김경종
1175 lines
32 KiB
Markdown
1175 lines
32 KiB
Markdown
<!-- source-page: 811 -->
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14.8 $f _ { 1 } = f _ { 3 } = 5 \mathrm { i n } ^ { 3 } / \mathrm { s } , \quad f _ { 2 } = 0$
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14.9 $p _ { 2 } = p _ { 3 } = 1 2 ~ \mathrm { m } , ~ p _ { 5 } = 1 1 ~ \mathrm { m }$
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# Chapter 15
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15.1 d2x ¼ 0:021 in., d3x ¼ 0:042 in., sx ¼ 0
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15.2 $d _ { 2 x } = 0 , ~ \sigma _ { x } = 5 0 . 4 ~ \mathrm { M P a }$
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15.3 $d _ { 1 x } = d _ { 1 y } = - 0 . 0 1 7 5 \mathrm { ~ i n . , } \quad \sigma ^ { ( 1 ) } = 4 3 5 0 \mathrm { ~ p s i \ : ( T ) }$
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$$
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\sigma^ {(2)} = - 6 1 5 0 \text { psi (C) }, \quad \sigma^ {(3)} = 4 3 5 0 \text { psi (T) }
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$$
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15.4 $d _ { 1 x } = - 0 . 0 2 9 1 \ \mathrm { i n . , } \quad d _ { 1 y } = - 0 . 0 0 9 5 \ \mathrm { i n . }$
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$$
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\sigma^ {(1)} = - 1 3 7 0 \text { psi (C) }, \quad \sigma^ {(2)} = 2 3 7 5 \text { psi (T) }, \quad \sigma^ {(3)} = - 1 3 7 0 \text { psi (C) }
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$$
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15.5 $d _ { 2 x } = 1 . 4 4 \times 1 0 ^ { - 4 } \mathrm { m } , ~ \sigma ^ { ( 1 ) } = - 2 0 . 2 \mathrm { M P a } \left( \mathrm { C } \right) , ~ \sigma ^ { ( 2 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 1 \mathrm { M P a } \left( \mathrm { C } \right)$
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15.6 $\begin{array} { r } { d _ { 1 x } = 0 , d _ { 1 y } = 6 . 0 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 5 \mathrm { M P a } ( \mathrm { C } ) } \end{array}$
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$$
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\sigma^ {(2)} = 1 8. 2 \mathrm{MPa(T)}
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$$
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15.7 $d _ { 1 x } = 0 , d _ { 1 y } = - 3 . 6 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 2 ) } = 0$
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15.8 $d _ { 2 x } = 0 . 0 1 7 3 \ \mathrm { i n . , } \quad \sigma _ { s t } = 8 4 0 \ \mathrm { p s i } \ ( \mathrm { T } ) , \quad \sigma _ { b r } = 1 6 8 0 \ \mathrm { p s i } \ ( \mathrm { C } )$
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15.12 $\mathbf { a } . \ - 0 . 0 0 1 9 0 7 \ \mathrm { i n } . \quad \mathbf { b } . \ \sigma _ { b r } = - 2 8 , 6 0 0 \ \mathrm { p s i } , \quad \sigma _ { m g } = - 1 9 , 0 6 7 \ \mathrm { p s i }$
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15.13 fT1x ¼ -4464 lb, fT1y ¼ -8929 lb, fT2x ¼ 4464 lb
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$$
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f _ {T 2 y} = - 8 9 2 9 \mathrm{lb}, \quad f _ {T 3 x} = 0, \quad f _ {T 3 y} = 1 7, 8 5 7 \mathrm{lb}
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$$
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15.14 $f _ { T 1 x } = - 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 1 y } = 0 , f _ { T 2 x } = 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 2 y } = - 8 6 . 2 5 0 ~ \mathrm { k N }$
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$$
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f _ {T 3 x} = 0, \quad f _ {T 3 y} = 8 6. 2 5 0 \mathrm{kN}
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$$
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15.15 $f _ { T 1 x } = - 6 0 . 0 \mathrm { ~ k i p } , f _ { T 1 y } = - 9 0 \mathrm { ~ k i p } , f _ { T 2 x } = 6 0 \mathrm { ~ k i p } , f _ { T 2 y } = 0 ,$
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$$
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f _ {T 3 x} = 0, \quad f _ {T 3 y} = 9 0 \mathrm{kip}
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$$
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15.16 $f _ { T 1 x } = 1 3 4 \ \mathrm { k N } , f _ { T 1 y } = 1 3 4 \ \mathrm { k N } , f _ { T 2 x } = - 1 3 4 \ \mathrm { k N } , f _ { T 2 y } = 0$
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$$
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f _ {T 3 x} = 0, \quad f _ {T 3 y} = - 1 3 4 \mathrm{kN}
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$$
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15.17 $\sigma _ { x } = \sigma _ { y } = - 8 9 2 9 \mathrm { p s i } ( \mathrm { C } ) , \tau _ { x y } = 0$
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15.18 $\sigma _ { x } = 6 7 . 2 ~ \mathrm { M P a } , \sigma _ { y } = 6 7 . 2 ~ \mathrm { M P a } , \tau _ { x y } = 0$
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15.19 $\{ f _ { T } \} = \frac { A E \alpha _ { 0 } } { 6 } \left\{ \begin{array} { r } { - 4 t _ { 1 } - 5 t _ { 2 } } \\ { 4 t _ { 1 } + 5 t _ { 2 } } \end{array} \right\}$
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15.20 $\frac { A E \alpha } { 2 } \bigg \{ { - t _ { 1 } - t _ { 2 } } \bigg \}$
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<!-- source-page: 812 -->
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15.21 $\{f_{T}\}=\frac{2\pi\bar{r}AE\alpha(\Delta T)[\bar{B}]^{T}}{1-2\nu}\left\{\begin{array}{l}1\\ 1\\ 1\\ 0\end{array}\right\}$
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15.22 $d_{2x} = 0.8 \times 10^{-3}$ in., $d_{3x} = 0$ , $d_{3y} = 0.8 \times 10^{-3}$ in.
