김경종
543 lines
25 KiB
Markdown
543 lines
25 KiB
Markdown
<!-- source-page: 81 -->
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(a) Translation in the direction of bar 2.
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(b) Rotation through a small angle about node 1 (the point $x = L_{1}, y = 0$ ).
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(c) Rotation through a small angle about the point $x = L_{3}, y = L_{1}$ .
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2.9 For the truss of Fig. 2.2-1, let bar lengths be $L_{1} = 4$ , $L_{2} = 5$ , and $L_{3} = 3$ . Now consider the rigid-body motion $\{\mathbf{D}\} = [1\quad 3\quad 4\quad 0\quad 0\quad -4]^{T}$ . Sketch the displaced truss. Explain why the product $[\mathbf{K}]\{\mathbf{D}\}$ is not zero.
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2.10 Consider the circular four-spring structure of Problem 2.2. Write the displacement vector $\{\mathbf{D}\}$ for each possible rigid-body motion, and show that $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{0}\}$ .
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2.11 Imagine that a $90^{\circ}$ curved-beam element is formulated using the six d.o.f. shown.
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(a) Does each row (or each column) of [k] sum to zero? Why or why not?
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(b) Sketch the approximate displaced shape if the d.o.f. are $\{\mathbf{d}\} = c[1\quad 1\quad 1\quad 1\quad 1\quad 1]^T$ , where $c$ is a small number.
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(c) Write $\{\mathbf{d}\}$ (all six terms) such that $[\mathbf{k}]\{\mathbf{d}\} = \{\mathbf{0}\}$ . There are infinitely many possibilities. Can you write three $\{\mathbf{d}\}$ 's that are linearly independent?
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<details>
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<summary>text_image</summary>
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y,v
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d5
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d4
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d6
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d1
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d3
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x,u
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d2
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R
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</details>
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Problem 2.11
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<details>
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<summary>text_image</summary>
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D₁
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D₃
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D₅
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D₂
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1
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2
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3
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D₆
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L₁
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L₂
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</details>
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Problem 2.12
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2.12 The beam shown contains two elements. Each node has two d.o.f., one in translation and one in rotation.
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(a) How many rigid-body motions are possible? Write a suitable d.o.f. vector $\{\mathbf{D}\}$ for each.
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(b) Let $\{\mathbf{D}\} = c[1\quad 1\quad 1\quad 1\quad 1\quad 1]^T$ , where $c$ is a small number. Sketch the deformed structure. Is $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{0}\}$ ? (Do not derive $[\mathbf{K}]$ .)
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(c) Let $\{\mathbf{D}\} = c[1 0 0 0 0 0]^T$ . Sketch the deformed structure and show qualitatively by properly directed arrows the nodal loads required.
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2.13 Consider the plane truss of Problem 2.6.
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(a) Impose support conditions implied by the sketch. That is, by discarding the appropriate rows and columns, obtain a smaller [K] that operates on only the active d.o.f.
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(b) For the loading by force $F$ shown, write the 4 by 1 vector $\{\mathbf{D}\}$ by inspection (not by solving simultaneous equations). Hence, find the nodal loads $\{\mathbf{R}\} = [\mathbf{K}]\{\mathbf{D}\}$ . Are these loads physically reasonable?
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2.14 The two-element structure shown is built of standard beam elements with two d.o.f. per node (see Fig. 1.2-2). By an error, the boundary conditions
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<details>
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<summary>text_image</summary>
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z,w
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1
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2
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3
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P
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x
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L/2
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L/2
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</details>
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Problem 2.14
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given to a computer program are $w_{1} = \theta_{1} = \theta_{2} = \theta_{3} = 0$ . The expected result, $w_{3} = PL^{3}/3EI$ , is not computed. What value of $w_{3}$ is in fact computed by the program? (The question can be answered by sketching the deformed structure and applying elementary beam theory.)
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2.15 (a) Let $k_{1} = k_{2} = k_{3} = k$ in Eq. 2.3-8. Solve for $u_{1}, v_{1}$ , and $v_{3}$ in terms of $P$ and $k$ .
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(b) Using the results of part (a), compute $p_2, q_2$ , and $p_3$ (Eq. 2.3-9). Show these forces applied to a free-body diagram of the truss, and check that static equilibrium conditions are satisfied.
