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?
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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?
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6.What is the maximum length an iron wire can have when hung vertically so that it does not break under its own weight?
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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)
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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.
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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.)
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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.
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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.
Please log in to view the 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.)
Please log in to view the solution.13.What must be the radius of a copper wire so that it does not break under a force of 500 N?
Please log in to view the solution.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.
Please log in to view the solution.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.
Please log in to view the solution.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.
Please log in to view the solution.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?
Please log in to view the solution.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?
Please log in to view the solution.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?
Please log in to view the solution.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?
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