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P317 assume that the bending stress is given by mzy x

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Unformatted text preview: thin an elastic uniformly loaded cantilever beam (Fig. P3.17): y p h h x L FIGURE P3.15. Problems 139 ch03.qxd 12/20/02 7:21 AM Page 140 FIGURE P3.17. x xy y p p3 15x2 + 2h22y + y 10I 3I px 2 =1h - y22 2I p = - 12h3 - 3h2y + y32 6I =- (P3.19) Here I = 2th3/3 and the body forces are omitted. Given p = 10 kN/m, L = 2 m, h = 100 mm, t = 40 mm, = 0.3, and E = 200 GPa, calculate the magnitude and direction of the maximum principal strain at point Q. 3.20. A prismatic bar is restrained in the x (axial) and y directions, but free to expand in z direction. Determine the stresses and strains in the bar for a temperature rise of T1 degrees. 3.21. Under free thermal expansion, the strain components within a given elastic solid are x = y = z = T and xy = yz = xz = 0. Show that the temperature field associated with this condition is of the form T = c1x + c2y + c3z + c4 in which the c’s are constants. 3.22. Redo Prob. 3.6 adding a temperature change T1, with all other conditions remaining unchanged. 3.23. Determine the axial force Px and moment Mz that the walls in Fig. 3.6b apply to the beam for T = a1y + a2, where a1 and a2 are constant. 3.24. A copper tube of 800-mm2 cross-sectional area is held at both ends as in Fig. P3.24. If at 20°C no axial force Px exists in the tube, what will Px be when the temperature rises to 120°C? Let E = 120 GPa and = 16.8 * 10-6per °C. FIGURE P3.24. 140 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 141 Secs. 3.8 through 3.10 3.25. Show that the case of a concentrated load on a straight boundary (Fig. 3.11a) is represented by the stress function P £=- r sin and derive Eqs. (3.45) from the result. 3.26. Verify that Eqs. (3.34) are determined from the equilibrium of forces acting on the elements shown in Fig. P3.26. 3.27. Demonstrate that the biharmonic equation §4 £ = 0 in polar coordinates can be written as ¢ 10 1 02 1 0£ 1 02 £ 02 02 £ + + 2 2≤ ¢ 2 + +2 ≤=0 2 r 0r r 0r r0 r 02 0r 0r 3.28. Show that the compatibility equation in polar coordinates, for the axisymmetrical problem of thermal elasticity, is given by 1d d£ ¢r ≤+E T=0 r dr dr (P3.28) 3.29. Assume that moment M acts in the plane and at the vertex of the wedge–cantilever shown in Fig. P3.29. Given a stress function £=- M1sin 2 - 2 cos 2 2 21sin 2 - 2 cos 2 2 (P3.29a) determine (a) whether £ satisfies the condition of compatibility; (b) the stress components r, , and r ; and (c) whether the expressions 2M sin 2 2M cos2 , = 0, = r r r2 r2 represent the stress field in a semi-infinite plate (that is, for (P3.29b) = FIGURE P3.26. Problems = /2 ). FIGURE P3.29. 141 ch03.qxd 12/20/02 7:21 AM Page 142 FIGURE P3.30. 3.30. Referring to Fig. P3.30, verify the results given by Eqs. (b) and (c) of Sec. 3.10. 3.31. Consider the pivot of unit thickness subject to force P per unit thickness at its vertex (Fig. 3.10a). Determine the maximum values of x and xy on a plane a distance L from the apex through the use of r given by Eq. (3.40) and the formulas of the elementary theory: (a) take = 15°; (b) take = 60°. Compare the results given by the two approaches. 3.32. Solve Prob. 3.31 for = 30°. 3.33. Redo Prob. 3.31 in its entirety for the wedge–cantilever shown in Fig. 3.10b. 3.34. A uniformly distributed load of intensity p is applied over a short distance on the straight edge of a large plate (Fig. P3.34). Determine stresses x, y, and xy in terms of p, 1, and 2, as required. [Hint: Let dP = pdy denote the load acting on an infinitesimal length dy = rd /cos (from geometry) and hence dP = prd /cos . Substitute this into Eqs. (3.47) and integrate the resulting expressions.] Secs. 3.11 through 3.13 3.35. Verify the result given by Eqs. (f) and (g) of Sec. 3.11 (a) by rewriting Eqs. (d) and (e) in the following forms, respectively, df1 1d d 1d ¢r ≤Rr = 0 br B r dr dr r dr dr d 1 d 3d 1 d 2 r 1r f22 R r ≤ = 0 ¢ B br dr r3 dr dr r3 dr (P3.35) FIGURE P3.34. 142 Chapter 3 Two-Dimensional Problems in Elasticity ch03.qxd 12/20/02 3.36. 3.37. 3.38. 3.39. 3.40. 3.41. 3.42. 7:21 AM Page 143 and by integrating (P3.35) and (b) by expanding Eqs. (d) and (e), setting t = ln r, and thereby transforming the resulting expressions into two ordinary differential equations with constant coefficients. Verify the results given in Fig. 3.15 by employing Eq. (3.52b) and the method of superposition. For the flat bar of Fig. C of Table 3.2, let b = 17h, c = 18h, and a = h (circular hole). Referring to Neuber’s nomograph (Fig. 3.17), determine the value of k for the bar loaded in tension. A 20-mm-thick steel bar with a slot (25 mm radii at ends) is subjected to an axial load P, as shown in Fig. P3.38. What is the maximum stress for P = 180 kN? For the flat bar in Fig. A of Table 3.2, let h = 3a and b = 15a. Referring to Neuber’s nomograph (Fig. 3.17), find the value of k for the bar subjected to (a) axial tensile load, and (b) bending. A thin-walled circular cylindrical vessel of diameter d and wall thickness t is subjected to...
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