Page 268 - Mechanical Behavior of Materials

270 Chapter 6 Review of Complex and Principal States of Stress and Strain

Table P6.11

Stress Components, MPa
Problem No. σ x σ y σ z τ xy τ yz τ zx
6.11 14 −56 70 0 0 0
6.12 200 0 250 100 0 0
6.13 345 138 −69 69 0 0
6.14 60 90 200 33 0 0

6.15 Consider the following state of plane stress: σ x = 200 MPa, σ y = 100 MPa and
τ xy = 0MPa.
(a) Determine the principal normal stresses and the maximum shear stress.
(b) Show that, for such a special case of x-y plane stress, where τ xy = 0 MPa, the in-plane
principal normal stress σ 1 and σ 2 are always the same as σ x and σ y as to both the values
and directions.
6.16 For the strain measurements on the surface of the mild steel part of Prob. 5.17, estimate
the maximum normal stress and the maximum shear stress. Assume that no yielding has
occurred.
6.17 For the strain measurements on the surface of the titanium alloy part of Prob. 5.18, estimate
the maximum normal stress and the maximum shear stress. Assume that no yielding has
occurred.
6.18 A spherical pressure vessel has a wall thickness of 2.5 mm and an inner diameter of 150 mm,
and it contains a liquid at 1.2 MPa pressure. Determine the maximum normal stress and the
maximum shear stress, and also describe the planes on which these act.
6.19 A pipe with closed ends has an outer diameter of 80 mm and wall thickness of 2.0 mm. It is
subjected to an internal pressure of 10 MPa and a bending moment of 2.0 kN·m. Determine
the maximum normal stress and the maximum shear stress. Neglect the localized effects of
the end closure.
6.20 Proceed as in Prob. 6.19, except let a torque of 3.0 kN·m be applied instead of the bending
moment given, while the 10 MPa pressure is still present.
6.21 A tube has an outer diameter of 60 mm and wall thickness of 3.0 mm. It is subjected to a
bending moment of 1.8 kN·m and a torque of 2.5 kN·m. Determine the maximum normal
stress and the maximum shear stress.
6.22 A solid shaft of diameter 50 mm is subjected to a bending moment M = 3.0 kN·m and
a torque T = 2.5 kN·m. Determine the maximum normal stress and the maximum shear
stress.
6.23 A solid shaft of diameter d is subjected to a bending moment M and a torque T .
(a) Derive an expression for the maximum shear stress as a function of d, M, and T .
(b) If M = 2.0 kN-m, what is the smallest diameter such that the maximum shear stress
does not exceed 100 MPa?
6.24 A thin-walled tube with closed ends has an inside radius of 80 mm and a wall thickness of
6 mm. It is subjected to an internal pressure of 20 MPa, a torque of 60 kN·m, and an axial
compressive force of 200 kN. Determine the maximum normal stress and the maximum shear
stress.