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# Section 2.4
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2.16 (a) Derive a 4 by 4 element stiffness matrix for a uniform plane truss member, using d.o.f. shown in the sketch.
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(b) Check that nodal loads $[k]\{d\}$ are zero for the following rigid-body motions: x-direction translation, y-direction translation, and a small counterclockwise rotation about node i.
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<details>
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<summary>text_image</summary>
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L
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u_j
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j
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v_j
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u_i
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β
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i
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v_i
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</details>
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Problem 2.16
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2.17 Consider a straight, uniform shaft of solid circular cross section, with a node at each end.
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(a) Let nodal d.o.f. be angular rotation vectors parallel to the bar, one at each end. Nodal loads are axially directed torque vectors. What is [k], in terms of the length, shear modulus, and radius of the cross section?
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(b) Let the bar be inclined at angle $\beta$ to the $x$ axis, with $\theta_{x}$ and $\theta_{y}$ as d.o.f. at each node (rotations about the $x$ and $y$ coordinate axes). What is [k]? As in part (a), consider torsional stiffness only.
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2.18 A uniform bar of axial stiffness $k = AE / L$ is arbitrarily oriented in space. Cosines of angles between the bar and the $x, y$ , and $z$ coordinate axes are $\ell, m$ , and $n$ . Nodal d.o.f. are translations $u, v$ , and $w$ at each end. Derive the 6 by 6 element stiffness matrix.
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# Section 2.5
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2.19 For each of the following structures, generate the structure stiffness matrix by writing element matrices “structure size” and assembling them.
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(a) The four-spring circular structure of Problem 2.2.
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(b) The four-bar truss of Problem 2.6 (without supports).
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# Section 2.6
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2.20 The uniform bar shown hangs under its own weight $W$ . Compute the deflection of the lower end in terms of $W$ , $L$ , $A$ , and $E$ . (Obtain $[\mathbf{K}_{11}]$ of Eq. 2.3-3 by retaining only active d.o.f. The uppermost node is fixed.)
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(a) Use one element of length $L$ .
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(b) Use two elements, each of length $L / 2$ .
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Problem 2.20
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<details>
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<summary>text_image</summary>
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y
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1
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2
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3
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x,u
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2L
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L
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</details>
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Problem 2.21
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2.21 The uniform bar shown is built of two elements. Both elements are uniformly heated T degrees. Obtain $[K_{11}]$ of Eq. 2.3-3 by retaining only the active d.o.f. $(u_{2}$ and $u_{3})$ . Solve for $u_{2}$ and $u_{3}$ in terms of $\alpha$ , L, and T.
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# Section 2.7
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2.22 As suggested in Section 2.7, permute node labels of both elements in Fig. 2.7-1 so that $j$ replaces $i, k$ replaces $j$ , and $i$ replaces $k$ . Maintain structure node labels 1, 2, 3, and 4 where they are shown in Fig. 2.7-1. Show that [K] of Eq. 2.7-4 is again produced.
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2.23 Change structure node labels in Fig. 2.7-1 from 1, 2, 3, and 4 to 1, 3, 4, and 7, respectively. Thus, the two elements shown are regarded as a fragment of a larger structure. To what row and column location in a 7 by 7 array [K] is each of the $a$ 's and $b$ 's in Eqs. 2.7-1 assigned?
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2.24 Add the following elements to Fig. 2.7-2. Show the locations of nonzero element coefficients in $\{R\}$ and in $[K]$ , as in Fig. 2.7-2.
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(a) Attach a triangular element 1-4-6 to nodes 1 and 4 of existing element 2.
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(b) Attach a rectangular element 3-5-7-8 to nodes 3 and 5 of existing element 1.
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2.25 Manually apply the assembly algorithm of Fig. 2.7-3 to matrix $[\mathbf{k}]_{\mathrm{I}}$ of Eq. 2.7-1. Specifically, by supplying numerical indexes for arrays, discover where the $a$ 's are placed in array S for
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(a) I = 1 in the DO 400 loop.
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(b) I = 2 in the DO 400 loop.
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(c) $l = 3$ in the DO 400 loop.
