OUTLINE CHTER 7: Fleural embers -Tpes of beams, loads and reations -Shear fores and bending moments -Shear fore and bending - -The fleure formula -The elasti urve -Slope and defletion b diret integration method Tpes of beams eams are usuall desribed b the manner in whih the are supported. Simpl supported beam Cantilevered beam Overhanging beam in support prevents translation but does not prevent rotation. Roller support prevents translation onl in the vertial diretion. Fied support prevents translation and rotation. Tpes of loads Conentrated load Distributed load oment [Referene # : age 55 8] Reations The first step in the analsis of beam is finding the reations. From the reations, the shear fore and bending moments an be found. For the statiall determinate beams, all reations an be found from free-bod diagrams and euilibrium euations. Conentrated load Distributed load oment Reation Reation 5 Shear fores and bending moments The stress resultants in statiall determinate beams an be alulated from euilibrium euations. F 0, 0 0, 0 6
Sign onvention positive shear fore ats lokwise against the material. positive bending moment ompresses the upper part of the beam. Homework # : Referene # roblem 6-7 Relationship between loads, shear fores and bending moments From euilibrium euations, +d d d +d + + + + 0 + + 0 8 Shear fore and bending Shear fore and bending D D 9 0 Shear fore and bending Draw the and D L/ b L b L/ L/ 0 D L/ L/ L/ L Homework # : Referene # roblem 6-0
[Referene # : age 8 0] Compression Tension Need Reinforement! D + Reinforement neutral ais ais Δ neutral surfae ais Δ Δs Δ Δ Undeformed element Δs' Δs lim Δs 0 Δs ( ) Δθ Δθ lim Δθ ais Δθ Δs Δ Deformed element Δ θ 0 5 6 Longitudinal normal strain will var linearl with from N ma / ma / ma ais Δ Normal strain distribution ma ma The fleure formula ma Normal strain variation ending stress variation FR F ; ma 0 df d ma d ma d ma The fleure formula ending stress variation d 0 7 ma d 8 ( R ) ; df ( d) ma d ma I I I The moment of inertia of the rosssetional area about N
Eample Eample If the beam has a suare ross setion of 00 mm on eah side, determine the absolute maimum bending stress in the beam.. kn 8 kn/m 0.8 m 0.8 m If the beam has a retangular ross setion with a width of 00 mm and a height of 00 mm, determine the absolute maimum bending stress in the beam. 0 kn m 8 kn 0 kn kn/m m 6 m 00 mm 00 mm 9 Homework # : Referene # roblem 6-6 0 The elasti urve Sign onvention The elasti urve the defletion diagram of the ais that passes though the entroid of eah ross-setional area of the beam. Fore oment Displaement Rotation or slope + + ositive internal moment - - Negative internal moment positive bending moment ompresses the upper part of the beam. [Referene # : age 569 589] Infletion point Infletion point D Elasti urve Infletion point Homework # : Referene # roblem -5 ais oment-urvature relationship ds efore deformation EI ais dθ ds fter deformation
Slope and displaement b integration Represent the urvature ( ) in terms of v and ; From ; EI / [ + ( dv / ) ] / / [ + ( dv / ) ] / EI Slope and displaement b integration EI From ; EI ( ) d From ; EI ( ) Simplif b ; EI d From w ; EI w( ) 5 6 + Sign onvention +w + + + ositive sign onvention +dv +v v If positive is direted to the left, then θ will be positive lokwise. + + + ds Elasti urve dθ +θ 7 oundar onditions Δ 0 Δ 0 Δ 0 Δ 0 Roller in Roller in θ 0 Δ 0 0 0 0 Fied end Free end Internal pin or hinge 8 Eample Eample Determine the euations of the elasti urve using the and oordinates. EI is onstant. Determine the maimum defletion of the beam and the slope at. EI is onstant. 0 0 a L b a a a 9 Homework # 5: Referene # roblem - 0