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$d_{4x} = d_{4y} = 0.8 \times 10^{-3}$ in.; stresses are zero
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15.23 $d_{2x} = 0.989 \times 10^{-3}$ in., $d_{3x} = -0.756 \times 10^{-3}$ in.,
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$$
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d _ {3 y} = 0. 9 8 9 \times 1 0 ^ {- 3} \text { in. }, \quad d _ {4 x} = 0. 1 3 2 \times 1 0 ^ {- 2} \text { in. },
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$$
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$$
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d _ {4 y} = 0. 2 0 4 5 \times 1 0 ^ {- 2} \text {in.}, \quad \sigma_ {1} ^ {(1)} = 1 7 \mathrm{ksi}, \quad \sigma_ {2} ^ {(2)} = - 1 7 \mathrm{ksi}
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$$
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# Chapter 16
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16.1 $[M] = \frac{\rho AL}{6}\left[ \begin{array}{ccc}2 & 1 & 0\\ 1 & 4 & 1\\ 0 & 1 & 2 \end{array} \right]$
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16.2 a. $[M] = \frac{\rho AL}{2}\left[ \begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 1 \end{array} \right]$
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$\mathbf{b.}[M] = \frac{\rho AL}{6}\left[ \begin{array}{cccc}2 & 1 & 0 & 0\\ 1 & 4 & 1 & 0\\ 0 & 1 & 4 & 1\\ 0 & 0 & 1 & 2 \end{array} \right]$
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16.3 $\omega_{1} = 0.806\sqrt{u},\omega_{2} = 2.81\sqrt{\mu}$
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16.4 $\omega_{1} = 5.368\times 10^{3}\mathrm{rad / s},\quad \omega_{2} = 17.556\times 10^{3}\mathrm{rad / s}$
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16.5 a. $t(\mathrm{s})$ $d_{i}(\mathrm{ft})$ $\dot{d}_i(\mathrm{ft / s})$ $\ddot{d}_i(\mathrm{ft / s^2})$
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<table><tr><td>0</td><td>0</td><td>0</td><td>25</td></tr><tr><td>0.03</td><td>0.01125</td><td>0.71</td><td>22.09</td></tr><tr><td>0.06</td><td>0.04238</td><td>1.03</td><td>-0.715</td></tr><tr><td>0.09</td><td>0.07287</td><td>0.67</td><td>-22.87</td></tr><tr><td>0.12</td><td>0.08278</td><td>-0.35</td><td>-45.28</td></tr><tr><td>0.15</td><td>0.05194</td><td>-1.43</td><td>-26.94</td></tr></table>
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<!-- source-page: 813 -->
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16.6 a. t ðsÞ $d _ { i }$ ðftÞ $\dot { d } _ { i } \left( \mathrm { f t } / \mathrm { s } \right)$ $\ddot { d } _ { i } \ ( \mathrm { f t } / \mathrm { s } ^ { 2 } )$
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<table><tr><td>0</td><td>0</td><td>0</td><td>10.00</td></tr><tr><td>0.02</td><td>0.0020</td><td>0.168</td><td>6.80</td></tr><tr><td>0.04</td><td>0.00672</td><td>0.256</td><td>1.968</td></tr><tr><td>0.06</td><td>0.01223</td><td>0.242</td><td>-3.338</td></tr><tr><td>0.08</td><td>0.01640</td><td>0.130</td><td>-7.84</td></tr><tr><td>0.10</td><td>0.01743</td><td>-0.053</td><td>-10.46</td></tr></table>
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<table><tr><td>b. $t$ (s)</td><td> $d_{i}$ (ft)</td><td> $\dot{d}_{i}$ (ft/s)</td><td> $\ddot{d}_{i}$ (ft/s2)</td><td> $F(t)$ (lb)</td></tr><tr><td>0.00</td><td>0.00000</td><td>0.000</td><td>10.000</td><td>20.0</td></tr><tr><td>0.02</td><td>0.00179</td><td>0.169</td><td>6.923</td><td>16.0</td></tr><tr><td>0.04</td><td>0.00625</td><td>0.263</td><td>2.248</td><td>12.0</td></tr><tr><td>0.06</td><td>0.0115</td><td>0.254</td><td>-2.945</td><td>8.0</td></tr><tr><td>0.08</td><td>0.0157</td><td>0.150</td><td>-7.458</td><td>4.0</td></tr><tr><td>0.10</td><td>0.0169</td><td>-0.0147</td><td>-10.251</td><td>0.0</td></tr></table>
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16.7 Node t ðsÞ di ðin:Þ $\dot { d } _ { i } \ ( \mathrm { i n . / s } )$ $\ddot { d } _ { i } \ ( \mathrm { i n } . / \mathrm { s } ^ { 2 } )$
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<table><tr><td>2</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>0.00025</td><td>2.6E-6</td><td>0.031</td><td>249.6</td></tr><tr><td></td><td>0.00050</td><td>3.4E-5</td><td>0.284</td><td>1768.9</td></tr><tr><td></td><td>0.00075</td><td>1.9E-4</td><td>1.085</td><td>4641.9</td></tr><tr><td></td><td>0.0010</td><td>6.36E-4</td><td>2.605</td><td>7519.3</td></tr><tr><td>3</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td></td><td>0.00025</td><td>6.59E-5</td><td>0.791</td><td>6328.8</td></tr><tr><td></td><td>0.00050</td><td>4.99E-4</td><td>2.817</td><td>9881.2</td></tr><tr><td></td><td>0.00075</td><td>1.51E-3</td><td>5.265</td><td>9701.7</td></tr><tr><td></td><td>0.0010</td><td>3.10E-3</td><td>7.369</td><td>7128.3</td></tr></table>
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16.8 Using Newmark’s method with $\begin{array} { r } { \gamma = \frac { 1 } { 2 } , \beta = \frac { 1 } { 6 } } \end{array}$
