김경종
583 lines
22 KiB
Markdown
583 lines
22 KiB
Markdown
<!-- source-page: 721 -->
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<details>
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<summary>text_image</summary>
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1
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2
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3
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L
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L
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</details>
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Figure P16–1
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<details>
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<summary>text_image</summary>
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1
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2
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3
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4
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L
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L
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L
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</details>
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Figure P16–2
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16.2 For the one-dimensional bar discretized into three elements as shown in Figure P16–2, determine the lumped- and consistent-mass matrices. Let the bar properties be $E , \rho ,$ and A throughout the bar.
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16.3 For the one-dimensional bar shown in Figure P16–3, determine the natural frequencies of vibration, $\omega ^ { \prime } \mathbf { s } ,$ using two elements of equal length. Use the consistentmass approach. Let the bar have modulus of elasticity $E ,$ mass density $\rho ,$ and crosssectional area A. Compare your answers to those obtained using a lumped-mass matrix in Example 16.3.
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<details>
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<summary>text_image</summary>
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2L
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</details>
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Figure P16–3
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<details>
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<summary>text_image</summary>
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60 in.
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</details>
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Figure P16–4
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16.4 For the one-dimensional bar shown in Figure P16–4, determine the natural frequencies of longitudinal vibration using first two and then three elements of equal length. Let the bar have $E = 3 0 \times 1 0 ^ { 6 }$ psi, r ¼ 0:00073 lb-s2/in4, A ¼ 1 in2, and $L = 6 0$ in.
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16.5 For the spring-mass system shown in Figure P16–5, determine the mass displacement, velocity, and acceleration for five time steps using the central difference method. Let $k = 2 0 0 0$ lb/ft and $m = 2$ slugs. Use a time step of $\Delta t = 0 . 0 3$ s. You might want to write a computer program to solve this problem.
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<details>
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<summary>text_image</summary>
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k
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m
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F(t)
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</details>
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<details>
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<summary>line</summary>
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| t (s) | F(t) (lb) |
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|---|---|
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| 0.09 | 100 |
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| 0.15 | 50 |
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</details>
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Figure P16–5
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<!-- source-page: 722 -->
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16.6 For the spring-mass system shown in Figure P16–6, determine the mass displacement, velocity, and acceleration for five time steps using (a) the central difference method, (b) Newmark’s time integration method, and (c) Wilson’s method. Let $k = 1 2 0 0$ lb/ft and m ¼ 2 slugs.
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Figure P16–6
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16.7 For the bar shown in Figure P16–7, determine the nodal displacements, velocities, and accelerations for five time steps using two finite elements. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi, $\rho = 0 . 0 0 0 7 3 ~ \mathrm { { l b } } \mathrm { { - s } } ^ { 2 } / \mathrm { { i n } } ^ { 4 } , \varLambda = 1 ~ \mathrm { { i n } } ^ { 2 } .$ , and $L = 1 0 0$ in.
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<details>
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<summary>text_image</summary>
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1
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2
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3
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F(t)
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L
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L
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</details>
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<details>
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<summary>line</summary>
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| t, s | F(t), lb |
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| ------ | -------- |
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| 0.001 | 1000 |
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| 0.002 | 0 |
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</details>
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Figure P16–7
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16.8 For the bar shown in Figure P16–8, determine the nodal displacements, velocities, and accelerations for five time steps using two finite elements. For simplicity of calculations, let $E = 1 \times 1 0 ^ { 6 }$ psi, r ¼ 1 lb-s2/in4, $A = 1 \mathrm { i n } ^ { 2 }$ , and $L = 1 0 0$ in. Use Newmark’s method and Wilson’s method.
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<details>
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<summary>text_image</summary>
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F(t)
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L L
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</details>
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<details>
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<summary>line</summary>
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| t (s) | F(t) (lb) |
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| :--- | :--- |
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| 0.0 | 2000 |
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| 0.5 | 0.5 |
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</details>
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Figure P16–8
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<!-- source-page: 723 -->
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16.9, Rework Problems 16.7 and 16.8 using a computer program.
