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14.8 $f _ { 1 } = f _ { 3 } = 5 \mathrm { i n } ^ { 3 } / \mathrm { s } , \quad f _ { 2 } = 0$   
14.9 $p _ { 2 } = p _ { 3 } = 1 2 ~ \mathrm { m } , ~ p _ { 5 } = 1 1 ~ \mathrm { m }$

# Chapter 15

15.1 d2x ¼ 0:021 in., d3x ¼ 0:042 in., sx ¼ 0   
15.2 $d _ { 2 x } = 0 , ~ \sigma _ { x } = 5 0 . 4 ~ \mathrm { M P a }$   
15.3 $d _ { 1 x } = d _ { 1 y } = - 0 . 0 1 7 5 \mathrm { ~ i n . , } \quad \sigma ^ { ( 1 ) } = 4 3 5 0 \mathrm { ~ p s i \ : ( T ) }$

$$
\sigma^ {(2)} = - 6 1 5 0 \text {   psi   (C) }, \quad \sigma^ {(3)} = 4 3 5 0 \text {   psi   (T) }
$$

15.4 $d _ { 1 x } = - 0 . 0 2 9 1 \ \mathrm { i n . , } \quad d _ { 1 y } = - 0 . 0 0 9 5 \ \mathrm { i n . }$

$$
\sigma^ {(1)} = - 1 3 7 0 \text {   psi   (C) }, \quad \sigma^ {(2)} = 2 3 7 5 \text {   psi   (T) }, \quad \sigma^ {(3)} = - 1 3 7 0 \text {   psi   (C) }
$$

15.5 $d _ { 2 x } = 1 . 4 4 \times 1 0 ^ { - 4 } \mathrm { m } , ~ \sigma ^ { ( 1 ) } = - 2 0 . 2 \mathrm { M P a } \left( \mathrm { C } \right) , ~ \sigma ^ { ( 2 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 1 \mathrm { M P a } \left( \mathrm { C } \right)$   
15.6 $\begin{array} { r } { d _ { 1 x } = 0 , d _ { 1 y } = 6 . 0 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 5 \mathrm { M P a } ( \mathrm { C } ) } \end{array}$

$$
\sigma^ {(2)} = 1 8. 2 \mathrm{MPa(T)}
$$

15.7 $d _ { 1 x } = 0 , d _ { 1 y } = - 3 . 6 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 2 ) } = 0$   
15.8 $d _ { 2 x } = 0 . 0 1 7 3 \ \mathrm { i n . , } \quad \sigma _ { s t } = 8 4 0 \ \mathrm { p s i } \ ( \mathrm { T } ) , \quad \sigma _ { b r } = 1 6 8 0 \ \mathrm { p s i } \ ( \mathrm { C } )$

15.12 $\mathbf { a } . \ - 0 . 0 0 1 9 0 7 \ \mathrm { i n } . \quad \mathbf { b } . \ \sigma _ { b r } = - 2 8 , 6 0 0 \ \mathrm { p s i } , \quad \sigma _ { m g } = - 1 9 , 0 6 7 \ \mathrm { p s i }$   
15.13 fT1x ¼ -4464 lb, fT1y ¼ -8929 lb, fT2x ¼ 4464 lb

$$
f _ {T 2 y} = - 8 9 2 9 \mathrm{lb}, \quad f _ {T 3 x} = 0, \quad f _ {T 3 y} = 1 7, 8 5 7 \mathrm{lb}
$$

15.14 $f _ { T 1 x } = - 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 1 y } = 0 , f _ { T 2 x } = 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 2 y } = - 8 6 . 2 5 0 ~ \mathrm { k N }$

$$
f _ {T 3 x} = 0, \quad f _ {T 3 y} = 8 6. 2 5 0 \mathrm{kN}
$$

15.15 $f _ { T 1 x } = - 6 0 . 0 \mathrm { ~ k i p } , f _ { T 1 y } = - 9 0 \mathrm { ~ k i p } , f _ { T 2 x } = 6 0 \mathrm { ~ k i p } , f _ { T 2 y } = 0 ,$

$$
f _ {T 3 x} = 0, \quad f _ {T 3 y} = 9 0 \mathrm{kip}
$$

15.16 $f _ { T 1 x } = 1 3 4 \ \mathrm { k N } , f _ { T 1 y } = 1 3 4 \ \mathrm { k N } , f _ { T 2 x } = - 1 3 4 \ \mathrm { k N } , f _ { T 2 y } = 0$

$$
f _ {T 3 x} = 0, \quad f _ {T 3 y} = - 1 3 4 \mathrm{kN}
$$

15.17 $\sigma _ { x } = \sigma _ { y } = - 8 9 2 9 \mathrm { p s i } ( \mathrm { C } ) , \tau _ { x y } = 0$   
15.18 $\sigma _ { x } = 6 7 . 2 ~ \mathrm { M P a } , \sigma _ { y } = 6 7 . 2 ~ \mathrm { M P a } , \tau _ { x y } = 0$   
15.19 $\{ f _ { T } \} = \frac { A E \alpha _ { 0 } } { 6 } \left\{ \begin{array} { r } { - 4 t _ { 1 } - 5 t _ { 2 } } \\ { 4 t _ { 1 } + 5 t _ { 2 } } \end{array} \right\}$   
15.20 $\frac { A E \alpha } { 2 } \bigg \{ { - t _ { 1 } - t _ { 2 } } \bigg \}$

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15.21 $\{f_{T}\}=\frac{2\pi\bar{r}AE\alpha(\Delta T)[\bar{B}]^{T}}{1-2\nu}\left\{\begin{array}{l}1\\ 1\\ 1\\ 0\end{array}\right\}$

15.22 $d_{2x} = 0.8 \times 10^{-3}$ in., $d_{3x} = 0$ , $d_{3y} = 0.8 \times 10^{-3}$ in.