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<table><tr><td>Node</td><td>t (s)</td><td> $d_i$ (in.)</td><td> $\dot{d}_i$ (in./s)</td><td> $\ddot{d}_i$ (in./s2)</td><td>F(t) (lb)</td></tr><tr><td rowspan="3">2</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>0.05</td><td>0.00172</td><td>0.103</td><td>4.131</td><td>0</td></tr><tr><td>0.10</td><td>0.01544</td><td>0.513</td><td>12.27</td><td>0</td></tr><tr><td rowspan="3">3</td><td>0</td><td>0</td><td>0</td><td>40.0</td><td>2000</td></tr><tr><td>0.05</td><td>0.0448</td><td>1.685</td><td>27.39</td><td>1800</td></tr><tr><td>0.10</td><td>0.1536</td><td>2.479</td><td>4.37</td><td>1600</td></tr></table>
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$\begin{array} { l l l } { { { \bf 1 6 . 1 1 } } } & { { { \bf a . } { \ \omega } \omega _ { 1 } = \displaystyle \frac { 3 . 1 5 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { \omega _ { 2 } = \displaystyle \frac { 1 6 . 2 4 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { { \bf c . } { \omega } \omega _ { 1 } = \displaystyle \frac { 9 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } \\ { { } } & { { { \bf d . } { \omega } = \displaystyle \frac { 1 4 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } & { { } } & { { { \bf d . } } } \end{array}$ EI 1=2 EI 1=2 EI 1=2
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<!-- source-page: 814 -->
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<table><tr><td rowspan="13">16.17</td><td colspan="2">Node:</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td></tr><tr><td>i</td><td>t (s)</td><td></td><td></td><td colspan="2">Temperature (°C)</td><td></td><td></td></tr><tr><td>0</td><td>0</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td><td>200</td></tr><tr><td>1</td><td>8</td><td>0</td><td>159.0095</td><td>191.4441</td><td>198.2110</td><td>199.6110</td><td>199.8444</td></tr><tr><td>2</td><td>16</td><td>0</td><td>135.5852</td><td>178.1491</td><td>193.6620</td><td>198.2112</td><td>199.1445</td></tr><tr><td>3</td><td>24</td><td>0</td><td>120.2309</td><td>165.7003</td><td>187.3485</td><td>195.5379</td><td>197.5152</td></tr><tr><td>4</td><td>32</td><td>0</td><td>109.1993</td><td>154.9587</td><td>180.4038</td><td>191.7446</td><td>194.8115</td></tr><tr><td>5</td><td>40</td><td>0</td><td>100.7600</td><td>145.7784</td><td>173.4129</td><td>187.1268</td><td>191.1242</td></tr><tr><td>6</td><td>48</td><td>0</td><td>94.00311</td><td>137.8529</td><td>166.6182</td><td>181.9599</td><td>186.6590</td></tr><tr><td>7</td><td>56</td><td>0</td><td>88.39929</td><td>130.9034</td><td>160.1012</td><td>176.4598</td><td>181.6395</td></tr><tr><td>8</td><td>64</td><td>0</td><td>83.61745</td><td>124.7101</td><td>153.8759</td><td>170.7856</td><td>176.2620</td></tr><tr><td>9</td><td>72</td><td>0</td><td>79.43935</td><td>119.1075</td><td>147.9316</td><td>165.0508</td><td>170.6822</td></tr><tr><td>10</td><td>80</td><td>0</td><td>75.71603</td><td>113.9733</td><td>142.2502</td><td>159.3352</td><td>165.0171</td></tr></table>
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<table><tr><td colspan="5">16.18</td></tr><tr><td rowspan="2">Time (s)</td><td rowspan="2">1</td><td colspan="2">Node</td><td rowspan="2">(using consistent capacitance matrix)</td></tr><tr><td>2</td><td>3</td></tr><tr><td colspan="5">Temperature (°C)</td></tr><tr><td>0</td><td>25</td><td>25</td><td>25</td><td></td></tr><tr><td>0.1</td><td>85</td><td>18.53611</td><td>26.36189</td><td></td></tr><tr><td>0.2</td><td>85</td><td>29.61303</td><td>21.63526</td><td></td></tr><tr><td>0.3</td><td>85</td><td>36.18435</td><td>22.42717</td><td></td></tr><tr><td>0.4</td><td>85</td><td>40.72491</td><td>25.30428</td><td></td></tr><tr><td>0.5</td><td>85</td><td>44.27834</td><td>28.85201</td><td></td></tr><tr><td>0.6</td><td>85</td><td>47.29072</td><td>32.49614</td><td></td></tr><tr><td>0.7</td><td>85</td><td>49.95809</td><td>36.01157</td><td></td></tr><tr><td>0.8</td><td>85</td><td>52.37152</td><td>39.31761</td><td></td></tr><tr><td>0.9</td><td>85</td><td>54.57756</td><td>42.39278</td><td></td></tr></table>
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<table><tr><td colspan="5">16.18</td></tr><tr><td rowspan="2">Time (s)</td><td rowspan="2">1</td><td colspan="2">Node</td><td rowspan="2">(using consistent capacitance matrix)</td></tr><tr><td>2</td><td>3</td></tr><tr><td colspan="5">Temperature (°C)</td></tr><tr><td>1</td><td>85</td><td>56.60353</td><td>45.23933</td><td></td></tr><tr><td>1.1</td><td>85</td><td>58.46814</td><td>47.86852</td><td></td></tr><tr><td>1.2</td><td>85</td><td>60.1859</td><td>50.29457</td><td></td></tr><tr><td>1.3</td><td>85</td><td>61.76908</td><td>52.53218</td><td></td></tr><tr><td>1.4</td><td>85</td><td>63.22852</td><td>54.59557</td><td></td></tr><tr><td>1.5</td><td>85</td><td>64.574</td><td>56.49814</td><td></td></tr><tr><td>1.6</td><td>85</td><td>65.81448</td><td>58.25235</td><td></td></tr><tr><td>1.7</td><td>85</td><td>66.95818</td><td>59.86974</td><td></td></tr></table>
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<!-- source-page: 815 -->
