Page 271 - Mechanical Behavior of Materials

Problems and Questions 273

Section 6.5
6.45 Determine the octahedral normal and shear stresses for the state of stress: σ x = 32 MPa,
σ y =−10 MPa, and τ xy =−20 MPa.
6.46 Determine the octahedral normal and shear stresses for the state of stress: σ x = 345 MPa,
σ y = 138 MPa, σ z =−69 MPa and τ xy = 69 MPa.
6.47 Consider a case of plane stress where the only nonzero components for the x-y-z coordinate
system chosen are σ x and τ xy . (For example, this situation occurs at the surface of a shaft under
combined bending and torsion.) Develop equations in terms of σ x and τ xy for the following:
maximum normal stress, maximum shear stress, and octahedral shear stress.
6.48 Consider the case of plane stress where the only nonzero components for the x-y-z system
chosen are σ x and σ y . (For example, this occurs in a thin-walled tube with internal pressure
and bending and/or axial loads.) Develop an equation for the octahedral shear stress in terms
of σ x and σ y .
6.49 Develop an equation for the octahedral shear stress in terms of the principal shear stresses.
6.50 Consider an internally pressurized thick-walled tube, as in Fig. A.6(a). Assume that the tube
has closed ends, but neglect the localized effects of the end closure. Develop an equation for
the octahedral shear stress τ h , expressing this as a function of the radii r 1 , r 2 , and R, and the
pressure p. Also, show that the maximum value of τ h occurs at the inner wall.
6.51 For the thick-walled tube of Prob. 6.30:
(a) Determine the maximum value of octahedral shear stress in the tube. Neglect the
localized effects of the end closure.
(b) Plot the variation of τ h versus radius and comment on the trend observed.
6.52 Derive the equations for the octahedral normal and shear stresses, Eqs. 6.32 and 6.33, on the
basis of equilibrium of the solid body shown in Fig. 6.15(a). (Suggestions: Note that the three
faces in the principal planes are acted upon by principal stresses, σ 1 , σ 2 , and σ 3 . Then sum
forces normal to the octahedral plane to get σ h , and parallel to this plane to get τ h .)

Section 6.6
6.53 For the strains measured on the free surface of a mild steel part in Prob. 5.17, determine the
principal normal strains and the principal shear strains. Assume that no yielding has occurred.
6.54 For pure planar shear, where only τ xy is nonzero, verify the principal stresses, strains, and
planes shown in Fig. 4.41.
6.55 A strain guage rosette of the type shown in Fig. 6.16(a) is employed to measure strains
on the free surface of an aluminum alloy part, with the result being ε x = 1900 × 10 −6 ,
ε y = 1250 × 10 −6 , and ε 45 = 2375 × 10 −6 . Determine the principal normal strains and the
principal shear strains. Assume that no yielding has occurred.
6.56 As mentioned in the previous problem, strains are measured using a strain gauge on the surface
of a polycarbonate plastic part as follows: ε x = 0.011, ε y = 0.0079, and γ xy = 0.0048.
Determine the principal normal strains and the principal shear strains. Assume that no yielding
has occurred. Poisson’s ratio can be taken from Table 5.2.
6.57 Consider a strain gage rosette mounted on the unloaded free surface of an engineering
component. The rosette is the type shown in Fig. 6.16(b), so that it measures strain in the x-
direction and in two additional directions that correspond to counterclockwise rotations from