김경종
551 lines
18 KiB
Markdown
551 lines
18 KiB
Markdown
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<details>
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<summary>natural_image</summary>
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Diagram of a mesh grid structure with circular nodes and directional arrows, no readable text or symbols
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</details>
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Max Principal
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<details>
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<summary>text_image</summary>
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H 1932.5
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—— 1689.9
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—— 1447.3
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—— 1204.7
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—— 962.06
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—— 719.44
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—— 476.82
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—— 234.21
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L -8.409
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</details>
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Figure 9–18 Principal stress plot for shaft of Figure 9–17
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developed in Chapter 6. Therefore, the two-dimensional element in commercial computer programs with the axisymmetric element selected will allow for the analysis of axisymmetric structures.
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Finally, note that other axisymmetric elements, such as a simple quadrilateral (one with four corner nodes and two degrees of freedom per node, as used in the steel die analysis of Figure 9–15) or higher-order triangular elements, such as in Reference [6], in which a cubic polynomial involving ten terms (ten a’s) for both u and w, could be used for axisymmetric analysis. The three-noded triangular element was described here because of its simplicity and ability to describe geometric boundaries rather easily.
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# d References
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[1] Utku, S., ‘‘Explicit Expressions for Triangular Torus Element Stiffness Matrix,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 6, No. 6, pp. 1174–1176, June 1968.
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[2] Zienkiewicz, O. C., The Finite Element Method, 3rd ed., McGraw-Hill, London, 1977.
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[3] Clough, R., and Rashid, Y., ‘‘Finite Element Analysis of Axisymmetric Solids,’’ Journal of the Engineering Mechanics Division, American Society of Civil Engineers, Vol. 91, pp. 71–85, Feb. 1965.
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[4] Rashid, Y., ‘‘Analysis of Axisymmetric Composite Structures by the Finite Element Method,’’ Nuclear Engineering and Design, Vol. 3, pp. 163–182, 1966.
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[5] Wilson, E., ‘‘Structural Analysis of Axisymmetric Solids,’’ Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 12, pp. 2269–2274, Dec. 1965.
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[6] Chacour, S., ‘‘A High Precision Axisymmetric Triangular Element Used in the Analysis of Hydraulic Turbine Components,’’ Transactions of the American Society of Mechanical Engineers, Journal of Basic Engineering, Vol. 92, pp. 819–826, 1973.
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[7] Greer, R. D., The Analysis of a Film Tower Die Utilizing the ANSYS Finite Element Package, M.S. Thesis, Rose-Hulman Institute of Technology, Terre Haute, IN, May 1989.
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[8] Gere, J. M., Mechanics of Materials, 5th ed., Brooks/Cole Publishers, Pacific Grove, CA, 2001.
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[9] Cook, R. D., Malkus, D. S., Plesha, M. E., and Witt, R. J., Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, New York, 2002.
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[10] Cook, R. D., and Young, W. C., Advanced Mechanics of Materials, Macmillan, New York, 1985.
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[11] Algor Interactive Systems, 150 Beta Drive, Pittsburgh, PA 15238.
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[12] Swanson, J. A. ANSYS-Engineering Analysis System’s User’s Manual, Swanson Analysis Systems, Inc., Johnson Rd., P.O. Box 65, Houston, PA 15342.
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# Problems
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9.1 For the elements shown in Figure P9–1, evaluate the stiffness matrices using Eq. (9.2.2). The coordinates are shown in the figures. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 2 5$ for each element.
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<details>
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<summary>text_image</summary>
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(0, 2)
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3
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(0, 0)
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1
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2
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(2, 0)
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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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(0, 0)
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1
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2
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3
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(2, 2)
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(2, 0)
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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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3
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(1, 2)
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(0, 0) 1 2 (2, 0)
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</details>
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Figure P9–1
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9.2 Evaluate the nodal forces used to replace the linearly varying surface traction shown in Figure P9–2. Hint: Use Eq. (9.1.34).
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<details>
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<summary>text_image</summary>
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z
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3
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p0
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1
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2
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r
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b
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h
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</details>
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Figure P9–2
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9.3 For an element of an axisymmetric body rotating with a constant angular velocity $\omega = 2 0$ rpm as shown in Figure P9–3, evaluate the body-force matrix. The coordinates of the element are shown in the figure. Let the weight density $\rho _ { w }$ be 0.283 lb/in3.