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16.10
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16.11 For the beams shown in Figure P16–11, determine the natural frequencies using first two and then three elements. Let E; r, and A be constant for the beams.
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(a)
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(b)
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（c）
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<details>
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<summary>text_image</summary>
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L
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</details>
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(d)
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Figure P16–11
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16.12 Rework Problem 16.11 using a computer program with $E = 3 \times 1 0 ^ { 7 }$ psi, $\rho = 0 . 0 0 0 7 3$ lb-s2/in4, A ¼ 1 in2, L ¼ 100 in., and $I = 0 . 0 8 3 3 \mathrm { i n } ^ { 4 }$ .
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16.13, For the beams in Figures P16–13 and P16–14 subjected to the forcing functions-16.14 shown, determine the maximum deflections, velocities, and accelerations. Use a computer program.
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<details>
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<summary>text_image</summary>
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A = 2 × 10⁻² m²
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F(t)
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4 m
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4 m
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</details>
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<details>
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<summary>line</summary>
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| t, s | F(t), kN |
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| ---- | -------- |
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| 0 | 5 |
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| 0.3 | 0 |
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</details>
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Figure P16–13
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<!-- source-page: 724 -->
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Figure P16–14
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16.15, For the rigid frames in Figures P16–15 and P16–16 subjected to the forcing functions 16.16 shown, determine the maximum displacements, velocities, and accelerations. Use a computer program.
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<details>
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<summary>text_image</summary>
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50 psf
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7
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0.5F(t)
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③
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④
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100 psf
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⑦
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8
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10 ft
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5
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0.8F(t)
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②
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⑤
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⑧
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100 psf
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6
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10 ft
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3
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F(t)
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①
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⑥
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⑨
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4
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10 ft
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2
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1
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30 ft
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(Bays on 25-ft centers)
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15 psf
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</details>
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For elements 1 and 9,
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$$
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A = 1 3 \mathrm{in} ^ {2}, I = 2 5 0 \mathrm{in} ^ {4}
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$$
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For elements 2,3,7,and 8,
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$$
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A = 6 \mathrm{in} ^ {2}, I = 1 0 0 \mathrm{in} ^ {4}
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$$
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For elements 4,5,and 6,
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$$
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A = 1 4 \mathrm{in} ^ {2}, I = 8 0 0 \mathrm{in} ^ {4}
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$$
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For all elements,
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$$
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E = 3 0 \times 1 0 ^ {6} \mathrm{psi}
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$$
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<details>
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<summary>line</summary>
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| t, s | F(t), k |
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| ---- | ------- |
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| 0 | 0 |
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| 0.3 | 10 |
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</details>
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Figure P16–15
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16.17 A marble slab with $k = 2 \ : \mathrm { W } / ( \mathrm { m } \cdot \mathrm { ° } \mathrm { C } ) , \rho = 2 5 0 0 \ : \mathrm { k g } / \mathrm { m } ^ { 3 }$ , and $c = 8 0 0 \mathrm { W \cdot s / ( k g \cdot ^ { \circ } C ) }$ is 2 cm thick and at an initial uniform temperature of $T _ { i } = 2 0 0 ^ { \circ } \mathrm { C } .$ . The left surface is suddenly lowered to $0 ^ { \circ } \mathrm { C }$ and is maintained at that temperature while the other surface is kept insulated. Determine the temperature distribution in the slab for 40 s. Use $\begin{array} { r } { \beta = \frac { 2 } { 3 } } \end{array}$ and a time step of ${ \mathrm { ~ ~ \nabla ~ } } 8 { \mathrm { ~ ~ s ~ } } .$ .