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16.18
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<table><tr><td rowspan="2">Time (s)</td><td colspan="3">Node</td></tr><tr><td>1</td><td>2</td><td>3</td></tr><tr><td></td><td colspan="3">Temperature (°C)</td></tr><tr><td>0</td><td>25</td><td>25</td><td>25</td></tr><tr><td>1.8</td><td>85</td><td>68.01265</td><td>61.36096</td></tr><tr><td>1.9</td><td>85</td><td>68.98485</td><td>62.73586</td></tr><tr><td>2</td><td>85</td><td>69.88121</td><td>64.0035</td></tr><tr><td>2.1</td><td>85</td><td>70.70765</td><td>65.17226</td></tr><tr><td>2.2</td><td>85</td><td>71.46961</td><td>66.24984</td></tr><tr><td>2.3</td><td>85</td><td>72.17214</td><td>67.24336</td></tr><tr><td>2.4</td><td>85</td><td>72.81986</td><td>68.15938</td></tr><tr><td>2.5</td><td>85</td><td>73.41705</td><td>69.00393</td></tr><tr><td>2.6</td><td>85</td><td>73.96766</td><td>69.78261</td></tr><tr><td>2.7</td><td>85</td><td>74.47531</td><td>70.50053</td></tr><tr><td>2.8</td><td>85</td><td>74.94336</td><td>71.16246</td></tr><tr><td>2.9</td><td>85</td><td>75.3749</td><td>71.77274</td></tr><tr><td>3</td><td>85</td><td>75.77277</td><td>72.33542</td></tr></table>
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# Appendix A
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A1. a. $\left[ { \begin{array} { r r } { 3 } & { 0 } \\ { - 3 } & { 1 2 } \end{array} } \right]$ b. Nonsense c. Nonsense
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$\left\{ { \begin{array} { l } { 1 1 } \\ { 9 } \\ { 1 1 } \end{array} } \right\}$ e. Nonsense f. $\left[ { \begin{array} { r r r } { 1 0 } & { 7 } & { 6 } \\ { 3 } & { - 1 } & { 7 } \end{array} } \right]$
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$\left[ \begin{array} { l l } { 1 } & { 0 } \\ { { \frac { 1 } { 4 } } } & { { \frac { 1 } { 4 } } } \end{array} \right]$
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A3. ${ \frac { 1 } { 1 7 } } \left[ { \begin{array} { r r r } { 1 2 } & { - 3 } & { - 8 } \\ { - 3 } & { 5 } & { 2 } \\ { - 8 } & { 2 } & { 1 1 } \end{array} } \right]$
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A4. Nonsense
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A5. $\left[ \begin{array} { l l } { { \frac { 1 } { 2 } } } & { 0 } \\ { { \frac { 1 } { 8 } } } & { { \frac { 1 } { 8 } } } \end{array} \right]$
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A6. Same as A3
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A8. $\left[ \begin{array} { c c } { \cos \theta } & { - \sin \theta } \\ { \sin \theta } & { \cos \theta } \end{array} \right]$
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# Appendix B
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B1. $x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2$
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B2. $x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2$
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<!-- source-page: 816 -->
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B3. $x_{1} = 2.5, x_{2} = -1, x_{3} = 0.5$
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B4. $x_{1} = 3, x_{2} = -1, x_{3} = -2$
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B5. a. $\left\{ \begin{array}{l}x_{1}\\ x_{2} \end{array} \right\} = \left[ \begin{array}{ll}2 & -1\\ 1 & -1 \end{array} \right]\left\{ \begin{array}{l}y_{1}\\ y_{2} \end{array} \right\}$ b. $\left\{ \begin{array}{l}z_1\\ z_2 \end{array} \right\} = \left[ \begin{array}{ll} - 3 & 2\\ 5 & -3 \end{array} \right]\left\{ \begin{array}{l}y_{1}\\ y_{2} \end{array} \right\}$
|
||
|
||
B6. $x_{1} = 0, x_{2} = 1, x_{3} = 2, x_{4} = 2, x_{5} = 0$
|
||
|
||
B7. $x_{1} = 3.15, x_{2} = 0.62$
|
||
|
||
B8. a. Unique b. Nonexistent c. Unique d. Nonunique
|
||
|
||
# Appendix D
|
||
|
||
D1. a. $f_{1y} = f_{2y} = -5 \mathrm{kip}$ , $m_{1} = -m_{2} = -100 \mathrm{k-ft}$
|
||
|
||
b. $f_{1v} = f_{2v} = -5 \mathrm{kip}, \quad m_1 = -m_2 = -18.75 \mathrm{k-ft}$
|
||
|
||
c. $f_{1y} = f_{2y} = -15 \mathrm{kip}, \quad m_1 = -m_2 = -75 \mathrm{k-ft}$
|
||
|
||
d. $f_{1v} = -18.75 \text{ kip}, \quad f_{2v} = -6.25 \text{ kip}, \quad m_1 = -58.3 \text{ k-ft}, \quad m_2 = 33.3 \text{ k-ft}$
|
||
|
||
e. $f_{1v} = -6$ kip, $f_{2v} = -14$ kip, $m_{1} = -26.67$ k-ft, $m_{2} = 40$ k-ft
|
||
|
||
f. $f_{1y} = -0.99 \, kN$ , $f_{2y} = -4.0 \, kN$ , $m_{1} = -2.04 \, kN \cdot m$ , $m_{2} = 5.10 \, kN \cdot m$
|
||
|
||
g. $f_{1y} = f_{2y} = -6 \, kN$ , $m_{1} = -m_{2} = -7.5 \, kN \cdot m$
|
||
|
||
h. $f_{1y} = f_{2y} = -10 \, kN$ , $m_{1} = -m_{2} = -6.67 \, kN \cdot m$
|
||
|
||
<!-- source-page: 817 -->
|
||
|
||
# Index
|
||
|
||
# A
|
||
|
||
Adaptive refinement, 355
|
||
|
||
Adjoint method, 718
|
||
|
||
Admissible variation, 55
|
||
|
||
Aluminum shapes, properties of, 759–772
|
||
|
||
Amplitude, defined, 649
|
||
|
||
Approximation functions, 72–74
|
||
|
||
compatible, 73
|
||
|
||
complete, 73–74
|
||
|
||
conforming, 73
|
||
|
||
