tively. From horizontal equilibrium we have gential displacement at a, respec- Let o(a) and u(a) be the axial stress and tan Aco -Ac(o +do Under the assumption of small displacements and a linearly elastic material we have where e is the modulus of elasticity and Ac is the area of the bar cross section Differentiating the constitutive equation and combining the two equations to eliminate o' we obtain the Poisson equation with f=p/(EAc) Note 5 String under transversal load Consider a string of unit length under tension T, which is subjected to a trans verse distributed load of magnitude p(a) per unit length p(a) da 6+d8 Let u(a) denote the transverse displacement at point a. Assuming small dis ements, so that the tension T can be taken as constant over the whole string and considering vertical equilibrium we have T(6+d6)-T6=pd The angle 8 can be related to the displacement u simply as dau Note the minus sign which is due to the fact that a positive u corresponds to a downwards displacement. Combining the two equations to eliminate the variable 8 we obtain the Poisson equation with f= p/T✙✛✚✢✜✤✣✦✥★✧✑✩✤✪✬✫✮✭✰✯✱✥✲✧✑✩✴✳✑✚✵✜☞✶✓✚✷✪✹✸✡✺✻✪✬✼✾✽✿✜☞❀❁✚❂✽❁✽✴✪❃✫✮✭✰✜❁✪✬✫✮❄❃✚✢✫❅✜❁✺❆✪❃✼❇✭✓✺❆✽☞❈✓✼✻✪❃❉❊✚❂❋●✚✢✫❅✜✤✪✹✜❍✧❏■✓❀❁✚❂✽☞❈✑✚❂❉✁❑ ✜☞✺❆▲❃✚❂✼◆▼P❖✦◗✓❀❁❘❃❋❙✶✮❘❃❀❁✺◆❚❂❘❃✫❅✜❁✪❃✼✑✚❱❯❅❲✓✺◆✼❆✺❆✳✓❀☞✺❆❲✓❋❨❳✴✚✵✶✮✪❩▲❃✚ ❬✝❭ ✣✄❪ ❬❫❭ ✥✲✣✰❴❛❵P✣✛✩❝❜❡❞❢❵❃✧ ❣✤✫❤✭✡✚✢❀✴✜☞✶✮✚✵✪P✽☞✽☞❲✓❋●❈✡✜❁✺◆❘P✫✐❘✬❥❦✽✿❋❧✪✬✼❆✼❇✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽♠✪❃✫✮✭✰✪♥✼◆✺❆✫✓✚❱✪✬❀❁✼◆▼●✚✢✼✻✪❃✽✿✜☞✺✻❉✝❋●✪✬✜☞✚❂❀☞✺✻✪✬✼✑❳♦✚ ✶✮✪❩▲P✚ ✣❢❜q♣ ✥★✯✗❴❛❵P✯❇✩❦❪r✯ ❵P✧ s ✥✲t✉❘✈❘❃✇❃✚P① ✽♠✼✻✪❩❳✉✩ ❳❍✶✓✚❂❀☞✚❫♣②✺✻✽③✜❁✶✓✚✝❋●❘✡✭✡❲✓✼❆❲✮✽✴❘✬❥④✚✢✼✻✪❃✽✿✜☞✺✻❉❊✺◆✜⑤▼●✪❃✫✮✭ ❬❫❭ ✺❆✽❝✜☞✶✮✚❫✪❃❀☞✚❱✪⑥❘✬❥✾✜❁✶✓✚❫✳✮✪✬❀♠❉❊❀❁❘P✽❁✽❝✽☞✚❂❉❊✜☞✺❆❘❃✫✛❖ ⑦✝✺⑨⑧❇✚✢❀❁✚✢✫❅✜☞✺✻✪✹✜❁✺◆✫✮❄❢✜☞✶✮✚✄❉✢❘❃✫✮✽✿✜☞✺◆✜☞❲✡✜❁✺◆▲P✚⑩✚❱❯❅❲✮✪✹✜❁✺◆❘P✫❛✪✬✫✮✭❛❉❊❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❢✜☞✶✮✚✰✜⑤❳✴❘❢✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❶✜☞❘ ✚✢✼❆✺❆❋✗✺❆✫✮✪✬✜☞✚✷✣✾❷✛❳✴✚✵❘❃✳✡✜❸✪✬✺❆✫✰✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫⑩✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✄❳❍✺⑨✜❁✶✏❺❻❜❼❞✾❽✡✥❾♣❬✝❭ ✩✁❖ ❿r➀✾➁✁➂❢➃ ➄❝➁✿➅❩➆❾➇❏➈➊➉❤➇③➋✱➂✬➅❢➁✿➅✹➌✡➇✱➍❅➎✡➂✬➅❩➍❱➌✡➏❍➏⑤➀✡➌❇➋ ➐❘❃✫❤✽✿✺✻✭✡✚✢❀♠✪✗✽⑤✜❁❀☞✺❆✫✓❄❶❘❃❥❏❲✓✫✓✺◆✜❍✼◆✚❂✫✓❄✬✜❁✶✰❲✓✫❤✭✡✚✢❀✴✜☞✚❂✫✮✽☞✺◆❘P✫✰➑⑥■✈❳❍✶✓✺✻❉❸✶✰✺✻✽♠✽✿❲✓✳✓➒⑤✚❂❉✁✜❁✚❂✭✐✜☞❘●✪⑥✜❁❀❁✪❃✫✮✽⑤❑ ▲❃✚❂❀❁✽☞✚✵✭✡✺❆✽✿✜☞❀❁✺❆✳✓❲✡✜☞✚❱✭✄✼◆❘❅✪❃✭⑩❘✬❥④❋❧✪❃❄❃✫✓✺◆✜☞❲✮✭✓✚❍❞④✥★✧❇✩✴❈❤✚❂❀✤❲✓✫✓✺◆✜✤✼❆✚✢✫✓❄❃✜☞✶✛❖ ✙✛✚✢✜⑥✯✱✥✲✧✑✩✷✭✓✚✢✫✓❘❃✜☞✚●✜☞✶✮✚❧✜☞❀❸✪✬✫✮✽☞▲❃✚❂❀❁✽☞✚●✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❶✪✹✜⑥❈✑❘❃✺❆✫❅✜❶✧❏❖✄➓✉✽❁✽✿❲✮❋✗✺❆✫✓❄❢✽☞❋❧✪✬✼❆✼❝✭✡✺✻✽⑤❑ ❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽❂■✹✽✿❘✉✜☞✶✮✪✬✜❏✜❁✶✓✚❝✜❁✚✢✫✮✽☞✺❆❘❃✫⑥➑❛❉❂✪✬✫⑥✳✑✚③✜❸✪✬✇P✚✢✫❶✪P✽④❉✢❘❃✫✮✽✿✜❁✪❃✫P✜④❘✹▲❃✚✢❀❏✜☞✶✓✚✴❳❍✶✓❘P✼◆✚✴✽⑤✜❁❀☞✺❆✫✓❄✮■ ✪✬✫❤✭⑩❉✢❘❃✫✮✽☞✺❆✭✓✚✢❀❁✺◆✫✓❄✗▲❃✚❂❀✿✜❁✺❆❉❂✪✬✼✾✚❱❯P❲✮✺◆✼❆✺◆✳✮❀☞✺❆❲✓❋❙❳✴✚✵✶✮✪❩▲❃✚ ➑❶✥★➔✉❴❛❵❧➔P✩❦❪r➑✤➔⑥❜❡❞✏❵P✧❏→ ➣♠✶✓✚✷✪✬✫✮❄❃✼❆✚❫➔❧❉❂✪✬✫✄✳❤✚✷❀❁✚✢✼✻✪✹✜❁✚❂✭✰✜☞❘●✜☞✶✮✚⑥✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❍✯❢✽✿✺❆❋●❈✓✼❆▼⑩✪❃✽ ➔❶❜②❪ ❵❃✯ ❵❃✧ → ↔✤❘❃✜☞✚✏✜❁✶✓✚❢❋●✺❆✫❅❲❤✽⑩✽☞✺◆❄P✫↕❳❍✶✓✺❆❉❸✶➙✺✻✽✰✭✓❲✓✚❢✜❁❘❡✜☞✶✓✚❢❥✲✪❃❉✁✜✰✜☞✶✮✪✬✜✄✪❡❈✑❘P✽☞✺⑨✜❁✺◆▲P✚✏✯➛❉✢❘❃❀❁❀☞✚❱✽✿❈✑❘❃✫✮✭✮✽ ✜☞❘❢✪✏✭✓❘✹❳❍✫❅❳♠✪✬❀❸✭✓✽✵✭✡✺✻✽✿❈✮✼❆✪P❉❊✚✢❋●✚❂✫P✜❱❖ ➐❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❻✜☞✶✓✚●✜⑤❳✴❘✏✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❫✜❁❘✏✚✢✼❆✺◆❋●✺❆✫✮✪✹✜❁✚❧✜☞✶✓✚ ▲✹✪✬❀❁✺❆✪❃✳✓✼❆✚✝➔●❳✴✚❫❘P✳✡✜❁✪❃✺◆✫✄✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫✰✚❂❯❅❲✮✪✹✜❁✺◆❘P✫⑩❳❍✺◆✜☞✶❢❺❻❜❡❞✾❽❩➑⑥❖ ➜