CHPTER STRESS ND STRIN EXERCISE, Page 95. If a oid tone i dropped into the ea and come to ret at a depth of 5000 m beow the urface of the ea, what wi be the tre in the tone? Take the denity of eawater 00 kg/m and g 9.8 m/. Stre in tone, σ ρgh 00 9.8 5000 50 MN/m. oid bar of ength m conit of three horter ection firmy joined together. uming the foowing appy, determine the change in ength of the bar when it i ubjected to an axia pu of 50 kn. ume Young moduu, E 0 N / m. Section Length (m) Diameter (mm) 0. 5 0. 0 0.5 0 5 0.770 m, 0 0.0 m and 0 0 7.090 m σ F 500.77 0 8 MN/m, σ F 50 0. 0 59. MN/m and σ F 50 0 7.09 0 70.7 MN/m δ 80 0. 59.0 0. 70.70 0.5 0
the change in ength of bar, δ.98 0 m 0.98 mm. If the bar of Probem were made from three different materia with the foowing eatic modui, determine the change in ength of the bar: Section E ( N / m ) 0 0 7 0 0 δ δ.8 8 0 0. 59. 0 0. 70.7 0 0.5 0 70 0 0 0 m.8 mm. circuar-ection oid bar of inear taper i ubjected to an axia pu of 0. MN, a how. If E 0 N / m, by how much wi the bar extend? W d u E d 0 5 7.850 m Hence, u 0. 0 5 7.85 0 0 8 7.9580 m u () bar extenion, u 0.79 mm Proof of formua for thi probem
t x, d 0 + 7 0 d 0 ( + 7x) W W t x, σ d / 0 7x 7W σ 7x 7W t x, ε 7x 0 x and δu Thu, u. 0 7x 8.0 8 7x.0 W W x 8 W 0 7x dx 8.0 0.0 7( 7 x) 8.0 0.0 5 7 0 u 7.9 0 m 0.79 mm 5. If a oid tone i dropped into the ea and come to ret at a depth of 000 m beow the urface of the ea, what wi be the tre in the tone? ume that denity of ea water 00 kg/m and g 9.8 m/.
Stre in tone, σ ρgh 00 9.8 000 0 MPa. oid bar of ength 0.7 m conit of three horter ection firmy joined together. uming the foowing appy, determine the change in ength of the bar when it i ubjected to an axia pu of 0 kn. ume that Young moduu, E 0 N / m. Section Length (m) Diameter (mm) 0. 0 0. 5 0. 0 0 0 and 0 0 7.850 5 m,.0 m 5 0.770 m σ F 00 7.850 5 8 MN/m, σ F 0 0.77 0 9.8 MN/m and σ F 0 0. 0 95.8 MN/m δ 80 0. 9.80 0. 95.80 0. 0 the change in ength of bar, δ 5.57 0 m 0.55 mm 7. If the bar of quetion were made from three different materia with the foowing eatic modui, determine the change in ength of the bar. Section EN / m 0 0 7 0 0
δ 8 0 0. 9.8 0 0. 95.8 0 0. 0 7 0 0 0 δ.058 0 m.058 mm 8. circuar-ection oid bar of inear taper i ubjected to an axia pu of 0. MN, a hown. If E 0 N / m, by how much wi the bar expand? From equation () of Probem, W d u E d 0 5 7.850 m Hence, u 0. 0 5 7.85 0 0 8.59 0 m bar extenion, u.59 mm
EXERCISE 5, Page 0. If the bar of quetion in Practie Exercie, page 95, were prevented from moving axiay by two rigid wa and ubjected to a temperature rie of 0 C, what woud be the maximum tre in the bar? ume the 0. MN oad i not acting. ume that coefficient of inear expanion, α 5 0 / C. Free expanion αt 5 0 0.50 m From equation () in the oution to Probem, Exercie, W u W T.5 0 u W T o that.5 0 7.9 0 0.0 from which, W T 88 N t the maer end, 88 maximum tre in bar, σ 5 7.850 0 MN/m. If the bar in Probem, Exercie, were prevented from moving axiay by two rigid wa and ubjected to a temperature rie of 0ºC, what woud be the maximum tre in the bar? ume the 0. MN oad i not acting, and α 50 / C. Free expanion αt 5 0 0.50 m From equation () in the oution to Probem, Exercie, W u W T.5 0 u W T o that.5 0 7.9 0 0. 0 from which, W T 788 N
t the maer end, 788 maximum tre in bar, σ 5 7.850 80 MN/m
EXERCISE, Page 08. n eectrica cabe conit of a copper core urrounded co-axiay by a tee heath, o that the two can be aumed to act a a compound bar. If the cabe hang down a vertica minehaft, determine the maximum permiibe ength of the cabe, auming the foowing appy: c 0 m E ectiona area of copper, c 0 N / m eatic moduu of copper, c 890 kg/m denity of copper, maximum permiibe tre in copper 0 MN/m, 0. 0 m E ectiona area of tee, 0 N / m eatic moduu of tee, 780 kg/m denity of tee, maximum permiibe tre in tee 00 MN/m g 9.8 m/ W (ρ c c +ρ ) g (0.89 + 0.57) 9.8 Compatibiity W 0. δ c δ ε c ε c E c E Ec 0 from which, σ c E 0 σ c 0.5σ