displacement, 72–74
|
||
|
||
interpolation, 74
|
||
|
||
Aspect ratio (AR), 351, 352–353
|
||
|
||
Axial symmetry, 100
|
||
|
||
Axis of revolution, 412
|
||
|
||
Axis of symmetry, 412
|
||
|
||
Axisymmetric element, 9, 412–442, 684–685
|
||
|
||
applications of, 428–433
|
||
|
||
body forces, 419–420
|
||
|
||
consistent-mass matrix, 684–685
|
||
|
||
defined, 9, 412
|
||
|
||
discretization, 423
|
||
|
||
displacement functions, 415–417
|
||
|
||
element type, selection of, 415
|
||
|
||
equations, 419–421
|
||
|
||
introduction to, 412
|
||
|
||
pressure vessel, solution of, 422–428
|
||
|
||
sti¤ness matrix, 412–422, 423–428
|
||
|
||
strain/displacement relationships, 417–419
|
||
|
||
stress/strain relationships, 417–419
|
||
|
||
surface forces, 420–421
|
||
|
||
# B
|
||
|
||
Banded-symmetric method, 735–741
|
||
|
||
Bar elements, 67–72, 92–100, 109–120, 120–124, 124–127, 127–131, 444–449, 665–669, 669–674. See also Truss equations
|
||
|
||
analysis of, 665–669, 669–674
|
||
|
||
collocation method, 129
|
||
|
||
consistent-mass matrix, 651–653
|
||
|
||
displacement function, 68, 446, 650
|
||
|
||
dynamic analysis of, 649–653, 665–669, 669–674
|
||
|
||
equations, 124–127, 447–449, 649–653
|
||
|
||
exact solution, 120–124
|
||
|
||
finite element solution, 120–124
|
||
|
||
Galerkin’s residual method, 124–127, 131
|
||
|
||
isoparametric formulation, 444–449
|
||
|
||
least squares method, 130
|
||
|
||
local coordinates for, 66–72
|
||
|
||
lumped-mass matrix, 651
|
||
|
||
mass matrix, 650–653
|
||
|
||
natural frequencies, 665–669
|
||
|
||
one-dimensional problems, 127–131, 665–669, 669–674
|
||
|
||
potential energy approach, 109–120
|
||
|
||
residual methods, 124–127, 127–131
|
||
|
||
selection of, 67, 444–446, 650
|
||
|
||
sti¤ness matrix, 66–72, 92–100, 444–449, 650–653
|
||
|
||
strain/displacement relationships, 69, 446–447, 650
|
||
|
||
stress, computation of, 82–83
|
||
|
||
stress/strain relationships, 69, 446–447, 650
|
||
|
||
subdomain method, 129–130
|
||
|
||
three-dimensional space, 92–100
|
||
|
||
time-dependent (dynamic) stress analysis, 649–653
|
||
|
||
time-dependent problem, 669–674
|
||
|
||
transformation matrix, 92–100
|
||
|
||
Beam element, 152–161, 161–163, 194–199, 214–218, 218–236, 255–269, 674–681
|
||
|
||
arbitrarily oriented, 214–218, 255–269
|
||
|
||
bending, 153–158, 255–260
|
||
|
||
boundary conditions, 161–163
|
||
|
||
defined, 152
|
||
|
||
deformations, 153–158
|
||
|
||
displacement function, 155–156
|
||
|
||
equations, 157–158, 161–163
|
||
|
||
mass matrices, 674–681
|
||
|
||
natural frequencies, 674–681
|
||
|
||
nodal hinge, 194–199
|
||
|
||
rigid plane frames, 218–236
|
||
|
||
selection of, 154
|
||
|
||
shape functions, 155–156
|
||
|
||
sign conventions, 152, 256–257
|
||
|
||
space, arbitrarily oriented in, 255–269
|
||
|
||
sti¤ness, 152–161
|
||
|
||
sti¤ness matrix, 153–158, 158–161
|
||
|
||
strain/displacement relationships, 156–157
|
||
|
||
stress/strain relationships, 156–157
|
||
|
||
transformation matrix, 216, 259–260
|
||
|
||
<!-- source-page: 818 -->
|
||
|
||
Beam element (Continued )
|
||
|
||
transverse shear deformations, 158–161
|
||
|
||
two-dimensional, arbitrarily oriented, 214–218
|
||
|
||
Beam equations, 151–213
|
||
|
||
bending deformations, 153–158
|
||
|
||
boundary conditions, 161–163
|
||
|
||
direct sti¤ness method, 163–175
|
||
|
||
displacement functions, 155–156
|
||
|
||
distributed loading, 175–188
|
||
|
||
Euler-Bernouli theory, 153–158
|
||
|
||
exact solution, 188–194
|
||
|
||
finite element solution, 188–194
|
||
|
||
fixed-end reactions, 175
|
||
|
||
Galerkin’s method, 201–203
|
||
|
||
introduction to, 151–152
|
||
|
||
load replacement, 177–178
|
||
|
||
nodal hinge, element with a, 194–199
|
||
|
||
potential energy approach, 199–201
|
||
|
||
sign conventions, 152
|
||
|
||
sti¤ness matrix, 153–158, 158–161, 161–163
|
||
|
||
sti¤ness of element, 152–161
|
||
|
||
strain/displacement relationships, 156–157
|
||
|
||
stress/strain relationships, 156–157
|
||
|
||
Timoshenko theory, 158–161
|
||
|
||
transverse shear deformations, 158–161
|
||
|
||
work-equivalence method, 176–177
|
||
|
||
Bending, 153–158, 255–260, 514–518
|
||
|
||
beam elements in arbitrary space, 255–260
|
||
|
||
deformations in beam elements, 153–158
|
||
|
||
plate element, 514–518
|
||
|
||
rigidity of a plate, 517
|
||
|
||
Body forces, 324–326, 419–420, 448, 460, 497–498
|
||
|
||
axisymmetric elements, 419–420
|
||
|
||
bar element, 448
|
||
|
||
centrifugal, 325
|
||
|
||
natural coordinate system, 448
|
||
|
||
plane element, 460
|
||
|
||
tetrahedral element, 497–498
|
||
|
||
treatment of, 324–326
|
||
|
||
Boundary conditions, 13–14, 34, 39–52, 103–109, 161–163, 320–322, 601
|
||
|
||
beam elements, 161–163
|
||
|
||
constant-strain triangular (CST) element, 320–322
|
||
|
||
fluid flow, 601
|
||
|
||
homogeneous, 39–40
|
||
|
||
inclined supports, 103–109
|
||
|
||
introduction to, 13–14, 34
|
||
|
||