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<!-- source-page: 725 -->
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<details>
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<summary>text_image</summary>
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F(t)
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E = 210 GPa
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I = 4 × 10⁻⁴ m⁴
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A = 2 × 10⁻² m²
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6 m
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6 m
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</details>
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<details>
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<summary>line</summary>
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| t, s | F(t), kN |
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| ---- | -------- |
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| 0 | 0 |
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| 0.2 | 25 |
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</details>
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Figure P16–16
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16.18 A circular fin is made of pure copper with a thermal conductivity of $k = 4 0 0$ W/ $( { \bf m } \cdot { \bf \tilde { \mu } C } ) , h = 1 5 0 ~ { \bf W } / ( { \bf m } ^ { 2 } \cdot { \bf \tilde { \mu } C } )$ , mass density $\rho = 8 9 0 0 ~ \mathrm { k g } / \mathrm { m } ^ { 3 }$ , and specific heat $c =$ 375 $\mathbf { J } / ( \mathrm { k g } \cdot \mathrm { ^ { \circ } C } )$ . The initial temperature of the fin is $2 5 ^ { \circ } \mathbf { C }$ . The fin length is 2 cm and the diameter is 0.4 cm. The right tip of the fin is insulated. See Figure P16–18. The base of the fin is then suddenly increased to a temperature of $8 5 ^ { \circ } \mathrm { C }$ and maintained at this temperature. Use the lumped form of the capacitance matrix, a time step of 0.1 s, and $\begin{array} { r } { \beta = \frac { 2 } { 3 } . } \end{array}$ . Use two elements of equal length. Determine the temperature distribution up to 3 s. Compare your results with Example 16.7, which used the consistent form of the capacitance matrix.
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<details>
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<summary>text_image</summary>
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T∞ = 25°C
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Insulated tip
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85°C
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2 cm
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</details>
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Figure P16–18
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16.19, Rework Problems 16.17 and 16.18 using a computer program.
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16.20
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<!-- source-page: 726 -->
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# Introduction
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In this appendix, we provide an introduction to matrix algebra. We will consider the concepts relevant to the finite element method to provide an adequate background for the matrix algebra concepts used in this text.
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# A.1 Definition of a Matrix
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A matrix is an $m \times n$ array of numbers arranged in m rows and n columns. The matrix is then described as being of order $m \times n .$ Equation (A.1.1) illustrates a matrix with m rows and n columns.
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$$
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[ a ] = \left[ \begin{array}{c c c c c c} a _ {1 1} & a _ {1 2} & a _ {1 3} & a _ {1 4} & \dots & a _ {1 n} \\ a _ {2 1} & a _ {2 2} & a _ {2 3} & a _ {2 4} & \dots & a _ {2 n} \\ a _ {3 1} & a _ {3 2} & a _ {3 3} & a _ {3 4} & \dots & a _ {3 n} \\ \vdots & \vdots & \vdots & \vdots & & \vdots \\ a _ {m 1} & a _ {m 2} & a _ {m 3} & a _ {m 4} & \dots & a _ {m n} \end{array} \right] \tag {A.1.1}
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$$
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If $m \neq n$ in matrix Eq. (A.1.1), the matrix is called rectangular. If $m = 1$ and $n > 1$ , the elements of Eq. (A.1.1) form a single row called a row matrix. If $m > 1$ and $n = 1$ , the elements form a single column called a column matrix. If $m = n ,$ the array is called a square matrix. Row matrices and rectangular matrices are denoted by using brackets ½ , and column matrices are denoted by using braces f g. For simplicity, matrices (row, column, or rectangular) are often denoted by using a line under a variable instead of surrounding it with brackets or braces. The order of the matrix should then be apparent from the context of its use. The force and displacement matrices used in structural analysis are column matrices, whereas the stiffness matrix is a square matrix.
