CHAPTER 4 STRESS AND STRAIN

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Frederica Crawford
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1 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 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) m, m and m σ F MN/m, σ F MN/m and σ F MN/m δ

2 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 ) δ δ 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 m Hence, u m u () bar extenion, u 0.79 mm Proof of formua for thi probem

3 t x, d 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 x.0 W W x 8 W 0 7x dx ( 7 x) u 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/.

4 Stre in tone, σ ρgh 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) and m,.0 m m σ F MN/m, σ F MN/m and σ F MN/m δ the change in ength of bar, δ 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

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.

5 δ δ 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 m Hence, u m bar extenion, u.59 mm

6 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 m From equation () in the oution to Probem, Exercie, W u W T.5 0 u W T o that from which, W T 88 N t the maer end, 88 maximum tre in bar, σ 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 m From equation () in the oution to Probem, Exercie, W u W T.5 0 u W T o that from which, W T 788 N

7 t the maer end, 788 maximum tre in bar, σ MN/m

8 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 ( ) 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

9 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. 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 Hence, cabe tretch, δ m mm Check: σ V (tee) 0 MN/m ε V (tee).5 0 and δ 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)

10 W 00, () However, W σ Equating equation () and () give: 0 0,000 N () 0,000 00, 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

11 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 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 ( ) 9.8 W 8.5

12 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 ( ) ( ) 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 Hence, cabe tretch, δ m 5.9 mm Check: σ V (tee) 0 MN/m ε V (tee).5 and δ 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?

13 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 ( (780 00) 8 0 g) W () However, W σ Equating equation () and () give: 0 0,000 N () 0, 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

14 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) ( x) ( x) ( x) and x x x 0.75

O -x 0. 4 kg. 12 cm. 3 kg

O -x 0. 4 kg. 12 cm. 3 kg Anwer, Key { Homework 9 { Rubin H andau 1 Thi print-out houd have 18 quetion. Check that it i compete before eaving the printer. Ao, mutipe-choice quetion may continue on the net coumn or page: nd a choice