Page 323 - Mechanical Behavior of Materials

324 Chapter 7 Yielding and Fracture under Combined Stresses

7.7 In an engineering component made of ductile cast iron A 536 (65-45-12) material, the
most severely stressed point is subjected to the following state of stress: σ x =−50, σ y =
120, σ z = 40, τ xy =−60, τ yz = 14, and τ zx = 0 MPa. Determine the safety factor against
yielding by (a) the maximum shear stress criterion, and (b) the octahedral shear stress
criterion.
7.8 In an engineering component, the most severely stressed point is subjected to the following
state of stress: σ x = 345, σ y = 138, τ xy = 69, and σ z = τ yz = τ zx = 0 MPa. What minimum
yield strength is required for the material if a safety factor of 2.5 against yielding is
required? Employ (a) the maximum shear stress criterion, and (b) the octahedral shear stress
criterion.
7.9 In an engineering component, the most severely stressed point is subjected to the following
state of stress: σ x = 14, σ y =−56, σ z = 70, and τ xy = τ yz = τ zx = 0 MPa. What minimum
yield strength is required for the material if a safety factor of 2.0 against yielding is required?
Employ (a) the maximum shear stress criterion, and (b) the octahedral shear stress criterion.
7.10 In Fig. 7.1, for each case (a), (b), (c), and (d) that is shown, sketch the three Mohr’s circles
corresponding to the principal shear stresses. Then, for each case, employ the maximum shear
stress criterion to determine σ y at yielding as a function of the uniaxial yield strength. Do you
confirm the predictions indicated?
7.11 Strains are measured on the surface of part made from 7075-T6 aluminum and are as follows:
ε x = 3600 × 10 −6 , ε y = 150 × 10 −6 , and γ xy = 700 × 10 −6 . Assume that no yielding has
occurred and also that no loading is applied directly to the surface, so that σ z = τ yz = τ zx =
0 MPa. What is the safety factor against yielding?
7.12 A strain gauge rosette, as in Ex. 6.9, is applied to the surface of a component made of Ti-6A1-
4V (solution treated and aged). Assume that no yielding has occurred and also that no loading
is applied directly to the surface, so that σ z = τ yz = τ zx = 0 MPa. Strains are measured as
follows: ε x = 1200 × 10 −6 , ε y =−650 × 10 −6 , and ε 45 = 1900 × 10 −6 . What is the safety
factor against yielding?
7.13 A strain gauge rosette, as in Ex. 6.9, is applied to the surface of a component made of
AISI 1020 steel (as rolled). Assume that no yielding has occurred, and also that no loading
is applied directly to the surface, so that σ z = τ yz = τ zx = 0 MPa. Strains are measured as
follows: ε x = 290 × 10 −6 ,ε y =−860 × 10 −6 , and ε 45 =−190 × 10 −6 . What is the safety
factor against yielding?
7.14 A solid circular shaft subjected to pure torsion must be designed to avoid yielding, with a
safety factor X. Find the required diameter as a function of the torque T and the yield strength
σ o , using (a) the maximum shear stress criterion, and (b) the octahedral shear stress criterion.
How much do these two sizes differ?
7.15 A solid circular shaft has a diameter of 50 mm and is made of AISI 1020 steel (as rolled). It
is subjected to a tensile axial force of 100 kN, a bending moment of 800 N·m, and a torque of
1500 N·m. Determine the safety factor against yielding.
7.16 A pipe with closed ends has an outer diameter of 80 mm and a wall thickness of 3.0 mm. It
is subjected to an internal pressure of 20 MPa and a bending moment of 2.0 kN·m. Determine
the safety factor against yielding if the material is 7075-T6 aluminum.
7.17 A thin-walled tube with closed ends has an inside radius r 1 = 80 mm and a wall thickness
◦
t = 6 mm. It is made of AISI 4142 steel tempered at 450 C and is subjected to an internal