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2.26 Imagine that coefficients in the plane truss element stiffness matrix are arranged to suit the order of d.o.f. $\{\mathbf{d}\} = \left[u_i \quad u_j \quad v_i \quad v_j\right]^T$ . If structure d.o.f. still have the order $\{\mathbf{D}\} = \left[u_1 \quad v_1 \quad u_2 \ldots u_N \quad v_N\right]^T$ , revise the assembly algorithm of Fig. 2.7-4 as required.
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2.27 Revise the assembly algorithm of Fig. 2.7-4 to deal with the following elements:
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(a) a plane frame element (three d.o.f. per node).
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(b) a space frame element (six d.o.f. per node).
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# Section 2.8
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2.28 (a) For each of the plane trusses shown, show the topology of the structure stiffness matrix, in the manner of Fig. 2.8-3a. However, let each X represent a 2 by 2 submatrix: thus, the sketch of [K] will have eight rows and eight columns of submatrices X.
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(b) How would your answer to part (a) change if the structure were a plane frame? Or a network of electrical resistors?
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<details>
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<summary>text_image</summary>
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2 4 6 8
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1 3 5 7
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</details>
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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1 --> 2
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2 --> 3
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3 --> 4
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4 --> 5
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5 --> 6
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6 --> 7
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7 --> 8
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1 --> 2
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2 --> 3
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3 --> 4
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4 --> 5
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5 --> 6
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6 --> 7
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7 --> 8
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8 --> 1
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```
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</details>
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Problem 2.28
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<details>
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<summary>text_image</summary>
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A
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B
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C D E F G H
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J
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I
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</details>
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(a)
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<details>
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<summary>text_image</summary>
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A
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H
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B
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G
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I
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C
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F
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D
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E
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</details>
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(b)
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<details>
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<summary>text_image</summary>
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A
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B
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C
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D
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E
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I
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H
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J
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K
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L
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F
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G
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N
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M
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</details>
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(c)
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<details>
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<summary>text_image</summary>
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H
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B
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C
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I
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G
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E
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D
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A
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F
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J
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</details>
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(d)
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Problem 2.29
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<!-- source-page: 85 -->
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2.29 Each of the structures shown is hypothetical and has one d.o.f. per node. Solid lines indicate connectivity between nodes. (Letters are for use in another problem.) Number the nodes so as to achieve minimum bandwidth of the coefficient matrix [K]. Sketch the topology of the assembled matrix [K] (as in Fig. 2.8-3a).
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2.30 In an element having many nodes, let $i$ and $j$ represent respectively the highest and lowest structure node numbers connected to that element. If $i - j$ happens to be the largest difference for any element of the structure, and if $n$ is the number of d.o.f. per node, what is semibandwidth $b$ in terms of $i, j$ , and $n$ ?
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2.31 Apply the formula for $b$ devised in Problem 2.30 to the following structures: (a) the plane truss in Fig. 2.2-1. (b) the first structure shown in Problem 2.28, regarded as a plane frame (three d.o.f. per node).
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2.32 Consider the structures of Problem 2.29. Assign node numbers by the following system. Pick a starting node (say A) and call it 1. Number as 2, 3, and so on, nodes that share an element with node 1 (thus, in (b), node numbers become H = 2 and B = 3). Next, number nodes that share an element with nodes 2, 3, and so on. (Figure 2.9-1 shows the results of such a scheme, but starting with the highest number and counting down.) For one d.o.f. per node, what semibandwidth b do you obtain?
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2.33 Reverse the node numberings found in Problem 2.32. For each structure, how many fills are there during equation solving, both in the original numbering of Problem 2.32 and in the reversed numbering?
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# Section 2.10
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2.34 For each of the plane trusses shown in Problem 2.28, write the "input" and "converted" forms of array ID (see Figs. 2.10-2 and 2.10-3). As support conditions, assume that all d.o.f. are set to zero at the upper left node and at the lower right node.
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2.35 Repeat Problem 2.34, but regard each structure as a plane frame (nodal d.o.f. u, v, and $\theta_{z}$ ).
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2.36 (a) The unsupported plane truss shown has eight d.o.f. Set up an 8 by 8 stiffness array [K]. Write a bar number in those positions of [K] that
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<details>
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<summary>text_image</summary>
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y,v
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3
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⑤
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4
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②
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③
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④
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1
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2
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x,u
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</details>
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(a)
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$$
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\left[ \begin{array}{c c c c c} 1 & & & & \\ & 5 & & 5 & \\ & & 2 & & \\ & 5 & & 3, 5 & 3 \\ & & & 3 & 3, 4 \end{array} \right]
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$$
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(b)
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Problem 2.36
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<!-- source-page: 86 -->
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receive nonzero stiffness contributions from that bar (e.g., write $K_{22} = K_{26} = K_{62} = K_{66} = 2$ , from bar 2).
