김경종
17 KiB
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10 in. 1000 lb/in. 10 in. 5 in. 10 in. 20 in.
Figure P7–4
7.5 When do problems occur using the smoothing (averaging of stress at the nodes from elements connected to the node) method for obtaining stress results?
7.6 What thickness do you think is used in computer programs for plane strain problems?
7.7 Which one of the CST models shown below is expected to give the best results for a cantilever beam subjected to an end shear load? Why?
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4 @ 1" = 4 in. 6 @ 2" = 12"
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4 in. 12 @ 1" = 12"
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6 @ 2" / 3 = 4 in. 6" 6"
(c)
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2 in. 2 in. 6" 6"
(d)
Figure P7–7
7.8 Show that Eq. (7.5.13) is obtained by static condensation of Eq. (7.5.12).
Solve the following problems using a computer program. In some of these problems, we suggest that students be assigned separate parts (or models) to facilitate parametric studies.
7.9 Determine the free-end displacements and the element stresses for the plate discretized into four triangular elements and subjected to the tensile forces shown in Figure P7–9. Compare your results to the solution given in Section 6.5 Why are these results different? Let E = 3 0 \times 1 0 ^ { 6 } psi, n ¼ 0:30, and t ¼ 1 in.
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5000 lb 10 in. 5000 lb 20 in.
Figure P7–9
7.10 Determine the stresses in the plate with the hole subjected to the tensile stress shown in Figure P7–10. Graph the stress variation \sigma _ { x } versus the distance y from the hole. Let E ¼ 200 GPa, n ¼ 0:25, and t ¼ 25 mm. (Use approximately 25, 50, 75, 100, and then 120 nodes in your finite element model.) Use symmetry as appropriate.
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10 kPa 500 mm 25 mm radius 10 kPa 500 mm y x
Figure P7–10
7.11 Solve the following problem of a steel tensile plate with a concentrated load applied at the top, as shown in Figure P7–11. Determine at what depth the effect of the load dies out. Plot stress \sigma _ { y } versus distance from the load. At distances of 1 in., 2 in., 4 in., 6 in., 10 in., 15 in., 20 in., and 30 in. from the load, list \sigma _ { y } versus these distances. Let the width of the plate be b = 4 in., thickness of the plate be t = 0 . 2 5 in., and length be L = 4 0 in. Look up the concept of St. Venant’s principle to see how it explains the stress behavior in this problem.
7.12 For the connecting rod shown in Figure P7–12, determine the maximum principal stresses and their location. Let E = 3 0 \times 1 0 ^ { 6 } psi, n ¼ 0:25, t ¼ 1 in., and P = 1 0 0 0 lb.
7.13 Determine the maximum principal stresses and their locations for the member with fillet subjected to tensile forces shown in Figure P7–13. Let E = 2 0 0 GPa and \nu = 0 . 2 5 . Then let E ¼ 73 GPa and \nu = 0 . 3 0 . Let t ¼ 25 mm for both cases. Compare your answers for the two cases.
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P = 1000 lb L σy = P/bt σy = P/bt σy = P/bt
Figure P7-11
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1 3/4 -in. radius 0.5 -in. radius 1 1/8 -in. 0.3125 in. 2 in. 0.3125 in. 2 in. 0.75 in. 7 1/8 in. 1 in. 40° P P P 1 5/8 in. Axis of symmetry
Figure P7–12
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Axis of symmetry 100 mm 100 mm 100 mm 50 mm 38 mm 36 N/mm 88 mm 6 mm radius
Figure P7–13
7.14 Determine the stresses in the member with a re-entrant corner as shown in Figure P7–14. At what location are the principal stresses largest? Let E = 3 0 \times 1 0 ^ { 6 } psi and \nu = 0 . 2 5 . Use plane strain conditions.
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10 in. 10 in. 1000 lb/in. 0.25 in. rad. 5 in. 10 in. 20 in.
Figure P7–14
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1 ft 100 lb/in. D 2D Axis of symmetry
Figure P7–15
7.15 Determine the stresses in the soil mass subjected to the strip footing load shown in Figure P7–15. Use a width of 2 D and depth of D , where D is 3 , 4 , 6 , 8 , , and 10 ft. Plot the maximum stress contours on your finite element model for each case. Compare your results. Comment regarding your observations on modeling infinite media. Let E = 3 0 { , } 0 0 0 psi and \nu = 0 . 3 0 . Use plane strain conditions.
