Page 331 - Mechanical Behavior of Materials

332 Chapter 7 Yielding and Fracture under Combined Stresses

(1) σ 1 = 200, σ 2 =−100 MPa
(2) σ 1 = 100, σ 2 =−600 MPa
(3) σ 1 =−300, σ 2 =−600 MPa
(d) Confirm the values from (c) by applying Eq. 7.66.
7.49 In a compression test, a cylinder of unreinforced concrete has an ultimate strength of
27.2 MPa, and the fracture is observed to occur on a plane inclined to the direction of
loading by an angle of approximately θ c = 25 . If this same concrete is subjected to lateral
◦
(compressive) stresses of σ 1 = σ 2 =−10 MPa, estimate the stress σ 3 necessary to cause
failure in compression. (Suggestion: If the tensile strength is needed, this may be estimated as
10% of the compressive strength.)
7.50 A cylinder of the mortar of Table 7.1 is subjected to an axial compressive stress σ z of 50 MPa,
along with equal lateral compressive stresses, σ x = σ y .
(a) What is the safety factor against fracture if the lateral compressive stresses are
12 MPa?
(b) Let σ z remain unchanged. But let the lateral compression be reduced, that is, |σ x |=
|σ y | < 12 MPa. Can σ x = σ y approach zero without fracture occurring? At what value
of σ x = σ y is fracture expected to occur?
7.51 A building column 400 mm in diameter is made of the sandstone of Table 7.1.
(a) What is the safety factor against fracture if the column is subjected to a compressive
force of 1250 kN?
(b) What is the safety factor against fracture if the column is subjected to a torque of
20 kN·m?
(c) What is the safety factor against fracture if the column is subjected at the same time to
both the 1250 kN compressive force and the 20 kN·m torque?
(d) Compare the safety factors calculated in (a), (b), and (c), and explain the trends in their
values.
7.52 A block of the concrete of Table 7.1 is loaded with a pressure p applied to all sides, and also
with a shear stress, τ xy = 30 MPa, as shown in Fig. P7.28.
(a) Will the block fracture if p = 40 MPa?
(b) What smallest value of p such that the block will not fracture?
7.53 A block of the mortar of Table 7.1 is loaded with a shear stress τ xy = 1.0 MPa, and also with
a normal stress σ z , as shown in Fig. P7.29.
(a) What is the safety factor against fracture if σ z = 0?
(b) What is the safety factor against fracture if σ z is 15 MPa compression?
(c) For the safety factor to be not less than 2.5, what is the most severe compressive σ z that
can be applied?
7.54 Consider a 50 mm diameter shaft of the gray cast iron of Table 7.1. If a safety factor of
3.0 against fracture is required, what is the largest torque that can be applied along with a
compressive axial force of 250 kN?
7.55 A building column 400 mm in diameter is made of the sandstone of Table 7.1. It resists a
compressive force P and a torque T = 34,000 N·m, and a safety factor of 4.0 against fracture
is required.
(a) What is the largest compressive force P that can be permitted?