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<!-- source-page: 727 -->
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To identify an element of matrix a, we represent the element by $a _ { i j } .$ where the subscripts i and j indicate the row number and the column number, respectively, of a. Hence, alternative notations for a matrix are given by
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$$
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\underline {{a}} = [ a ] = [ a _ {i j} ] \tag {A.1.2}
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$$
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Numerical examples of special types of matrices are given by Eqs. (A.1.3)– (A.1.6). A rectangular matrix a is given by
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$$
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\underline {{a}} = \left[ \begin{array}{l l} 2 & 1 \\ 3 & 4 \\ 5 & 4 \end{array} \right] \tag {A.1.3}
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$$
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where a has three rows and two columns. In matrix a of Eq. (A.1.1), if m ¼ 1, a row matrix results, such as
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$$
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\underline {{a}} = \left[ \begin{array}{l l l l} 2 & 3 & 4 & - 1 \end{array} \right] \tag {A.1.4}
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$$
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If n ¼ 1 in Eq. (A.1.1), a column matrix results, such as
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$$
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\underline {{a}} = \left\{ \begin{array}{l} 2 \\ 3 \end{array} \right\} \tag {A.1.5}
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$$
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If m ¼ n in Eq. (A.1.1), a square matrix results, such as
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$$
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\underline {{a}} = \left[ \begin{array}{l l} 2 & - 1 \\ 3 & - 2 \end{array} \right] \tag {A.1.6}
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$$
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Matrices and matrix notation are often used to express algebraic equations in compact form and are frequently used in the finite element formulation of equations. Matrix notation is also used to simplify the solution of a problem.
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# A.2 Matrix Operations
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We will now present some common matrix operations that will be used in this text.
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# Multiplication of a Matrix by a Scalar
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If we have a scalar k and a matrix c, then the product $\underline { { a } } = k \underline { { c } }$ is given by
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$$
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\underline {{a}} _ {i j} = k \underline {{c}} _ {i j} \tag {A.2.1}
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$$
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—that is, every element of the matrix c is multiplied by the scalar k. As a numerical example, consider
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$$
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\underline {{c}} = \left[ \begin{array}{c c} 1 & 2 \\ 3 & 1 \end{array} \right] \qquad k = 4
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$$
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The product $\underline { { a } } = k \underline { { c } }$ is
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$$
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\underline {{a}} = 4 \left[ \begin{array}{c c} 1 & 2 \\ 3 & 1 \end{array} \right] = \left[ \begin{array}{c c} 4 & 8 \\ 1 2 & 4 \end{array} \right]
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$$
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Note that if c is of order $m \times n ,$ then a is also of order $m \times n$
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<!-- source-page: 728 -->
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# Addition of Matrices
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Matrices of the same order can be added together by summing corresponding elements of the matrices. Subtraction is performed in a similar manner. Matrices of unlike order cannot be added or subtracted. Matrices of the same order can be added (or subtracted) in any order (the commutative law for addition applies). That is,
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$$
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\underline {{{c}}} = \underline {{{a}}} + \underline {{{b}}} = \underline {{{b}}} + \underline {{{a}}} \tag {A.2.2}
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$$
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or, in subscript (index) notation, we have
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$$
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[ c _ {i j} ] = [ a _ {i j} ] + [ b _ {i j} ] = [ b _ {i j} ] + [ a _ {i j} ] \tag {A.2.3}
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$$
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As a numerical example, let
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$$
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\underline {{a}} = \left[ \begin{array}{c c} - 1 & 2 \\ - 3 & 2 \end{array} \right] \qquad \underline {{b}} = \left[ \begin{array}{c c} 1 & 2 \\ 3 & 1 \end{array} \right]
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$$
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The sum a þ b ¼ c is given by
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$$
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\underline {{c}} = \left[ \begin{array}{c c} - 1 & 2 \\ - 3 & 2 \end{array} \right] + \left[ \begin{array}{c c} 1 & 2 \\ 3 & 1 \end{array} \right] = \left[ \begin{array}{c c} 0 & 4 \\ 0 & 3 \end{array} \right]
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$$
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Again, remember that the matrices a, b, and c must all be of the same order. For instance, a $2 \times 2$ matrix cannot be added to a 3 3 matrix.
