Deformation of solids
1.What should we know about the deformation of solids?
Solution:
Deformation is a change in the shape of a body due to external forces.
Deformation:
- a.) elastic – reversible, temporary
- b.) plastic – non-elastic, permanent
Deformation can be caused by tension, compression, bending, shear, torsion, or a combination.
Tensile deformation:
Absolute elongation:
Δl = l – l0
Relative elongation:
Normal stress:
Hooke's law:
Hooke's law holds only up to the elastic limit.
Stress–strain curve:
σu = proportional limit
σd = σE = elastic limit (allowable stress)
σk = yield point
σp = ultimate strength
E = Young's modulus in tension
2.Determine whether an iron wire with a diameter of 2 mm will break if it is tensioned by a force of 1 kN. (σE = 314 MPa)
Solution:
Given/Analysis:
d = 2 mm, r = 1 mm = 10-3 m, F = 103 N, σE = 314 MPa
Because σn > σE, the iron wire will break.
3.Calculate the relative elongation of a rope with an original length of 10 m if it elongates by 40 mm during deformation.
Solution:
Given/Analysis:
l0 = 10 m, Δl = 40 mm = 0.04 m
The relative elongation of the rope is ε = 0.4%.
4.How does the normal stress in a wire change if the tensile force acting on the wire increases 4 times and the wire diameter 2 times?
Solution:
Given/Analysis:
The normal stress in the wire does not change.
5.What must be the radius of a copper wire (σp = 2·108 Pa) so that it does not break under a force of 500 N?
Solution:
Given/Analysis:
F = 500 N, σp = 2·108 Pa
The wire radius must satisfy r >= 0.89 mm.
6.What is the maximum length an iron wire can have when hung vertically so that it does not break under its own weight?
Solution:
Given/Analysis:
g = 9.81 m·s-2, ρ(Fe) = 7800 kg·m-3 = 7.8·103 kg·m-3, σp = 314·106 Pa
The length of the wire can be at most 4104 m.
7.What is the tensile force in a guitar string of length 0.65 m and cross-sectional area 0.325 mm2 if it elongated by 5 mm during tightening? (E = 220 GPa)
Solution:
Given/Analysis:
l0 = 0.65 m, Δl = 5·10-3 m, S = 0.325·10-6 m2, E = 220·109 Pa
The guitar string is tensioned by a force F = 550 N.
8. An aluminum wire with original length 4.2 m and diameter 1.1 mm elongated by 15.2 mm under a force of 0.23 kN. Determine the Young's modulus in tension E.
Solution:
Given/Analysis:
l0 = 4.2 m, r = 0.55·10-3 m, F = 0.23·103 N, Δl = 15.2·10-3 m
The Young's modulus in tension for aluminum is E(Al) = 67 GPa.
9.At the end of a vertical steel rod (E = 220 GPa) of length 1.5 m, a 500 kg weight is to be suspended. What rod diameter should we choose if we want the elongation after loading to be no more than 0.3 mm? (Neglect the rod’s own weight.)
Solution:
Given/Analysis:
l0 = 1.5 m, m = 500 kg, Δl = 3·10-4 m, E = 220 GPa = 220·109 Pa, g = 10 m·s-2
We choose a rod diameter d = 12 mm.
10.Determine the work required to elastically stretch a steel rod (E = 220 GPa) of length 1 m and cross-section 1 cm2 by 1 mm in tension.
Solution:
Given/Analysis:
l0 = 1 m, Δl = 1 mm = 1·10-3 m, S = 1 cm2 = 1·10-4 m2, E = 220·109 Pa
To stretch the steel rod, the required work is W = 22 J.
11.A steel test rod with a diameter of 15 mm broke under a force of 1.63·105 N. Determine the tensile strength of the steel.
Solution:
12.A passenger elevator with a mass of 500 kg is held by 3 steel cables, each with a diameter of 1 cm. Calculate the stress in each steel cable. (Neglect the weight of the cable.)
Solution:
The stress in a steel cable is 20.83 MPa.
13.What must be the radius of a copper wire so that it does not break under a force of 500 N?
Solution:
The radius of the copper wire must be equal to or greater than 0.892 mm.
14.A steel wire with a length of 3 m and a cross-section of 1.2 mm2 elongates by 8 mm when subjected to deforming forces. Calculate the magnitude of the deforming forces.
Solution:
The deforming force is 704 N.
15.During the production of prestressed reinforced concrete components, steel rods 6 meters long were tensioned with a force of 60 000 N. Calculate the elongation of the steel rods if their diameter is 10 mm.
Solution:
The steel rods elongate by 20.8 mm.
16.A brass wire 1.1 m long with a cross-section of 4·10–6 m2 was stretched by a force of 80 N, causing it to elongate by 0.2 mm. Calculate the Young’s modulus of brass.
Solution:
Analysis:
The Young’s modulus of brass is 110 GPa.
17.A steel rod is fixed at one end. How will its temperature change if a compressive stress of 5 MPa is applied at the free end?
Solution:
The steel rod heats up by approximately 20C.
18.A lid with a diameter of 32 cm must be attached to the opening of a pressure vessel using 24 screws. The gas pressure inside the vessel is 6 MPa. What cross-sectional area should the screws have?
Solution:
The cross-sectional area of a screw is 4 cm2.
19.A two-column hydraulic press compresses material with a force of 600 kN. The columns are made of steel and have a diameter of 80 mm. What is the normal stress in a column?
Solution:
The normal stress is 60 MPa.
20.A load with a mass of 4000 kg is suspended on a steel cable with a cross-sectional area of 2 cm2. What is the relative elongation of the cable?
Solution:
The relative elongation of the cable is 0.09%.