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(b) After support conditions are imposed by use of array ID, a similarly diagrammed matrix [K] appears as shown. Sketch the truss, showing the supports implied.
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2.37 Imagine that the structure in Fig. 2.10-1 is a plane frame, for which nodal d.o.f. are $u, v,$ and $\theta_{z}$ . Supports apply no nodal moments.
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(a) Modify the ID arrays in Figs. 2.10-2 and 2.10-3 as required.
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(b) Modify the assembly algorithm of Fig. 2.10-5 as required.
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2.38 Consider the 2 by 2 system of equations
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$$
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\left[ \begin{array}{l l} K _ {1} & K _ {2} \\ K _ {3} & K _ {4} \end{array} \right] \left\{ \begin{array}{l} x \\ y \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ b \end{array} \right\}
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$$
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Use the penalty method to impose the result $x = c$ . Show that exact values of $x$ and $y$ are approached as the added stiffness approaches infinity.
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2.39 How would you impose a prescribed relative displacement by the penalty method? Consider, for example, imposing $u_4 - u_2 = c$ in Fig. 2.11-1a, where $c$ is a constant. Give a physical explanation, then state exactly which coefficients in the equations $[\mathbf{K}]\{\mathbf{D}\} = \{\mathbf{R}\}$ must be changed and how you would change them. (This procedure is not recommended; see the Caution in Section 2.10.)
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2.40 Consider the axially loaded structure in Fig. 2.11-1a and the $[K]\{D\} = \{R\}$ equation in Fig. 2.11-1b. Impose the displacement $u_{3} = 6$ and solve for $u_{2}$ and $u_{4}$ .
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(a) Use the penalty method of Fig. 2.10-6.
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(b) Use the “zero-one” procedure of Fig. 2.10-7.
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(c) Use Eq. 2.10-1, and determine the supplementary terms on the right-hand side by summing element contributions, as suggested below Eq. 2.10-1.
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2.41 Consider again the four-spring circular structure of Problem 2.2. No forces are applied, but displacements $u_{2} = u_{4} = c$ are prescribed, where c is a constant. Impose these displacements and solve for $u_{1}$ and $u_{3}$ . Use the “zero-one” procedure of Fig. 2.10-7.
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2.42 From the “converted” ID arrays found in Problem 2.34, compute semi-bandwidth b by applying the method described at the end of Section 2.10.
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2.43 Write a Fortran algorithm that calculates semibandwidth $b$ according to the procedure outlined at the end of Section 2.10.
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# Section 2.11
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2.44 Let $k_{1} = k_{2} = k_{3} = k$ in Eq. 2.3-8. Calculate $u_{1}, v_{1}$ , and $v_{3}$ in terms of $P$ and $k$ by applying the Gauss elimination method.
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2.45 Consider the circular four-spring structure of Problem 2.2. Without imposing any support condition, show the four modified [K]'s produced by successive steps of Gauss elimination (as in Fig. 2.11-1).
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2.46 A one-element cantilever beam is shown. Also shown is the stiffness matrix that operates on the unrestrained d.o.f. $w_{2}$ and $\theta_{2}$ . In parts (a) and (b) carry
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<!-- source-page: 87 -->
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$$
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[ K ] = \left[ \begin{array}{c c} 1 2 E I / L ^ {3} & - 6 E I / L ^ {2} \\ - 6 E I / L ^ {2} & 4 E I / L \end{array} \right] \begin{array}{l} w _ {2} \\ \theta_ {2} \end{array}
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$$
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Problem 2.46
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out one step of Gauss elimination, and explain the physical meaning of the diagonal coefficient that remains.
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(a) Eliminate $w_{2}$ (reduction of [K] to upper triangular form).
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(b) Eliminate $\theta_{2}$ (reduction of [K] to lower triangular form).
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2.47 Consider reduction of [K] to upper triangular form by Gauss elimination. Coefficients that are initially zero may become nonzero in this process. In the following matrices, which zeros above the diagonal become nonzero?
