Chapter 5 Bending Moments and Shear Force Diagrams for Beams

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1 Chpter 5 ending Moments nd Sher Force Digrms for ems n ddition to illy loded brs/rods (e.g. truss) nd torsionl shfts, the structurl members my eperience some lods perpendiculr to the is of the bem nd will cse only sher nd bending in the bem. The current chpter together with Chpters 6 to 8 will focus on such n issue. 5. SHER FORCE ND ENDNG MOMENTS DGRMS FOR EMS Sher Force Digrm (SFD)indictes how force pplied perpendiculr to the is (i.e. prllel to cross section) of bem is trnsmitted long the length of tht bem. ending Moment Digrm (MD) will show how the pplied lods to bem crete moment vrition long the length of the bem. These digrms re used to determine the norml nd sher stresses s well s deflection nd slopes in the following chpters. 5. EM SGN CONVENTON (S&4 th :56-57; 5 th :56-57) t ny point long its length, bem cn trnsmit bending moment M() nd sher force V(). f loded bem is cut, the definitions of positive distributed lod, sher force nd positive bending moment re s Fig. 5. below: ositive internl bending moment ositive distributed lod ositive internl sher force Fig. 5. em sher force nd bending moment sign convention Where distributed lod cts downwrd on the bem; internl sher force cuses clockwise rottion of the bem segment on which it cts; nd the internl moment cuses compression in the top fibers of the segment, or to bend the segment so tht it holds wter. 5. RETONSH ETWEEN EM ODNGS (S&4 th :64-68; 5 th :64-68) bem (Fig. 5.) is loded with verticl forces F i, bending moments M i nd distributed lods w(). F F w() F..D. of element d w() M() M() dm() d d M M V() d V() dv() d d d Fig. 5. Trnsversely loded bem nd free body digrm of element d ecture Notes of Mechnics of Solids, Chpter 5

2 ook t the FD of n elementl length d of the bove loded bem (Fig. 5.). s it hs n infinitesiml length, the distributed lod cn be considered s Uniformly Distributed od (UD) with constnt mgnitude w() over the differentil length d. t is now necessry to equte the equilibrium of the element. Strting with verticl equilibrium dv ( ) ( ) ( ) ( ) F y = = V w d V d = d (5.) Dividing by d in the limit s d, dv ( ) = w( ) d (5.) Tking moments bout the right hnd edge of the element: d dm ( ) ( ) ( ) ( ) ( ) M R.H. Edge = = M V d w d M d = d (5.3) Dividing by d in the limit s d, dm ( ) = V ( ) d (5.4) Eqs. (5.) & (5.3) re importnt when we hve found one nd wnt to determine the others. 5.3 ENDNG MOMENT ND SHER FORCE EQUTONS ntroductionry Emple - Simply Supported em y using the free body digrm technique, the bending moment nd sher force distributions cn be clculted long the length of the bem. et s tke simply supported bem, Fig. (5.3), s n emple to shown the solutions: F..D. (globl equilibrium) F..D. (method of section -) M() o V() R Y =(-/) R Y =/ R Y =(-/) Fig. 5.3 FD of bem cut before force Step : Cut bem just before the force (i.e. Section -), nd drw free body digrm including the unknown sher force nd bending moment s in Fig Tke moments bout the right hnd end (O): M o = = M ( ) = M ( ) = (5.5) To determine the sher force, use Eq. (5.4), giving tht: dm ( ) ( ) V = = (5.6) d To verify Eq. (5.6), equte verticl equilibrium: ecture Notes of Mechnics of Solids, Chpter 5