$d_{4x} = d_{4y} = 0.8 \times 10^{-3}$ in.; stresses are zero

15.23 $d_{2x} = 0.989 \times 10^{-3}$ in., $d_{3x} = -0.756 \times 10^{-3}$ in.,

$$
d _ {3 y} = 0. 9 8 9 \times 1 0 ^ {- 3} \text {   in. }, \quad d _ {4 x} = 0. 1 3 2 \times 1 0 ^ {- 2} \text {   in. },
$$

$$
d _ {4 y} = 0. 2 0 4 5 \times 1 0 ^ {- 2} \text {in.}, \quad \sigma_ {1} ^ {(1)} = 1 7 \mathrm{ksi}, \quad \sigma_ {2} ^ {(2)} = - 1 7 \mathrm{ksi}
$$

# Chapter 16

16.1 $[M] = \frac{\rho AL}{6}\left[ \begin{array}{ccc}2 & 1 & 0\\ 1 & 4 & 1\\ 0 & 1 & 2 \end{array} \right]$

16.2 a. $[M] = \frac{\rho AL}{2}\left[ \begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 1 \end{array} \right]$

$\mathbf{b.}[M] = \frac{\rho AL}{6}\left[ \begin{array}{cccc}2 & 1 & 0 & 0\\ 1 & 4 & 1 & 0\\ 0 & 1 & 4 & 1\\ 0 & 0 & 1 & 2 \end{array} \right]$

16.3 $\omega_{1} = 0.806\sqrt{u},\omega_{2} = 2.81\sqrt{\mu}$

16.4 $\omega_{1} = 5.368\times 10^{3}\mathrm{rad / s},\quad \omega_{2} = 17.556\times 10^{3}\mathrm{rad / s}$

16.5 a. $t(\mathrm{s})$ $d_{i}(\mathrm{ft})$ $\dot{d}_i(\mathrm{ft / s})$ $\ddot{d}_i(\mathrm{ft / s^2})$
<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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16.6 a. t ðsÞ $d _ { i }$ ðftÞ $\dot { d } _ { i } \left( \mathrm { f t } / \mathrm { s } \right)$ $\ddot { d } _ { i } \ ( \mathrm { f t } / \mathrm { s } ^ { 2 } )$ 

<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>

<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>

16.7 Node t ðsÞ di ðin:Þ $\dot { d } _ { i } \ ( \mathrm { i n . / s } )$ $\ddot { d } _ { i } \ ( \mathrm { i n } . / \mathrm { s } ^ { 2 } )$ 

<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>

16.8 Using Newmark’s method with $\begin{array} { r } { \gamma = \frac { 1 } { 2 } , \beta = \frac { 1 } { 6 } } \end{array}$ 

<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>

$\begin{array} { l l l } { { { \bf 1 6 . 1 1 } } } & { { { \bf a . } { \ \omega } \omega _ { 1 } = \displaystyle \frac { 3 . 1 5 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { \omega _ { 2 } = \displaystyle \frac { 1 6 . 2 4 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { { \bf c . } { \omega } \omega _ { 1 } = \displaystyle \frac { 9 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } \\ { { } } & { { { \bf d . } { \omega } = \displaystyle \frac { 1 4 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } & { { } } & { { { \bf d . } } } \end{array}$ EI 1=2 EI 1=2 EI 1=2

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<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>

<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>

<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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16.18 

<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>

# Appendix A

A1. a. $\left[ { \begin{array} { r r } { 3 } & { 0 } \\ { - 3 } & { 1 2 } \end{array} } \right]$ b. Nonsense c. Nonsense   
$\left\{ { \begin{array} { l } { 1 1 } \\ { 9 } \\ { 1 1 } \end{array} } \right\}$ e. Nonsense f. $\left[ { \begin{array} { r r r } { 1 0 } & { 7 } & { 6 } \\ { 3 } & { - 1 } & { 7 } \end{array} } \right]$

$\left[ \begin{array} { l l } { 1 } & { 0 } \\ { { \frac { 1 } { 4 } } } & { { \frac { 1 } { 4 } } } \end{array} \right]$

A3. ${ \frac { 1 } { 1 7 } } \left[ { \begin{array} { r r r } { 1 2 } & { - 3 } & { - 8 } \\ { - 3 } & { 5 } & { 2 } \\ { - 8 } & { 2 } & { 1 1 } \end{array} } \right]$

A4. Nonsense

A5. $\left[ \begin{array} { l l } { { \frac { 1 } { 2 } } } & { 0 } \\ { { \frac { 1 } { 8 } } } & { { \frac { 1 } { 8 } } } \end{array} \right]$

A6. Same as A3

A8. $\left[ \begin{array} { c c } { \cos \theta } & { - \sin \theta } \\ { \sin \theta } & { \cos \theta } \end{array} \right]$

# Appendix B

B1. $x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2$

B2. $x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2$

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B3. $x_{1} = 2.5, x_{2} = -1, x_{3} = 0.5$

B4. $x_{1} = 3, x_{2} = -1, x_{3} = -2$

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. $f_{1y} = f_{2y} = -10 \, kN$ , $m_{1} = -m_{2} = -6.67 \, kN \cdot m$

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# 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

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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

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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

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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