nonhomogeneous, 39, 40–41
|
||
|
||
penalty method, 50–52
|
||
|
||
skewed supports, 103–109
|
||
|
||
sti¤ness method, 39–52
|
||
|
||
#
|
||
|
||
Castigliano’s theorem, 12
|
||
|
||
Central di¤erence method, 653, 654–659
|
||
|
||
Centrifugal body force, 325
|
||
|
||
Circular frequency, natural, 649
|
||
|
||
Coarse-mesh generation, 310
|
||
|
||
Coe‰cient matrix, inversion of, 726
|
||
|
||
Coe‰cient of thermal expansion, 618
|
||
|
||
Cofactor method, 716–717
|
||
|
||
Collocation method, 129
|
||
|
||
Column matrices, 4, 708
|
||
|
||
Compatibility, 35, 363–367, 746–748
|
||
|
||
condition of, 748
|
||
|
||
equations, 746–748
|
||
|
||
finite element results, 363–367
|
||
|
||
requirement, 35
|
||
|
||
Compatible displacements, 755
|
||
|
||
Compatible functions, 73
|
||
|
||
Complete, approximation functions, 73–74
|
||
|
||
Computer programs, 6–7, 23–24, 374–380, 524–528, 693–701
|
||
|
||
finite element method, 23–24
|
||
|
||
plate bending element, solution for, 524–528
|
||
|
||
role of, 6–7
|
||
|
||
step-by-step solutions, 374–380
|
||
|
||
structural dynamics, 693–701
|
||
|
||
Concentrated loads, 360–361
|
||
|
||
Condensation, see Static condensation
|
||
|
||
Conduction,535–538,542–546,557–558
|
||
|
||
element conduction matrix, 542–546, 557–558
|
||
|
||
heat, one-dimensional, 535–537
|
||
|
||
heat, two-dimensional, 537–538
|
||
|
||
Conforming functions, 73
|
||
|
||
Connecting (mixing) di¤erent kinds of elements, 361–362
|
||
|
||
Consistent-mass matrix, 651–653, 682–685
|
||
|
||
Constant-strain triangular (CST) element, 304–305, 310–324, 324–329, 342, 406–408
|
||
|
||
body forces, 324–326
|
||
|
||
boundary conditions, 320–322
|
||
|
||
coarse-mesh generation, 310 defects, 342
|
||
|
||
displacement function, 311–315
|
||
|
||
equations, 310–324
|
||
|
||
forces (stresses), 322–324
|
||
|
||
global equations, 320–322
|
||
|
||
introduction to, 304–305
|
||
|
||
LST elements, comparison of, 406–408
|
||
|
||
matrix, 310–324, 329–331
|
||
|
||
nodal displacements, 322
|
||
|
||
penalty formulation, 331
|
||
|
||
selection of, 310–311
|
||
|
||
strain/displacement relationships, 315–320
|
||
|
||
stress/strain relationships, 315–320
|
||
|
||
surface forces, 326–329
|
||
|
||
Constitutive law, 11
|
||
|
||
Constitutive matrix, 309, 522
|
||
|
||
Continuity, 35, 73
|
||
|
||
requirement, 35 symbol, 73
|
||
|
||
Convection, heat transfer with, 538–539, 540
|
||
|
||
Convergence of finite element solution, 367–368
|
||
|
||
Coordinates, 66–72, 444–446
|
||
|
||
bar elements, 67–72, 444–446
|
||
|
||
intrinsic system, 444 natural system, 444
|
||
|
||
Coulomb-Mohr theory, 342
|
||
|
||
Cramer’s rule, 724–725
|
||
|
||
CST, see Constant-strain triangular (CST) element
|
||
|
||
Cubic elements, 9
|
||
|
||
Curvature matrix, 521–522
|
||
|
||
# D
|
||
|
||
D’Alembert’s principle, 755–756
|
||
|
||
Defects, CST elements, 342
|
||
|
||
Deformation, 33, 153–158, 158–161, 514–518
|
||
|
||
bending in beams, 153–158
|
||
|
||
bending rigidity of a plate, 517 defined, 33
|
||
|
||
Kirchho¤ assumptions, 515–516
|
||
|
||
plate bending, 514–518
|
||
|
||
potential energy, 518
|
||
|
||
stress/strain relationships, 517–518
|
||
|
||
transverse shear in beams, 158–161
|
||
|
||
Degrees of freedom, 14, 15, 29 defined, 15
|
||
|
||
spring element, 29
|
||
|
||
unknown, 14
|
||
|
||
Determinant, defined, 716
|
||
|
||
Di¤erential equations, 535–538, 594–596, 744–746
|
||
|
||
elasticity theory, 744–746
|
||
|
||
equilibrium, 744–746
|
||
|
||
fluid flow, 594–598
|
||
|
||
heat transfer, 535–538
|
||
|
||
Direct equilibrium method, 11
|
||
|
||
Direct integration, 653
|
||
|
||
Direct sti¤ness method, 2–4, 13–14, 28, 37–39, 163–175.
|
||
|
||
See also Superposition
|
||
|
||
beam analysis using, 163–175
|
||
|
||
history of, 2–4, 28
|
||
|
||
total sti¤ness matrix, assembly by, 37–39
|
||
|
||
use of, 13–14
|
||
|
||
<!-- source-page: 819 -->
|
||
|
||
Direction cosines, 85, 95–96
|
||
|
||
Directional sti¤ness bias, 371
|
||
|
||
Discontinuities, natural subdivisions at, 354, 357
|
||
|
||
Discretization, 1, 8–10, 331–332, 423 axisymmetric element, 423
|
||
|
||
finite element method, 1, 8–10, 331–332
|
||
|
||
plane stress, 331–332
|
||
|
||
Displacement function, 11, 31–32, 68, 155–156, 311–315, 399–401, 446, 450–451, 455–456, 494–496, 519–521
|
||
|
||
bar element, 68, 446
|
||
|
||
beam element, 155–156
|
||
|
||
constant-strain triangular (CST) element, 311–315
|
||
|
||
Hermite cubic interpolation, 155–156
|
||
|
||
interpolation, 32
|
||
|
||
isoparametric function, 446, 450–451, 455–456
|
||
|
||
linear-strain triangle (LST), 399–401
|
||
|
||
plane element, 455–456
|
||
|
||
plane stress element, 450–451
|
||
|
||
plate bending element, 519–521
|
||
|
||
selection of, 11
|
||
|
||
shape, 32, 155–156
|
||
|
||
spring element, 31–32
|
||
|
||
tetrahedral element, 494–496
|
||
|
||
Displacement method, 7, 28–64. See also Sti¤ness method introduction to, 28–64 use of, 7
|
||
|
||
