rdge esgn Revew of nfluence Lnes / 7 Sprng 0 nfluence dagrams can be constructed for beams for three types of forces : a) a partcular reacton b) the shear at a partcular locaton n the beam c) the bendng moment at a partcular locaton n the beam. nfluence dagrams for beams can be constructed by applyng a unt deformaton at the beam locaton of f nterest, based on the recprocal deflecton equaton. The equaton for trusses s: Adaptng ths equaton for beams for each of the types of forces: R reacton R = + f s n nches, s untless shear Note: slopes of segments are equal =+ f s n nches, s untless moment =+ Eamples Typcally s n feet so that s n feet.
rdge esgn Revew of nfluence Lnes / 7 Sprng 0. nfluence agrams for etermnate eams (quanttatve) hnge C E 6 roblem. Calculate the mamum reacton at due to the AASHTO HS0 truc, shown below. 8 3 3 4 4 R raw the nfluence lne for the reacton at by dsplacng the beam one unt upwards at. The deflected shape s shown above. b 4 0.6 nfluence agram for Reacton at 3 3 8 oston of AASHTO Truc to cause ma. Reacton at To calculate the reacton at due to the truc, place the 3 rear ale at the le end of the brdge, and the 3 drve ale 4 to the rght (4 to the rght of ). The reacton at s then: R ( 3 )( ) ( 3 )( 0.6) 83.
rdge esgn Revew of nfluence Lnes 3 / 7 Sprng 0 roblem. Calculate the shear just to the rght of due to the AASHTO unform load plus concentrated load shown below. 8 0.64 lf raw the nfluence lne for shear at by breang the beam just to the rght of and dsplacng the two ends unt, as shown. The beam segments on ether sde of the cut should have the same slope. The dsplacements at the other locatons can be calculated by smlar trangles. For eample, the dsplacement at C (labeled a here) s calculated a, a 0.467 0' 4' 0.467 C a = 0.467 6 0.5 nfluence agram for Shear just to Rght of 8 0.64 lf 0.64 lf AASHTO Lane Loadng to cause ma. Shear just to Rght of 8 ( 8 )( ) 8 lf.64 0.64 lf 0 ( 0.467) 4 ( ) 0.35 lf 8 0.64 8 0.35 8.35
rdge esgn Revew of nfluence Lnes 4 / 7 Sprng 0 roblem 3. Calculate the moment at E due to the AASHTO unform load plus concentrated load. raw the nfluence lne for moment at E by breang the beam at E and rotatng the rght end an relatve to the le end, as shown. Snce the beam segments are both feet on ether end of the brea, the angles of each end are equal and equal to one half of.0 rad = 0.5 rad. The ordnate of the nfluence lne at E s calculated from the followng equaton: a tan ' tan for 0.5 rad small dsplaceme nts a, a 6' ' The other ordnates are calculated usng smlar trangles. 5 5 E 0.5 rad a=6 6 3 nfluence agram for oment at E 8 0.64 lf 0.64 lf AASHTO Lane Loadng to cause ma. oment at E The moment at E due to the concentrated load s mamum when the load s placed at E. The moment due to the 8 load s then calculated from: 8 E ( 8 )( 6 ) 08 The moment at E due to the unform load s mamum when the load s placed at on the beam where the nfluence dagram s postve. The moment due to the 0.64 lf load s then calculated from: 0.64 lf E ( 0.64 lf )( (0')( 5 )) ( 0.64 lf )( (4 )( 6 )) 6. E 08 6. 70.
rdge esgn Revew of nfluence Lnes 5 / 7 Sprng 0. nfluence agrams for ndetermnate eams (qualtatve) roblem 4. etermne the locaton(s) to place a unform lve load to cause the mamum reacton at Support. raw the shape of the nfluence lne by applyng a unt vertcal dsplacement at Support. 3 4 Then, apply loads to Spans and to cause ma. reacton at Support. roblem 5. etermne the locaton(s) to place a unform lve load to cause the mamum shear just to the rght of Support 3. raw the shape of the nfluence lne by applyng a unt vertcal relatve dsplacement just to the rght of Support 3. Therefore, load Spans and 3 to cause mamum shear at Support 3.
rdge esgn Revew of nfluence Lnes 6 / 7 Sprng 0 roblem 6. etermne the locaton(s) to place a unform lve load to cause the mamum moment at the mddle of Span. The nfluence dagram s: Therefore, load Spans and 3 to cause mamum postve moment at mddle of Span. roblem 7. etermne the locaton(s) to place a unform lve load to cause the mamum moment at Support 3. The nfluence dagram s: Therefore, load Spans and to cause mamum negatve moment at Support.
rdge esgn Revew of nfluence Lnes 7 / 7 Sprng 0 3. nfluence Lnes for Frames roblem 8. etermne the load pattern to cause mamum moment at the mddle of Span on Level. The nfluence dagram s: Level 3 Level And the span load pattern to cause ma postve moment at the mddle of Span on Level s therefore: Level 3 Level