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# Multiplication of Matrices
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For two matrices a and $\underline b$ to be multiplied in the order shown in Eq. (A.2.4), the number of columns in a must equal the number of rows in $\underline { { b } } .$ For example, consider
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$$
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\underline {{c}} = \underline {{a}} \underline {{b}} \tag {A.2.4}
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$$
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If a is an m n matrix, then $\underline b$ must have n rows. Using subscript notation, we can write the product of matrices a and $\underline b$ as
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$$
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[ c _ {i j} ] = \sum_ {e = 1} ^ {n} a _ {i e} b _ {e j} \tag {A.2.5}
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$$
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where n is the total number of columns in a or of rows in b. For matrix a of order $2 \times 2$ and matrix $\underline b$ of order $2 \times 2$ , after multiplying the two matrices, we have
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$$
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\left[ c _ {i j} \right] = \left[ \begin{array}{l l} a _ {1 1} b _ {1 1} + a _ {1 2} b _ {2 1} & a _ {1 1} b _ {1 2} + a _ {1 2} b _ {2 2} \\ a _ {2 1} b _ {1 1} + a _ {2 2} b _ {2 1} & a _ {2 1} b _ {1 2} + a _ {2 2} b _ {2 2} \end{array} \right] \tag {A.2.6}
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$$
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For example, let
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$$
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\underline {{a}} = \left[ \begin{array}{c c} 2 & 1 \\ 3 & 2 \end{array} \right] \qquad \underline {{b}} = \left[ \begin{array}{c c} 1 & - 1 \\ 2 & 0 \end{array} \right]
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$$
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The product ab is then
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$$
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\underline {{a}} \underline {{b}} = \left[ \begin{array}{c c} 2 (1) + 1 (2) & 2 (- 1) + 1 (0) \\ 3 (1) + 2 (2) & 3 (- 1) + 2 (0) \end{array} \right] = \left[ \begin{array}{c c} 4 & - 2 \\ 7 & - 3 \end{array} \right]
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$$
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<!-- source-page: 729 -->
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In general, matrix multiplication is not commutative; that is,
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$$
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\underline {{a}} \underline {{b}} \neq \underline {{b}} \underline {{a}} \tag {A.2.7}
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$$
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The validity of the product of two matrices $\underline { { \boldsymbol { a } } }$ and $\underline b$ is commonly illustrated by
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$$
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\begin{array}{c} \underline {{a}} \\ (i \times e) \end{array} \begin{array}{c} \underline {{b}} \\ (e \times j) \end{array} = \begin{array}{c} \underline {{c}} \\ (i \times j) \end{array} \tag {A.2.8}
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$$
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where the product matrix $\underline { { \boldsymbol { \mathcal { C } } } }$ will be of order $i \times j ;$ that is, it will have the same number of rows as matrix $\underline { { \boldsymbol { a } } }$ and the same number of columns as matrix $\underline { { b } } .$
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# Transpose of a Matrix
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Any matrix, whether a row, column, or rectangular matrix, can be transposed. This operation is frequently used in finite element equation formulations. The transpose of a matrix $\underline { { \boldsymbol { a } } }$ is commonly denoted by $\underline { { a } } ^ { T }$ . The superscript $T$ is used to denote the transpose of a matrix throughout this text. The transpose of a matrix is obtained by interchanging rows and columns; that is, the first row becomes the first column, the second row becomes the second column, and so on. For the transpose of matrix a,
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$$
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\left[ a _ {i j} \right] = \left[ a _ {j i} \right] ^ {T} \tag {A.2.9}
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$$
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For example, if we let
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$$
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\underline {{a}} = \left[ \begin{array}{c c} 2 & 1 \\ 3 & 2 \\ 4 & 5 \end{array} \right]
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$$
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then
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$$
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\underline {{a}} ^ {T} = \left[ \begin{array}{c c c} 2 & 3 & 4 \\ 1 & 2 & 5 \end{array} \right]
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$$
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where we have interchanged the rows and columns of $\underline { { \boldsymbol { a } } }$ to obtain its transpose.