Displacements, 34, 70, 72–74, 755–758. See also Strain/ displacement relationships
|
||
|
||
approximation functions for, 72–74
|
||
|
||
compatible, 755
|
||
|
||
nodal, 34, 70
|
||
|
||
virtual work, principles of, 755–758
|
||
|
||
Distributed loading, 175–188 beams, 175–188
|
||
|
||
e¤ective global nodal forces, 181–182
|
||
|
||
fixed-end reactions, 175
|
||
|
||
general formulation of, 178–179
|
||
|
||
load replacement, 177–178
|
||
|
||
work-equivalence method, 176–177
|
||
|
||
Dynamics, 647–707
|
||
|
||
axisymmetric element, analysis of, 684–685
|
||
|
||
bar element equations, 649–653
|
||
|
||
beam element mass matrices, 674–681
|
||
|
||
central di¤erence method, 653, 654–659
|
||
|
||
computer program example solutions, 693–701
|
||
|
||
introduction to, 647
|
||
|
||
mass matrices, 650–653, 674–681, 681–685
|
||
|
||
natural frequencies, 649, 665–669, 674–681
|
||
|
||
Newmark’s method, 659–663
|
||
|
||
numerical integration in time, 653–665, 687–693
|
||
|
||
one-dimensional bar analysis, 665–669, 669–674
|
||
|
||
plane frame element, analysis of, 682–683
|
||
|
||
plane stress/strain element, analysis of, 683–684
|
||
|
||
spring-mass system, 647–649
|
||
|
||
structural, 647–707
|
||
|
||
tetrahedral (solid) element mass matrices, analysis of, 685
|
||
|
||
time, numerical integration in, 653–665, 687–693
|
||
|
||
time-dependent heat transfer, 686–693
|
||
|
||
time-dependent stress analysis, 649–653, 669–674
|
||
|
||
truss element, analysis of, 681–682
|
||
|
||
Wilson’s (Wilson-Theta) method, 664–665
|
||
|
||
# E
|
||
|
||
E¤ective stress, 341
|
||
|
||
Elasticity theory, 744–751
|
||
|
||
compatibility equations, 746–748
|
||
|
||
condition of compatibility, 748
|
||
|
||
di¤erential equations of equilibrium, 744–746
|
||
|
||
equilibrium, di¤erential equations of, 744–746
|
||
|
||
introduction to, 744
|
||
|
||
modulus of elasticity, 748
|
||
|
||
strain/displacement, 746–748
|
||
|
||
stress/strain relationships, 748–751
|
||
|
||
Elements, 8–10, 11, 13–14, 30–34, 65–150, 151–213, 304–305, 310–324, 342, 351–362, 398–403, 444–449, 449–452, 480–482, 493–500, 501–508, 514–533
|
||
|
||
aspect ratio (AR), 351
|
||
|
||
axisymmetric, 9
|
||
|
||
bar, 65–150, 444–449
|
||
|
||
beam, 151–213
|
||
|
||
coarse-mesh generation, 310
|
||
|
||
connecting (mixing), modeling, 361–362
|
||
|
||
constant-strain triangular (CST), 304–305, 310–324, 342
|
||
|
||
cubic, 9
|
||
|
||
defects, CST, 324
|
||
|
||
equations, 11, 13–14, 34, 69–70, 402–403, 451–452, 522–523
|
||
|
||
finite, 8
|
||
|
||
forces, 34, 70
|
||
|
||
heterosis, 523
|
||
|
||
isoparametric, 446
|
||
|
||
LaGrange, 482
|
||
|
||
linear hexahedral, 501–504
|
||
|
||
linear-strain triangle (LST), 398–403
|
||
|
||
plane stress, 449–452
|
||
|
||
plate bending, 514–533
|
||
|
||
Q8, 480
|
||
|
||
Q9, 482
|
||
|
||
quadratic, 9
|
||
|
||
quadratic hexahedral, 504–508
|
||
|
||
refinement, methods of, 355–356, 358–359
|
||
|
||
selection of, 8–10, 30–31, 310–311, 399, 444–446, 449, 519
|
||
|
||
serendipity, 481
|
||
|
||
shapes, modeling, 351
|
||
|
||
sizing, 355–356, 358–359
|
||
|
||
spring, 30–34
|
||
|
||
sti¤ness matrix, 11, 33–34, 66–72, 402–403, 447–449, 451–452, 522–523
|
||
|
||
tetrahedral, 493–500
|
||
|
||
transition triangles, 359–360
|
||
|
||
Energy method, 12
|
||
|
||
Equations, 11, 13–14, 34, 52–60, 65–149, 151–213, 214–237, 238–255, 310–324, 398–411, 419–422, 447–449, 451–452, 459–460, 497–498, 522–523, 535–538, 542–546, 557–558, 594–596, 599–601, 608, 659–661, 664–665, 722–743, 744–751.
|
||
|
||
See also Elasticity theory;
|
||
|
||
Simultaneous linear equations
|
||
|
||
axisymmetric element, 419–422
|
||
|
||
bar element, 124–127, 447–449
|
||
|
||
beam, 151–213
|
||
|
||
beam element, 199–201, 201–203
|
||
|
||
compatibility, 746–748
|
||
|
||
constant-strain triangular (CST) element, 310–324
|
||
|
||
di¤erential, 535–538, 594–596, 744–745
|
||
|
||
element, 11, 13–14, 69–70
|
||
|
||
element conduction, 542–546, 557–558
|
||
|
||
finite element, 111
|
||
|
||
fluid flow, 599–601, 608
|
||
|
||
frame, 214–237
|
||
|
||
global, 13–14, 34, 70, 161–163, 546, 601
|
||
|
||
grid, 214, 238–255
|
||
|
||
heat transfer, 535–538
|
||
|
||
isoparametric formulation, 447–449, 459–460
|
||
|
||
Jacobian function, 447
|
||
|
||
<!-- source-page: 820 -->
|
||
|
||
Equations (Continued )
|
||
|
||
linear-strain triangle (LST), 398–411
|
||
|
||
Newmark’s, 659–661
|
||
|
||
one-dimensional, 124–127, 131, 542–546
|
||
|
||
plane element, 459–460
|
||
|
||
plane stress element, 451–452
|
||
|
||
plate bending element, 522–523
|
||
|
||
simultaneous linear, 722–743
|
||
|
||
spring element, 52–60
|
||
|
||
tetrahedral element, 497–498
|
||
|
||
total, 13–14, 70
|
||
|
||
truss, 65–149
|
||
|
||
two-dimensional, 557–558
|
||
|
||
Wilson’s, 664–665
|
||
|
||
Equilibrium, 363–367, 744–746
|
||
|
||
compatibility and, 363–367
|
||
|
||
di¤erential equations 744–746
|
||
|
||
finite element results, 363–367
|
||
|
||
Equivalent stress, 341
