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 800mm2 crosssectional 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 * 106per °C. FIGURE P3.24. 140 Chapter 3 TwoDimensional 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 semiinfinite 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 TwoDimensional 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 20mmthick 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 thinwalled circular cylindrical vessel of diameter d and wall thickness t
is subjected to...
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 Force, Stress, Eqs., plane stress

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