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<details>
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<summary>text_image</summary>
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ω
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3 (6, 6)
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1 (4, 4) 2 (6, 4)
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Axis of symmetry
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z
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r
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</details>
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Figure P9–3
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9.4 For the axisymmetric elements shown in Figure P9–4, determine the element stresses. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 2 5$ . The coordinates (in inches) are shown in the figures, and the nodal displacements for each element are $u _ { 1 } = 0 . 0 0 0 1$ in., $w _ { 1 } = 0 . 0 0 0 2$ in., $u _ { 2 } = 0 . 0 0 0 5$ in., $w _ { 2 } = 0 . 0 0 0 6$ in., $u _ { 3 } = 0$ , and $w _ { 3 } = 0$ .
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<details>
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<summary>text_image</summary>
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3 (1, 3)
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1 (0, 0) 2 (2, 0)
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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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1
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(1, 0)
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3
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(3, 3)
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2
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(3, 0)
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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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3 (0, 2)
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1 (0, 0) 2 (2, 0)
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</details>
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(c)
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Figure P9–4
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9.5 Explicitly show that the integration of Eq. (9.1.35) yields the j surface forces given by Eq. (9.1.36).
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9.6 For the elements shown in Figure P9–6, evaluate the stiffness matrices using Eq. (9.2.2). The coordinates (in millimeters) are shown in the figures. Let $E = 2 1 0$ GPa and $\nu = 0 . 2 5$ for each element.
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<details>
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<summary>line</summary>
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| Point | r | z |
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|---|---|---|
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| 1 | 0 | 0 |
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| 2 | 50 | 0 |
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| 3 | 0 | 50 |
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</details>
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(a)
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<details>
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<summary>line</summary>
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| Point | r | z |
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| 1 | 0 | 0 |
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| 2 | 60 | 0 |
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| 3 | 60 | 60 |
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</details>
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(b)
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<details>
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<summary>line</summary>
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| Point | r | z |
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|---|---|---|
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| 1 | 0 | 0 |
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| 2 | 2 | 0 |
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| 3 | 1 | 2 |
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</details>
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（c）
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Figure P9–6
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<!-- source-page: 454 -->
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9.7 For the axisymmetric elements shown in Figure P9–7, determine the element stresses. Let E ¼ 210 GPa and $\nu = 0 . 2 5$ . The coordinates (in millimeters) are shown in the figures, and the nodal displacements for each element are
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$$
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u _ {1} = 0. 0 5 \mathrm{mm} \quad w _ {1} = 0. 0 3 \mathrm{mm}
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$$
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$$
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u _ {2} = 0. 0 2 \mathrm{mm} \quad w _ {2} = 0. 0 2 \mathrm{mm}
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$$
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$$
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u _ {3} = 0. 0 \mathrm{mm} \quad w _ {3} = 0. 0 \mathrm{mm}
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$$
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<details>
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<summary>line</summary>
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| Point | r | z |
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| (0, 0) | 0 | 3 |
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| (50, 0) | 0 | 2 |
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| (0, 50) | 0 | 3 |
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</details>
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(a）
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<details>
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<summary>radar</summary>
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| Vertex | r | z |
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|---|---|---|
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| 1 | 0 | 1 |
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| 2 | 0 | 2 |
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| 3 | 0 | 3 |
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| 4 | 0 | 4 |
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</details>
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(b)
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<details>
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<summary>line</summary>
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| Point | r | z |
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|---|---|---|
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| 1 | 0 | 0 |
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| 2 | 2 | 0 |
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| 3 | 1 | 2 |
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</details>
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Figure P9–7
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9.8 Can we connect plane stress elements with axisymmetric ones? Explain.
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9.9 Is the three-noded triangular element considered in Section 9.1 a constant strain element? Why or why not?
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9.10 How should one model the boundary conditions of nodes acting on the axis of symmetry?
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9.11 How would you evaluate the circumferential strain, $\varepsilon _ { \boldsymbol { \theta } } .$ , at $r = 0 ?$ What is this strain in terms of the a’s given in Eq. (9.1.3). Hint: Elasticity theory tells us that the radial strain must equal the circumferential strain at $r = 0$ .
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9.12 What will be the stresses $\sigma _ { r }$ and $\sigma _ { \theta }$ at $r = 0 ?$ Hint: Look at Eq (9.1.2) after considering problem 9.11.