σ c i the deign criterion, σ c 0 MN/m and σ 0 MN/m Equiibrium W σ c c + σ or 0. (0 0 0 + 0 0 0. 0 ) (0 0 0 0 0 0. 0 ) 0. the maximum permiibe ength of the cabe, 0.5 m. How much wi the cabe of quetion tretch, owing to ef-weight? verage tre in copper 5 MN/m Therefore, average train 50 0.50 Hence, cabe tretch, δ.5 0 0.5 0.0098 m mm Check: σ V (tee) 0 MN/m ε V (tee).5 0 and δ.5 0 0.5 0.0 m mm. If a weight of 00kN were owered into the ea, via a tee cabe of cro-ectiona area 8 0 m, what woud be the maximum permiibe depth that the weight coud be owered if the foowing appy? Denity of tee 780 kg/m, denity of ea water 00 kg/m, maximum permiibe tre in tee 00MN/m, g 9.8 m/ ny buoyancy acting on the weight itef may be negected. W (00,000 + (780 00) 8 0 g)
W 00,000 + 5.8 () However, W σ 00 0 8 Equating equation () and () give: 0 0,000 N () 0,000 00,000 + 5.8 the maximum permiibe depth, 0000/5.8 8 m. weighte rigid horizonta beam i upported by two vertica wire, a hown. If the foowing appy, determine the poition from the eft that a weight W can be upended, o that the bar wi remain horizonta when the wire tretch. Left wire: cro-ectiona area, eatic moduu E, ength Right wire: cro-ectiona area, eatic moduu E, ength δ δ ε ε E E or E Taking moment about the right wire give: F W ( x) E () F W ( x) 5
σ σ W ( x) W ( x) 7 () F + F W σ + σ W W ( x) E W from equation () and () E W ( x) W ( x) E W E 0.5 ( x) ( x) and x x x 0.75 5. n eectrica cabe conit of a copper core urrounded co-axiay by a tee heath, o that the two can be aumed top act a a compound bar. If the cabe hang down a vertica minehaft, determine the maximum permiibe ength of the cabe, auming the foowing appy: c c 0 m E ectiona area of copper, c 0 N / m eatic moduu of copper, 890kg / m denity of copper, maximum permiibe tre in copper 0MN / m, 0. 0 m E ectiona area of tee, 0 N / m eatic moduu of tee, 780kg / m denity of tee, maximum permiibe tre in tee 00MN / m, g 9.8 m/ W (ρ c c +ρ ) g (.79 + 0.078) 9.8 W 8.5
Compatibiity δ c δ ε c ε c E c E Ec 0 from which, σ c E 0 σ c 0.5σ σ c i the deign criterion, σ c 0 MN/m and σ 0 MN/m Equiibrium W σ c c + σ or 8.5 (0 0 0 + 0 0 0. 0 ) (0 0 0 0 0 0. 0 ) 8.5 the maximum permiibe ength of the cabe, 59.7 m. How much wi the cabe of quetion 5 tretch, owing to ef-weight? verage tre in copper 5 MN/m Therefore, average train 50 0.50 Hence, cabe tretch, δ.5 0 59.7 0.059 m 5.9 mm Check: σ V (tee) 0 MN/m ε V (tee).5 and δ.5 0 0 59.7 0.059 m 5.9 mm
7. If a weight of 95 kn were owered into the ea via a tee cabe of cro-ectiona area 8 0 m what woud be the maximum permiibe depth that the weight coud be owered if the foowing appy? Denity of tee 780 kg/m Denity of ea water 00 kg/m Maximum permiibe tre in tee 00MN / m g 9.8 m/ ny buoyancy acting on the weight itef may be negected. W (95000 + (780 00) 8 0 g) W 95000 + 5.8 () However, W σ 00 0 8 Equating equation () and () give: 0 0,000 N () 0,000 95000 + 5.8 the maximum permiibe depth, 5000/5.8 m 8. weighte rigid horizonta beam i upported by two vertica wire, a hown. If the foowing appy, determine the poition from the eft that a weight W can upended, o that the bar wi remain horizonta when the wire tretch. Left wire: cro-ectiona area, eatic moduu E, ength Right wire: cro-ectiona area.5, eatic moduu E, ength δ δ
ε ε 8 E E or E E 9 () Taking moment about the right wire give: F W ( x) F W ( x) 0 σ σ W ( x) W ( x) () Reoving verticay give: F + F W σ + σ W W ( x) E W from equation () and () E W ( x) W ( x) E W E 0.5 or ( x) ( x) ( x) 0.50.5 0.5 ( x) 0.50.5 0.5 ( x) ( x) and x x x 0.75