|
||
|
||
Euler-Bernouli theory, 153–158
|
||
|
||
Exact solution, 120–124, 188–194
|
||
|
||
bar element, 120–124
|
||
|
||
beams, 188–194
|
||
|
||
finite element solution, comparison to, 120–124, 188–194
|
||
|
||
Explicit numerical integration method, 689
|
||
|
||
F
|
||
|
||
Field problems, 52
|
||
|
||
Finite element, defined, 8
|
||
|
||
Finite element method, 1–26, 120–124, 350–363, 540–555, 555–564, 566–568, 569–574, 598–606, 606–610. See also Modeling
|
||
|
||
advantages of, 19–22
|
||
|
||
applications of, 15–19
|
||
|
||
boundary conditions, 13–14
|
||
|
||
computer, role of, 6–7
|
||
|
||
computer programs for, 23–24
|
||
|
||
constitutive law, 11
|
||
|
||
defined, 1, 8
|
||
|
||
degrees of freedom, 14, 15
|
||
|
||
direct equilibrium method, 11
|
||
|
||
direct sti¤ness method, 2–3, 13–14
|
||
|
||
discretization, 1, 8–10
|
||
|
||
displacement function, selection of, 11
|
||
|
||
displacement method, 7
|
||
|
||
element conduction matrix, 542–546, 557–558
|
||
|
||
element types, selection of, 8–10, 541, 555, 598
|
||
|
||
energy method, 12
|
||
|
||
exact solution, comparison to, 120–124
|
||
|
||
flexibility method, 7
|
||
|
||
fluid flow, 598–606, 606–610
|
||
|
||
force method, 7
|
||
|
||
functional, 12
|
||
|
||
generalized displacements, 14
|
||
|
||
global equations, 13–14
|
||
|
||
gradient/potential relationship, 599, 607
|
||
|
||
heat flux/temperature gradient relationship, 542, 556–557
|
||
|
||
heat transfer, 540–555, 555–564, 566–568, 569–574
|
||
|
||
history of, 2–4
|
||
|
||
introduction to, 1–26
|
||
|
||
matrix notation, 4–6
|
||
|
||
modeling, 350–363
|
||
|
||
one-dimensional, 540–555, 569, 598–606
|
||
|
||
potential function, 598–599, 607
|
||
|
||
primary unknowns, 14
|
||
|
||
results, interpretation of, 14
|
||
|
||
steps of, 7–14
|
||
|
||
sti¤ness method, 7
|
||
|
||
strain/displacement relationships, 11
|
||
|
||
stress/strain relationships, 11, 14
|
||
|
||
temperature function, 541, 556
|
||
|
||
temperature gradient/temperature relationships, 542, 556–557
|
||
|
||
three-dimensional, 566–568
|
||
|
||
total equations, 13–14
|
||
|
||
truss equations, 120–124
|
||
|
||
two-dimensional, 555–564, 606–610
|
||
|
||
variational method, 540–555
|
||
|
||
velocity/gradient relationship, 599, 607
|
||
|
||
weighted residuals, methods of, 12–13
|
||
|
||
work method, 12
|
||
|
||
Finite element solution, 120–124, 188–194, 331–342, 363–367, 367–369
|
||
|
||
approximations in, 364–367
|
||
|
||
bar element, 120–124
|
||
|
||
beams, 188–194
|
||
|
||
compatibility of results, 363–367
|
||
|
||
convergence of, 367–368
|
||
|
||
CST defects, 342
|
||
|
||
discretization, 331–332
|
||
|
||
equilibrium of results, 363–367
|
||
|
||
exact solution, comparison to, 120–124, 188–194
|
||
|
||
plane stress, 305–309
|
||
|
||
sti¤ness matrix, assemblage of, 332–342
|
||
|
||
Fixed-end forces, 229–230
|
||
|
||
Fixed-end reactions, 175
|
||
|
||
Flexibility method, 7
|
||
|
||
Flowcharts, 374, 574, 611, 656, 661
|
||
|
||
central di¤erence method, 656
|
||
|
||
fluid flow, 611
|
||
|
||
heat transfer, 574
|
||
|
||
Newmark’s equations, 661
|
||
|
||
numerical integration, 656
|
||
|
||
plane stress/strain, 374
|
||
|
||
Fluid flow, 593–616
|
||
|
||
boundary conditions, 601
|
||
|
||
di¤erential equations, 594–598
|
||
|
||
equations, 599–601, 608
|
||
|
||
finite element formulation, 598–606, 606–610
|
||
|
||
flowchart for, 611
|
||
|
||
global equations, 601
|
||
|
||
gradient/potential relationship, 599, 607
|
||
|
||
introduction to, 593
|
||
|
||
nodal potentials, 601
|
||
|
||
one-dimensional, 598–601
|
||
|
||
pipes, 596–598
|
||
|
||
porous medium, 594–596
|
||
|
||
potential function, 589
|
||
|
||
program, example of, 611–612
|
||
|
||
solid bodies, around, 596–598
|
||
|
||
sti¤ness matrix, 599–601, 608
|
||
|
||
two-dimensional, 606–610
|
||
|
||
velocities, 602
|
||
|
||
velocity/gradient relationship, 599, 607
|
||
|
||
volumetric flow rates, 602
|
||
|
||
Force, 7, 34, 36, 70, 178–182, 229–230, 232–233, 322–324, 324–329, 419–421, 448–449, 460, 497–498, 752–754
|
||
|
||
axisymmetric elements, 419–421
|
||
|
||
bar element, 70, 448–449
|
||
|
||
body, 324–326, 419–420, 448, 460, 497–498
|
||
|
||
centrifugal body, 325
|
||
|
||
constant-strain triangular (CST)
|
||
|
||
element, 322–324, 324–329
|
||
|
||
equivalent nodal, 178–180, 752–754
|
||
|
||
fixed-end, 229–230
|
||
|
||
global nodal matrix, 36
|
||
|
||
method, 7
|
||
|
||
nodal, 178–182, 232–233
|
||
|
||
plane element, 460
|
||
|
||
rigid plane frames, 229–230, 232–233
|
||
|
||
spring element, 34
|
||
|
||
stresses, 322–324
|
||
|
||
surface, 326–329, 420–421, 448–449, 460, 498
|
||
|
||
tetrahedral element, 497–498
|
||
|
||
Forced convection, 538, 540
|
||
|
||
Frame equations, 214–237
|
||
|
||
e¤ective nodal forces, 232–233
|
||
|
||
fixed-end forces, 229–230
|
||
|
||
inclined supports, 237
|
||
|
||
introduction to, 214
|
||
|
||
rigid plane frames, 218–236
|
||
|
||
skewed supports, 237
|