} = 72.7\mathrm{MPa},\quad \sigma_{y} = 169.6\mathrm{MPa},\quad \sigma_{z} = 72.7\mathrm{MPa}$

$$
\tau_ {x y} = 5 9. 2 \mathrm{MPa}, \quad \tau_ {y z} = 3 2. 3 \mathrm{MPa}, \quad \tau_ {z x} = 9 1. 5 \mathrm{MPa}
$$

11.10 $N_{2} = \frac{(1 - s)(1 - t)(1 - z^{\prime})}{8},\quad N_{3} = \frac{(1 - s)(1 + t)(1 - z^{\prime})}{8},$

$$
N _ {4} = \frac {(1 - s) (1 + t) (1 + z ^ {\prime})}{8},
$$

$$
N _ {5} = \frac {(1 + s) (1 - t) (1 + z ^ {\prime})}{8}, \quad N _ {6} = \frac {(1 + s) (1 - t) (1 - z ^ {\prime})}{8},
$$

$$
N _ {7} = \frac {(1 + s) (1 + t) (1 - z ^ {\prime})}{8}, \quad N _ {8} = \frac {(1 + s) (1 + t) (1 + z ^ {\prime})}{8}
$$

11.11 $N_{1}=\frac{(1-s)(1-t)(1+z')(-s-t+z'-2)}{8}$ ,

$$
N _ {2} = \frac {(1 - s) (1 - t) (1 - z ^ {\prime}) (- s - t - z ^ {\prime} - 2)}{8}
$$

11.13 $d_{max} = -0.662$ in. under the load

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# Chapter 13

13.1 $t_2 = 166.7^{\circ}\mathrm{C}, t_3 = 233.3^{\circ}\mathrm{C}$   
13.2 $t_2 = 150^{\circ}\mathrm{F},\quad t_3 = 100^{\circ}\mathrm{F},\quad t_4 = 50^{\circ}\mathrm{F}$   
13.3 $t_2 = 875^{\circ}\mathrm{F},\quad t_3 = 1250^{\circ}\mathrm{F},\quad F_1 = -180\mathrm{Btu / h}$   
13.4 $t_1 = 151^{\circ}\mathrm{F}, t_2 = 148^{\circ}\mathrm{F}, t_3 = 140^{\circ}\mathrm{F}, t_4 = 125^{\circ}\mathrm{F}$   
13.5 $t_2 = 183^\circ \mathrm{F}, t_3 = 267^\circ \mathrm{F}, t_4 = 350^\circ \mathrm{F}, t_5 = 433^\circ \mathrm{F}$   
13.6 $t_{2}=421^{\circ}C,\quad t_{3}=121^{\circ}C,\quad q^{(3)}=3975\ W/m^{2}$   
13.7 $t_2 = 418.2^{\circ}\mathrm{C}, t_3 = 527.3^{\circ}\mathrm{C}$   
13.8 $t_{2}=20^{\circ}C,\quad t_{3}=20^{\circ}C,\quad\bar{q}_{\max}=0.0009\mathrm{W},\quad\bar{q}_{\min}=-0.0009\mathrm{W}$   
13.9 6°C at center of wall, $\bar{q}_{max} = 5.54$ W, $\bar{q}_{min} = -5.54$ W   
13.11 185°C at right end, $\bar{q}_{max} = 439$ W   
13.14 $t_0 = 92.25^\circ \mathrm{C}, t_1 = 88.575^\circ \mathrm{C}, t_2 = 84.9^\circ \mathrm{C}, t_3 = 80^\circ \mathrm{C}$   
13.16 $\underline{k}=\frac{AK_{xx}}{L}\begin{bmatrix}1&-1\\ -1&1\end{bmatrix}$   
13.18 $\underline{k}=\left[\begin{array}{ccc}39.57 & 7.076 & -5.417 \\ & 35.82 & -1.667 \\ & & 7.083\end{array}\right]$ , $\underline{f}=\left\{\begin{array}{c}2936 \\ 2936 \\ 50\end{array}\right\}$ Btu/h   
13.19 $\underline{f}=\left\{\begin{array}{c}1291\\27.3\\1254\end{array}\right\}$ W   
13.22 $t_4 = 75^{\circ}\mathrm{F},\quad t_5 = 25^{\circ}\mathrm{F}$   
13.36 12°C at 2.5 cm from top, 25°C 1.25 cm from top, $\bar{q}_{max} = 1416$ W, $\bar{q}_{min} = -1083$ W   
13.41 $\bar{q}_{\mathrm{max}} = 3457\mathrm{W},\bar{q}_{\mathrm{min}} = -3848\mathrm{W}$

# Chapter 14

14.1 $p_{2} = 4.545 \mathrm{~m}, \quad p_{3} = 1.818 \mathrm{~m}, \quad v_{x}^{(1)} = 10.91 \mathrm{~m/s}, \quad Q_{f}^{(1)} = 21.82 \mathrm{~m}^{3}/\mathrm{s}$   
14.2 $p_{2} = -15 \, m,\quad p_{3} = -40 \, m,\quad p_{4} = -65 \, m,\quad v_{x}^{(1)} = 25 \, m/s,\quad Q_{1} = 50 \, m^{3}/s$   
14.3 $p_2 = 8.182 \text{ in.}, \quad p_3 = 5.455 \text{ in.}, \quad v_x^{(1)} = 0.182 \text{ in./s}, \quad v_x^{(2)} = 0.273 \text{ in./s}, \quad v_x^{(3)} = 0.545 \text{ in./s}, \quad Q_f^{(1)} = 1.091 \text{ in}^3/\text{s}$   
14.4 $p_{2} = -3\mathrm{cm},\quad p_{3} = -8\mathrm{cm},\quad v_{x}^{(1)} = 1.2\mathrm{cm / s},\quad v_{x}^{(2)} = 2\mathrm{cm / s},$ $Q_{1} = Q_{2} = 6\mathrm{cm}^{3} / \mathrm{s}$   
14.6 $v^{(1)} = 2.0$ in./s, $v^{(2)} = 4.0$ in./s, $Q^{(1)} = Q^{(2)} = 4$ in $^3$ /s   
14.7 $\underline{f}_{Q}=\left\{\begin{array}{l}54.76\\28.57\\16.67\end{array}\right\}m^{3}/s$