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Solve the following axisymmetric problems using a computer program.
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9.13 The soil mass in Figure P9–13 is loaded by a force transmitted through a circular footing as shown. Determine the stresses in the soil. Compare the values of $\sigma _ { r }$ using an
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<details>
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<summary>text_image</summary>
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6000 lb total force
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1 ft
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Soil mass
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</details>
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Figure P9–13
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<!-- source-page: 455 -->
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axisymmetric model with the $\sigma _ { y }$ values using a plane stress model. Let $E = 3 0 0 0$ psi and $\nu = 0 . 4 5$ for the soil mass.
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9.14 Perform a stress analysis of the pressure vessel shown in Figure P9–14. Let $E = 5 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 1 5$ for the concrete, and let $E = 2 9 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 2 5$ for the steel liner. The steel liner is 2 in. thick. Let the pressure $p$ equal 500 psi.
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<details>
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<summary>text_image</summary>
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Model of a nuclear reactor
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Steel liner
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Concrete
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13 in.
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16 in.
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p
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2 in.
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30 in.
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3-in-radius hole
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7.5 in.
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50 in.
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25 in.
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</details>
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Figure P9–14
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9.15 Perform a stress analysis of the concrete pressure vessel with the steel liner shown in Figure P9–15. Let $E = 3 0$ GPa and $\nu = 0 . 1 5$ for the concrete, and let $E = 2 0 5$ GPa and $\nu = 0 . 2 5$ for the steel liner. The steel liner is 50 mm thick. Let the pressure p equal 700 kPa.
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<details>
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<summary>text_image</summary>
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Steel liner
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Concrete
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400 mm
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p
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325 mm
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750 mm
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1250 mm
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Figure P9-15
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</details>
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<!-- source-page: 456 -->
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9.16 Perform a stress analysis of the disk shown in Figure P9–16 if it rotates with constant angular velocity of $\omega = 5 0$ rpm. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi, $\nu = 0 . 2 5$ , and the weight density $\rho _ { w } = 0 . 2 8 3 ~ \mathrm { 1 b } / \mathrm { i n } ^ { 3 }$ (mass density, $\rho = \rho _ { w } / ( g = 3 8 6 ~ \mathrm { i n . / s ^ { 2 } } )$ . (Use 8 and then 16 elements symmetrically modeled similar to Example 9.4. Compare the finite element solution to the theoretical circumferential and radial stresses given by
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$$
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\sigma_ {\theta} = \frac {3 + v}{8} \rho \omega^ {2} a ^ {2} \left(1 - \frac {1 + 3 v r ^ {2}}{3 + v a ^ {2}}\right), \quad \sigma_ {r} = \frac {3 + v}{8} \rho \omega^ {2} a ^ {2} \left(1 - \frac {r ^ {2}}{a ^ {2}}\right)
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$$
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<details>
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<summary>text_image</summary>
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a
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ω
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</details>
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<details>
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<summary>text_image</summary>
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Axis of symmetry
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12 in.
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3 in.
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</details>
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Figure P9–16
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9.17 For the die casting shown in Figure P9–17, determine the maximum stresses and their locations. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 2 5$ . The dimensions are shown in the figure.
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<details>
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<summary>text_image</summary>
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Fixed edge
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1.625 in.
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1.0625 in.
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0.5-in. radius
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0.75 in.
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4.175 in.
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0.1875 in.
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0.125-in.
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radius
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P = 10,000 lb
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30°
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P
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0.625 in.
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1.2 in.
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62°
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0.4375 in.
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2.5 in.
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8.5 in.
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Axis of symmetry
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</details>
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Figure P9–17
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<!-- source-page: 457 -->
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9.18 For the axisymmetric connecting rod shown in Figure P9–18, determine the stresses $\sigma _ { z } , \sigma _ { r } , \sigma _ { \theta } ,$ and $\tau _ { r z }$ . Plot stress contours (lines of constant stress) for each of the normal stresses. Let $E = 3 0 \times 1 0 ^ { 6 }$ psi and $\nu = 0 . 2 5$ . The applied loading and boundary conditions are shown in the figure. A typical discretized rod is shown in the figure for illustrative purposes only.
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<details>
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<summary>text_image</summary>
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1 3/4 -in. radius
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1 1/8 in. 3/4 in. 2 in. 1 in
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10°
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1000 lb 1 in.