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(a) [K] of Eq. 2.2-6.
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(b) [K] of Fig. 2.7-2.
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2.48 Imagine that no boundary conditions are imposed, so that too many structure d.o.f. remain active. A solution for the d.o.f. by Gauss elimination is started, but fails during the attempt to eliminate the $n$ th d.o.f. For the following structures, what is $n$ , and why?
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(a) The plane truss of Fig. 2.8-2a (allow two d.o.f. per node).
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(b) Imagine that Fig. 2.8-2a represents a plane frame (allow three d.o.f. per node).
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(c) The network of Fig. 2.9-1 (allow one d.o.f. per node).
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(d) The beam of Problem 2.12 (allow the six d.o.f. shown).
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# Section 2.12
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2.49 Consider the uniform hanging bar of Problem 2.20. Assume that finite element analysis yields nodal displacements that are exact. Plot the correct distribution of axial stress (from $W / A$ at the top to zero at the bottom). On the same plot show the stress distribution predicted by finite elements using
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(a) one element.
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(b) two identical elements.
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(c) four identical elements.
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2.50 The uniform bar shown is built of two identical bar elements and is loaded by axially directed forces $P_{2}$ and $P_{3}$ at nodes 2 and 3, respectively. Impose
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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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P₂
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P₃
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L
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L
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</details>
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Problem 2.50
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<details>
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<summary>text_image</summary>
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3
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3
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4
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y,v
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2
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1
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x,u
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</details>
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Problem 2.51
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<!-- source-page: 88 -->
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the displacement $u_{2} = \overline{u}_{2}$ by the penalty method described in Section 2.10, using $k_{s} = 1000AE/L$ . Solve for $u_{2}$ and $u_{3}$ , then compute loads $\{R\} = [K]\{D\}$ , where [K] pertains to the original structure. Interpret the result for the special cases $\overline{u}_{2} = 0$ and $P_{3} = 0$ .
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2.51 Let $AE / L = 2(10)^6$ N/m for each bar of the two-bar truss shown.
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(a) Set up the 2 by 2 structure stiffness matrix that operates on $\{\mathbf{D}\} = |u_1, v_1|^T$ .
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(b) Let there be a prescribed downward displacement of 0.0001 m at node 1. No horizontal load is applied and no horizontal displacement is prescribed at node 1. Modify the structure equations using the penalty method procedure described in Section 2.10, using $k_{s} = 1000 \, AE/L$ .
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(c) Solve for $u_{1}$ by Gauss elimination.
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(d) Solve for the vertical force applied to node 1.
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# Section 2.13
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2.52 Repeat the example given in Section 2.13, but allow node 1 to move axially, so that the active d.o.f. are $u_{1}, u_{2}$ , and $u_{3}$ .
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2.53 Analyze the truss shown for nodal displacements and element stresses. Follow the steps used in the example problem of Section 2.13. Let $E = 200$ GPa for each bar.
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<details>
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||
<summary>text_image</summary>
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P=10,000 N
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400 mm
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y,v
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1
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A=200 mm²
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2
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500 mm
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A=140 mm²
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x,u
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P
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4
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3
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</details>
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Problem 2.53
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2.54 Using the steps listed in Section 2.13 as a guide, write a computer program for the analysis of plane trusses. An algorithm for equation solving is given in Appendix B.
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<!-- source-page: 89 -->
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# STATIONARY PRINCIPLES, THE RAYLEIGH-RITZ METHOD, AND INTERPOLATION
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The equilibrium configuration of a system is found by analysis of its potential energy. Expressions for potential energy are presented. These and other integral expressions, called functionals, are introduced as a starting point for an approximation technique—namely, the Rayleigh–Ritz method—whose modern form is the finite element method. Interpolation, necessary to the method, is described.
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# 3.1 INTRODUCTION
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In preceding chapters, element stiffness matrices [k] have been formulated by direct physical argument. This is easily done for truss and beam elements by activating d.o.f. in turn and computing the nodal loads required to maintain the deformation state. Finite elements obtained by discretization of a continuum are not as easily formulated. (For example, is there an easy way to find nodal forces that appear in response to displacement $u_{3}$ in Fig. 1.1-2c?) A systematic and general way of obtaining [k] is needed. One of the best ways is the Rayleigh–Ritz method. An alternative, the method of weighted residuals, is discussed in Chapter 15.
