Page 359 - Introduction to Continuum Mechanics

Problems 343

and all other TIJ — 0
(a) Demonstrate that the equilibrium equations are identically satisfied for any choice of V-
(b) Show that if ip satisfies the equation

then the stress will correspond to a compatible strain field for simply-connected cross-sectional
areas.

(c) Show that the lateral boundary condition requires that V^> be in the same direction as the
outward normal. In other words, the values of t/> on the outer boundary is a constant.
5.62. A beam of circular cross-section is subjected to pure bending. The magnitude of each
end couple is 14,000 N-m. If the maximum normal stress is not to exceed 0.124 GPa, what
should be the diameter?
5.63. The rectangular beam of Example 5.15.1 has a width b and a height 1.26. If the right-hand
couple is given by M = 24,000e2 ft-lb (32,500 N • m), determine the dimension b in order that
the maximum shearing stress does not exceed 600 psi (4.14 MPa).

5.64. Let the beam of Example 5.15.1 be loaded by both the indicated bending moment and a
centroidally applied tensile force P. Determine the magnitude of P in order that 7\i>0.
5.65. Verify that if <p (x\, X2) satisfy Eq. (5.16.7), than it does correspond to a compatible strain
field.
5.66. Show that if the bending moment applied to a bar in pure bending is not referred to
principal axes, then the flexural stress will be

5,67. Figure P5.5 shows the cross-section of a beam subjected to pure bending. If the end
4
couples are given by ± 10 N • m, find the maximum normal stress.