Period #8: Axial Load/Deformation in Indeterminate Members

Transcription

1 ENGR:75 Meh. Def. odie Period #8: ial oad/deformation in Indeterminate Member. Review We are onidering aial member in tenion or ompreion in the linear, elati regime of behavior. Thu the magnitude of aial tree in the member are le than the material yield tree, and the magnitude of aial train are le than the material yield train. 8. The Univerity of Iowa E aial aial aial d E d E E ontant fore, area

2 . Priniple of linear uperpoition In olid mehani and trutural mehani, it i ometime very onvenient to onider multiple load ating on a ytem one at a time, and to then add their effet. Thi i known a uperpoition. Thi an generally be done only when the truture behave linearly i.e. feature a mathematially linear relationhip between applied fore and reulting diplaement. Two neeary ondition for the priniple of linear uperpoition to apply are:. the material in the truture mut be in the linear elati regime of behavior; and;. the deformation in the truture diplaement, rotation, and train mut be mall. onider thi truture uing and not uing uperpoition ENGR:75 Meh. Def. odie The Univerity of Iowa 8.

3 nalyi w/o Superpoition - from to : / ompreive from to : tenile / from to : / / ENGR:75 Meh. Def. odie The Univerity of Iowa 8.3

4 nalyi w/ Superpoition ae from to : from to : from to : / / ae / / from to : / ombining ae & : ae ae - ENGR:75 Meh. Def. odie The Univerity of Iowa 8.4

5 . ppliation of linear uperpoition to analyze indeterminate aially loaded member. Statially indeterminate trutural ytem are thoe where the number of unknown upport reation eeed the number of global equilibrium equation. In uh ytem it i not poible to olve for the upport reation uing only the equilibrium equation of tati. To olve for the reation fore at the redundant upport, we an ue knowledge of how muh the truture diplae at the upport to olve for the unknown reation. In thi illutrative eample, there i only one relevant equation of equilibrium but there are two upport reation at and to olve for. Thu the ytem i tatially indeterminate. To olve for both upport reation, we need an additional ondition. Here, ine the upport at both and are taken to be rigid, we an ay that / = or that overall the aial member doe not hange it length. P ENGR:75 Meh. Def. odie The Univerity of Iowa 8.5

6 Now, letting remain fied, we ll onider the two load ating on the Now, letting remain fied, we ll onider the two load ating on on the ytem: the known fore of magnitude P ating at ; and the ytem: the known fore of of magnitude P P ating at at ; ; and the unknown reation fore at. unknown reation fore at at.. P P P ae : / P : P / ae : / / ae : ae : : / / / / Putting the two load ae together : Putting the two load ae together : : P P P / / / / P / / / / P onluion: Indeterminate problem an be olved! rom Stati, Solve for rom Stati, Solve P P P P P P P : for : P P : ENGR:75 Meh. Def. odie The Univerity of Iowa 8.6

7 D. Thermal epanion/ontration Mot material undergo epanion when their temperature inreae and ontration when the temperature dereae. In unretrained aial member the aial train due to a temperature hange T i given by the epreion: t at where a i the material oeffiient of thermal epanion. Often time member are retrained and annot epand or ontrat when the temperature hange. In uh ae, fore and aoiated tre build up in the member and thee are alled thermal tree. rod with Young modulu E and oeffiient of thermal epanion a i poitioned between two rigid upport at initial temperature T o. When the temperature inreae to T what i the aial fore or tre in the rod? + = =-a T T=T o T=T o =T -T o a o = o / The Univerity of Iowa ENGR:75 Meh. Def. odie 8.7

8 E. Eample Problem Eample #8.: The olumn i ontruted from high-trength onrete and four -36 teel reinforing rod. If it i ubjeted to an aial fore of 8kN, determine the required diameter of eah rod o that one-fourth of the load i arried by the teel, and three-fourth by the onrete. E t =GPa, and E =5GPa. Solution: 8kN 8kN 8kN.3m 4 d 4 rod d 4.9m 4.9m d m 33.85mm 8kN 8kN 8kN 4.9m m 8kN kn 6kN E kn5gpa E E E GPa6kN 4 E E The Univerity of Iowa ENGR:75 Meh. Def. odie 7.8

9 Eample #8.: Two -36 teel wire are ued to upport the 65-lb engine. Originally, i 3 in. long and i 3.8 in. long. Determine the fore upported by eah wire when the engine i upended from them. Eah wire ha a roetional area of. in. 65lb rom Stati : ' 3 65lb 3".8".8" ' 3.8" 3" 65lb ' ' E E 3".8" ' ' E ' ' E 3".8" 65lb ' ' E 3" 3" 65lb.65kip.8" 3 E 9 kip in.in 3" 3".3986" '.3986" E 3".3986".8" ' 3" 3 9 ki.in 36.3lb 3 9 ki.in 88.7lb The Univerity of Iowa ENGR:75 Meh. Def. odie 8.9

10 Eample #8.3: The bronze 86 pipe ha an inner radiu of.5 in. and a wall thikne of. in. If the ga flowing through it hange the temperature of the pipe uniformly from T = at to T =6 at, determine the aial fore it eert on the wall. The pipe wa fitted between the wall when T=6. Solution: Propertie of 86 bronze: E=5 3 ki; a=9.6-6 / rom tati : Ue uperpoition: ae : epanion of the pipe due to thermal train aume pipe i free to etend ae : ompreion of the pipe due to retraining wall fore =. T 4 at t t d a 4 d.645".645".645".645" a ki 7.6kip 96" The Univerity of Iowa ENGR:75 Meh. Def. odie 8.