7.16 For the tooth implant subjected to loads shown in Figure P7–16, determine the maximum principal stresses. Let E = 1 . 6 \times 1 0 ^ { 6 } psi and \nu = 0 . 3 for the dental restorative
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10 lb 15 lb 8 in. 1/16 in. 6" 16 6" 16 9" 16 -in. radius 0 17/32 in. 35/32 in. 37/32 in. 55/32 in. 36/16 in. 2¼ in.
Figure P7–16
implant material (cross-hatched), and let E = 1 \times 1 0 ^ { 6 } psi and \nu = 0 . 3 5 for the bony material. Let X ¼ 0:05 in., 0.1 in., 0.2 in., 0.3 in., and 0.5 in., where X represents the various depths of the implant beneath the bony surface. Rectangular elements are used in the finite element model shown in Figure P7–16. Assume the thickness of each element to be t ¼ 0:25 in.
7.17 Determine the middepth deflection at the free end and the maximum principal stresses and their location for the beam subjected to the shear load variation shown in Figure P7–17. Do this using 64 rectangular elements all of size 12 in. 1 in.; then all of size 6 in. 1 in.; then all of size 3 in. 2 in. Then use 60 rectangular elements all of size 2.4 in. \times 2 \frac { 2 } { 3 } in.; then all of size 4.8 in. \times 1 \frac { 1 } { 3 } in. Compare the free-end deflections and the maximum principal stresses in each case to the exact solution. Let E = 3 0 \times 1 0 ^ { 6 } psi, n ¼ 0:3, and t ¼ 1 in. Comment on the accuracy of both displacements and stresses.
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40,000-lb total shear load parabolically distributed 8 in. 48 in.
Figure P7–17
7.18 Determine the stresses in the shear wall shown in Figure P7–18. At what location are the principal stresses largest? Let E ¼ 21 GPa, n ¼ 0:25, t _ { \mathrm { w a l l } } = 0 . 1 0 m, and t _ { \mathrm { b e a m } } = 0:20 m.
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50 kN/m 8 m 1 m Beam 2 m 4 m 4 m 10 m
Figure P7–18
7.19 Determine the stresses in the plates with the round and square holes subjected to the tensile stresses shown in Figure P7–19. Compare the largest principal stresses for each plate. Let E = 2 1 0 \mathrm { { G P a } , \nu = 0 . 2 5 . } , and t = 5 \mathrm { m m } .
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25-mm radius 500 mm 1 kN/m² 500 mm 1 kN/m²
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1 mm rod 25 mm 1 kN/m² 500 mm 1 kN/m²
Figure P7–19
7.20 For the concrete overpass structure shown in Figure P7–20, determine the maximum principal stresses and their locations. Assume plane strain conditions. Let E = 3 . 0 \times 1 0 ^ { 6 } psi and \nu = 0 . 3 0 .
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2 k/ft 11.5-ft radius 8 ft 18 ft 10 ft 10 ft 10 ft 10 ft
Figure P7–20
7.21 For the steel culvert shown in Figure P7–21, determine the maximum principal stresses and their locations and the largest displacement and its location. Let E _ { \mathrm { s t e e l } } = 2 1 0 \mathrm { G P a } and let \nu = 0 . 3 0 .
7.22 For the tensile member shown in Figure P7–22 with two holes, determine the maximum principal stresses and their locations. Let E = 2 1 0 \mathrm { G P a } , \nu = 0 . 2 5 , and t = 1 0 mm. Then let E = 7 0 \mathrm { G P a } and \nu = 0 . 3 0 . Compare your results.
7.23 For the plate shown in Figure P7–23, determine the maximum principal stresses and their locations. Let E = 2 1 0 \mathrm { G P a } and \nu = 0 . 2 5 .
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1.5 m → 20 kN 0.25 m 3 m Figure P7-21 3 m
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0.3 m 0.4 m 0.3 m 0.75 m 75-mm radius 20 kN t = 10 mm Figure P7-22 1 m
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25-mm radius 0.3 m 0.15 m 0.1 m 5 kN t = 10 mm 0.25 m 0.25 m 0.25 m
Figure P7–23
7.24 For the concrete dam shown subjected to water pressure in Figure \mathrm { P } 7 { - } 2 4 , determine the principal stresses. Let E = 3 . 5 \times 1 0 ^ { 6 } psi and \nu = 0 . 3 0 . Assume plane strain conditions. Perform the analysis for self-weight and then for hydrostatic (water) pressure against the dam vertical face as shown.
