김경종
29 KiB
\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
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
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}
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}
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}
\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
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.
\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
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
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