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The Rayleigh–Ritz method has a classical form and a finite element form. In the classical form, an approximating field is defined over the entire region of interest. In the finite element form, the approximating field is defined in piecewise fashion. As degrees of freedom, finite elements use nodal values of the field (and perhaps nodal values of one or more spatial derivatives of the field as well). By degrees of freedom (d.o.f.) we mean independent quantities used to define a configuration of a system that violates neither compatibility conditions nor constraints such as support conditions. Using more general terms, one can say that d.o.f. are quantities used to define the spatial variation of an approximating field.
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In order to analyze a continuum by use of the Rayleigh–Ritz method, one must have a functional. A functional is an integral expression that implicitly contains differential equations that describe the problem. In structural mechanics the most widely used functional is the expression for potential energy. Functionals are also available for problems of heat conduction, acoustic modes in cavities, certain types of fluid flow, and other problems. We will present some of these functionals and will show how they are used to produce finite element formulations.
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Differential equations are said to state a problem in the strong form. An integral expression such as a functional that implicitly contains the differential equations is called the weak form. The strong form states conditions that must be met at every material point, whereas the weak form states conditions that must be met
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|
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only in an average sense.
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|
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diff eq—evolypic
|
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inv. eq—imavex
|
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|
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<!-- source-page: 90 -->
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A functional, such as that for potential energy $\Pi_p$ , contains integrals that span the line, area, or volume of interest. After applying the Rayleigh-Ritz method, the $\Pi_p$ expression contains no integrals and is no longer called a functional. Rather, $\Pi_p$ is then a function of a finite number of d.o.f. Indeed, for an initially discrete structure such as a truss, no integrals need be invoked in writing the $\Pi_p$ expression. We will consider these “initially discrete” forms first, then return to integral forms later in this chapter.
|
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|
||
Physical insight was responsible for the early rapid development of the finite element method and for its ready appeal to stress analysts. A more mathematical approach augments physical understanding by placing the finite element method on a sound foundation, thus allowing statements to be made regarding bounds and convergence, and suggesting solution tactics that are not apparent from physical reasoning alone.
|
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|
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# 3.2 PRINCIPLE OF STATIONARY POTENTIAL ENERGY
|
||
|
||
In the present section we consider time-independent problems of structural mechanics. We define a system as the physical structure and the loads applied to it. The configuration of a system is the set of positions of all particles of the structure. Let the system have a reference configuration $C_R$ and a displaced configuration $C_D$ . A system is called conservative if work done by internal forces and work done by external loads are each independent of the path taken between $C_R$ and $C_D$ . In an elastic structure, work done by internal forces is equal in magnitude to the change in strain energy.
|
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|
||
The loaded spring of Fig. 3.2-1 is a case in point. Let $C_R$ and $C_D$ refer to unstretched and stretched configurations, respectively. If the spring dissipates no energy, then the work of internal forces (i.e., strain energy in the spring) depends only on stretch $D$ , not on whether the passage from $C_R$ to $C_D$ is via path $A$ or path $B$ . Similarly, if external load $P$ has constant magnitude and constant direction, it does negative work of magnitude $PD$ regardless of the path taken from $C_R$ to $C_D$ . We conclude that because internal forces and external loads are both conservative, so is the system.
|
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|
||
Boundary conditions are of two types: essential (or principal) and nonessential (often called natural). In the finite element method, essential boundary conditions are prescribed values of nodal d.o.f., and nonessential boundary conditions are prescribed values of higher derivatives of the field quantity than are usually used as nodal d.o.f. For example, if standard beam elements are used, nodal d.o.f. are lateral deflection w and its first derivative, $w_{xx}$ . When these elements are used to analyze the beam of Fig. 3.2-2a, essential boundary conditions (which can also be called geometric or kinematic in this problem) are that w = 0 and $w_{xx} = 0$ .
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
k
|
||
C_R
|
||
P
|
||
C_D
|
||
B
|
||
A
|
||
D
|
||
</details>
|
||
|
||
Figure 3.2-1. A linear spring of stiffness k loaded by a constant force P that acts parallel to the x axis. Hypothetical displacement paths A and B of the loaded point are shown by dashed lines.
|