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Another important relationship that involves the transpose is
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$$
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(\underline {{a}} \underline {{b}}) ^ {T} = \underline {{b}} ^ {T} \underline {{a}} ^ {T} \tag {A.2.10}
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$$
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That is, the transpose of the product of matrices $\underline { { \boldsymbol { a } } }$ and $\underline b$ is equal to the transpose of the latter matrix $\underline b$ multiplied by the transpose of matrix $\underline { { \boldsymbol { a } } }$ in that order, provided the order of the initial matrices continues to satisfy the rule for matrix multiplication, Eq. (A.2.8). In general, this property holds for any number of matrices; that is,
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$$
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(\underline {{a}} \underline {{b}} \underline {{c}} \dots \underline {{k}}) ^ {T} = \underline {{k}} ^ {T} \dots \underline {{c}} ^ {T} \underline {{b}} ^ {T} \underline {{a}} ^ {T} \tag {A.2.11}
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$$
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Note that the transpose of a column matrix is a row matrix.
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As a numerical example of the use of Eq. (A.2.10), let
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$$
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\underline {{a}} = \left[ \begin{array}{c c} 1 & 2 \\ 3 & 4 \end{array} \right] \qquad \underline {{b}} = \left\{ \begin{array}{c} 5 \\ 6 \end{array} \right\}
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$$
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First, $\begin{array} { c } { { \underline { { { a } } } \underline { { { b } } } = \left[ \begin{array} { l l } { { 1 } } & { { 2 } } \\ { { 3 } } & { { 4 } } \end{array} \right] \left\{ \begin{array} { l } { { 5 } } \\ { { 6 } } \end{array} \right\} = \left\{ \begin{array} { l } { { 1 7 } } \\ { { 3 9 } } \end{array} \right\} } } \\ { { \displaystyle ( \underline { { { a } } } \underline { { { b } } } ) ^ { T } = [ 1 7 \quad 3 9 ] } } \end{array}$
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Then, ðA:2:12Þ
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<!-- source-page: 730 -->
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Because $\underline { { b } } ^ { T }$ and $\underline { { a } } ^ { T }$ can be multiplied according to the rule for matrix multiplication, we have
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$$
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\underline {{b}} ^ {T} \underline {{a}} ^ {T} = [ 5 \quad 6 ] \left[ \begin{array}{l l} 1 & 3 \\ 2 & 4 \end{array} \right] = [ 1 7 \quad 3 9 ] \tag {A.2.13}
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$$
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Hence, on comparing Eqs. (A.2.12) and (A.2.13), we have shown (for this case) the validity of Eq. (A.2.10). A simple proof of the general validity of Eq. (A.2.10) is left to your discretion.
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# Symmetric Matrices
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If a square matrix is equal to its transpose, it is called a symmetric matrix; that is, if
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$$
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\underline {{a}} = \underline {{a}} ^ {T}
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$$
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then a is a symmetric matrix. As an example,
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$$
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\underline {{a}} = \left[ \begin{array}{l l l} 3 & 1 & 2 \\ 1 & 4 & 0 \\ 2 & 0 & 3 \end{array} \right] \tag {A.2.14}
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$$
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is a symmetric matrix because each element $a _ { i j }$ equals $a _ { j i }$ for $i \neq j .$ In Eq. (A .2.14), note that the main diagonal running from the upper left corner to the lower right corner is the line of symmetry of the symmetric matrix a. Remember that only a square matrix can be symmetric.
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# Unit Matrix
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The unit (or identity) matrix I is such that
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$$
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\underline {{a}} \underline {{I}} = \underline {{I}} \underline {{a}} = \underline {{a}} \tag {A.2.15}
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$$
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The unit matrix acts in the same way that the number one acts in conventional multiplication. The unit matrix is always a square matrix of any possible order with each element of the main diagonal equal to one and all other elements equal to zero. For example, the 3 3 unit matrix is given by
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$$
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\underline {{I}} = \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]
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$$
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# Inverse of a Matrix
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The inverse of a matrix is a matrix such that
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$$
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\underline {{a}} ^ {- 1} \underline {{a}} = \underline {{a}} \underline {{a}} ^ {- 1} = \underline {{I}} \tag {A.2.16}
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$$
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where the superscript, 1, denotes the inverse of $\underline { { \boldsymbol { a } } }$ as $\underline { { \boldsymbol { a } } } ^ { - 1 }$ . Section A.3 provides more information regarding the properties of the inverse of a matrix and gives a method for determining it.
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