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1000 lb
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1 5/8 in. 7 1/8 in. Axis of symmetry
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</details>
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Figure P9–18
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9.19 For the thick-walled open-ended cylindrical pipe subjected to internal pressure shown in Figure P9–19, use five layers of elements to obtain the circumferential stress, $\sigma _ { \theta } ,$
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<details>
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<summary>text_image</summary>
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Axis of symmetry
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0.06 m 0.06 m 0.06 m 0.06 m 0.06 m
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1 2 3 4 5 6
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① ② ③ ④ ⑤
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7 12
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p = 35 MPa
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⑥ ⑩
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13 18
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⑪ ⑮
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19 24
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⑯ ⑳
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1.20 m 0.3 m
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25 30
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1.5 m
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</details>
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Figure P9–19
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<!-- source-page: 458 -->
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and the principal stresses and maximum radial displacement. Compare these results to the exact solution. Let $E = 2 0 5 \mathrm { G P a }$ and $\nu = 0 . 3$ .
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9.20 A steel cylindrical pressure vessel with flat plate end caps is shown in Figure P9–20 with vertical axis of symmetry. Addition of thickened sections helps to reduce stress concentrations in the corners. Analyze the design and identify the most critically stressed regions. Note that inside sharp re-entrant corners produce infinite stress concentration zones, so refining the mesh in these regions will not help you get a better answer unless you use an inelastic theory or place small fillet radii there. Recommend any design changes in your report. Let the pressure inside be 1000 kPa.
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<details>
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<summary>text_image</summary>
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25 18.75
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30°
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60°
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300 225
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250 dia.
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310 dia.
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Dimensions in millimeters
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</details>
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Figure P9–20
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9.21 For the cylindrical vessel with hemispherical ends (heads) under uniform internal pressure of intensity $p = 5 0 0$ psi shown in Figure P9–21, determine the maximum von Mises stress and where it is located. The material is ASTM—A242 quenched and tempered alloy steel. Use a factor of safety of 3 against yielding. The inner radius is $a = 1 0 0$ inches and the thickness t ¼ 2 in.
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<details>
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<summary>text_image</summary>
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Figure P9-21
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</details>
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9.22 For the cylindrical vessel with ellipsoidal heads shown in Figure P9–22a under loading $p = 5 0 0$ psi, determine if the vessel is safe against yielding. Use the same material and factor of safety as in previous problem, 9.21. Now let $a = 1 0 0$ in. and $b = 5 0$ in. Which vessel has the lowest hoop stress? Recommend the preferred head shape of the two based on your answers.
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<!-- source-page: 459 -->
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For modeling purposes, the equation of an ellipse is given by $b ^ { 2 } x ^ { 2 } + a ^ { 2 } y ^ { 2 } = a ^ { 2 } b ^ { 2 }$ , where a is the major axis and b is the minor axis of the ellipse shown in Figure P9–22(b).
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Figure P9–22
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9.23 The syringe with plunger is shown in Figure P9–23. The material of the syringe is glass with $E = 6 9 \ { \mathrm { G P a } } , \ \nu = 0 . 1 5 ,$ , and tensile strength of 5 MPa. The bottom hole is assumed to be closed under test conditions. Determine the deformation and stresses in the glass. Compare the maximum principal stress in the glass to the ultimate tensile strength. Do you think the syringe is safe? Why?
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<details>
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<summary>text_image</summary>
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45 N
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15 mm
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25 mm
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20 mm
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90 mm
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Plunger
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Liquid
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Glass syringe
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45°
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8 mm
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4 mm
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8 mm
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4 mm
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12 mm
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</details>
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Figure P9–23
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<!-- source-page: 460 -->
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9.24 For the tapered solid circular shaft shown, a semicircular groove has been machined into the side. The shaft is made of a hot rolled 1040 steel alloy with yield strength of 71,000 psi. The shaft is subjected to a uniform axial pressure of 4000 psi. Determine the maximum principal stresses and von Mises stresses at the fillet and at the semicircular groove. Is the shaft safe from failure based on the maximum distortion energy theory?
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<details>
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<summary>text_image</summary>
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||
|
||
R = 0.5 in.
|
||
R = 1 in.
|
||
3 in.
|
||
1 in.
|
||
4000 psi
|
||
30 in.
|
||
30 in.
|
||
20 in.
|
||
</details>
|
||
|
||
Figure P9–24
|