7.25 Determine the stresses in the wrench shown in Figure P7–25. Let E = 2 0 0 ~ \mathrm { G P a } and \nu = 0 . 2 5 , and assume uniform thickness t = 1 0 \ \mathrm { m m } .
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260 ft 30 ft 25 ft 100 ft 30 ft 180 ft
Figure P7–24
7.26 Determine the principal stresses in the blade implant and the bony material shown in Figure P7–26. Let Eblade ¼ 20 GPa, nblade ¼ 0:30, Ebone ¼ 12 GPa, and \nu _ { \mathrm { b o n e } } = 0 . 3 5 . Assume plane stress conditions with t ¼ 5 mm.
7.27 Determine the stresses in the plate shown in Figure P7–27. Let E = 2 1 0 ~ \mathrm { G P a } and \nu = 0 . 2 5 . The element thickness is 10 mm.
7.28 For the 0.5 in. thick canopy hook shown in Figure P7–28, used to hold down an aircraft canopy, determine the maximum von Mises stress and maximum deflection. The hook is subjected to a concentrated upward load of 22,400 lb as shown. Assume boundary conditions of fixed supports over the lower half of the inside hole diameter. The hook is made from AISI 4130 steel, quenched and tempered at 4 0 0 ~ ^ { \circ } \mathrm { F } . (This problem is compliments of Mr. Steven Miller.)
7.29 For the \textstyle { \frac { 1 } { 4 } } in. thick L-shaped steel bracket shown in Figure P7–29, show that the stress at the 90 degree re-entrant corner never converges. Try models with increasing numbers of elements to show this while plotting the maximum principal stress in the bracket. That is, start with one model, then refine the mesh around the re-entrant corner and see what happens, say, after two refinements. Why? Then add a fillet, say, of radius \textstyle { \frac { 1 } { 2 } } in. and see what happens as you refine the mesh. Again plot the maximum principal stress for each refinement.
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700 N 50 mm 250 mm 15 mm 40 mm 20 mm 70 mm 40 mm 160 mm 50 mm 20 mm 40 mm 75 mm 40 mm
Figure P7–25
Use a computer program to help solve the design-type problems, 7.30–7.36.
7.30 The machine shown in Figure P7–30 is an overload protection device that releases the load when the shear pin S fails. Determine the maximum von Mises stress in the upper part ABE if the pin shears when its shear stress is 40 MPa. Assume the upper part to have a uniform thickness of 6 mm. Assume plane stress conditions for the upper part. The part is made of 6061 aluminum alloy. Is the thickness sufficient to prevent failure based on the maximum distortion energy theory? If not, suggest a better thickness. (Scale all dimensions as needed.)
7.31 The steel triangular plate 1 in. thick shown in Figure P7–31 is bolted to a steel column with { \scriptstyle { \frac { 3 } { 4 } } - \mathrm { i n } } . .-diameter bolts in the pattern shown. Assuming the column and bolts are very rigid relative to the plate and neglecting friction forces between the column and plate, determine the highest load exerted on any bolt. The bolts should not be included
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66 N 66 N 6 mm 6 mm Bony material Blade implant R = 6 mm 6 mm Bony material 0.7 mm 7 mm 3 mm 1 mm 6 mm 22 mm
Figure P7–26
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3 kN 40 mm 20 mm 60 mm 3 kN
Figure P7–27
in the model. Just fix the nodes around the bolt circles and consider the reactions at these nodes as the bolt loads. If { \frac { 3 } { 4 } } \mathrm { - i n } .-diameter bolts are not sufficient, recommend another standard diameter. Assume a standard material for the bolts. Compare the reactions from the finite element results to those found by classical methods.
7.32 \begin{array} { r } { \mathbf { A } \frac { 1 } { 4 } } \end{array} in. thick machine part supports an end load of 1000 lb as shown in Figure P7–32. Determine the stress concentration factors for the two changes in geometry located at the radii shown on the lower side of the part. Compare the stresses you get to classical beam theory results with and without the change in geometry, that is, with a uniform depth of 1 in. instead of the additional material depth of 1.5 in. Assume standard mild steel is used for the part. Recommend any changes you might make in the geometry.