3 F y = V ( ) = V ( ) = which is the sme eqution s (5.6). These then, re the equtions for the bending moment nd sher force vrition in the rnge of. To find out the rest of the bending moment nd sher force distributions, it is necessry to now crry out similr nlysis, but cutting the bem just before the end (Section -). Step : Cut bem just before the right hnd end (RHE) F..D. (Section -) M() o V() R Y =( -/) Fig. 5.4 FD of bem cut before the right hnd end Equte moments bout the right side: M = = ( ) M ( ) = giving: M ( ) = ( ) = (5.7) nd using Eq. (5.4), the sher force eqution is : dm ( ) ( ) V = = (5.8) d These epressions for the bending moment nd sher force cn now be plotted ginst to produce the Sher Force nd ending Moment Digrms s Fig. 5.5: ecture Notes of Mechnics of Solids, Chpter 5 3

4 oding Digrm (-/) / V() (-/) Sher Force Digrm -/ M() (-/) ending Moment Digrm Fig. 5.5 Sher Force nd ending Moment Digrms for simply supported bem Mculey's Nottion (4 th :59-599; 5 th :59-599) The two sets of equtions for V() nd M(), Eqs. (5.5), (5.6), (5.7) nd (5.8), cn be condensed to just one set of equtions if we use specil type of nottion clled Mculey's Nottion. The bove equtions would look like this (to be derived in Emple 5.) V ( ) = (5.9) M ( ) = (5.) Where the nottion hs the following mening: n = ( ) n for < for ( n ) (5.) when differentiting: n n = n for n for n = for n = (5.) Remrks To derive the bending moment eqution by using Mculey's nottion, you my need to do the following: ) Determine the ground rections from globl equilibrium; ) Cut the bem just before the right hnd end; ecture Notes of Mechnics of Solids, Chpter 5 4

5 3) Equte the cut FD to equilibrium bout the right hnd end; 4) ll length terms in the bending moment/sher force equtions MUST be written using Mculey's nottion; 5) lwys indicte the powers, even if they re or. Emple 5.: s in the introductory emple, determine the sher force nd bending moment equtions nd plot them for simply-supported bem s in the introductory emple. Step : Determine the ground rections; F..D. (globl equilibrium) We hve R Y =(-/) R Y = (-/) nd R Y = / R Y =/ Step : Drw FD of bem cut just before the RHS (Section -). F..D. (Section -) o R Y =( -/) V() M() Step 3: Equilibrium for FD of bem cut just before the RHS (Section -). Tke moments bout RHS: M O = = M ( ) = M ( ) = nd differentiting w.r.t. '', s Eq. (5.4), gives the sher force eqution s: dm ( ) ( ) V = = d Step 4: lotting the Sher Force nd ending Moment Digrms ecture Notes of Mechnics of Solids, Chpter 5 5

6 ccording to M() nd V() to depict the digrms, ook t the equtions segment by segment When M ( ) = ( ) = nd V ( ) = ( ) = To plot this segment in the digrm, firstly look t the boundry points s =,M ( ) = nd =,M ( ) = ( / ). Drw two points nd then connect them becuse the eqution gives line. ikewise, one cn plot Sher Force Digrm in this region. When M ( ) = ( ) ( ) = = nd V ( ) = ( ) ( ) = = Remrks: lese drw globl FD of the bem firstly nd follow by its Ser Force nd ending Moment Digrms. The reson for doing this is tht when you get sufficient eperience, you my be ble to directly plot the Sher Force Digrm by observing the eternl forces s well s plot ending Moment Digrms by observing the Sher Force Digrm. Nevertheless you MUST still work out nd indicte the loctions nd vlues (including or ve) t ll turning points in the digrms in detil. t is lso interesting to note tht concentrted forces (e.g. rection forces nd eternl forces) correspond to inclined line in MD nd horizontl line in SFD. oding Digrm (-/) / V() (-/) Sher Force Digrm -/ M() (-/) ending Moment Digrm Emple 5.: Determine the sher force nd bending moment equtions nd plot them for simply-supported bem loded with UD. ecture Notes of Mechnics of Solids, Chpter 5 6

7 Step : Determine the ground rections; From globl equilibrium the ground rection forces cn be found to be both equl to w/ s, w F..D. (globl equilibrium) w/ w/ M = = R Y ( w) = R Y = w / ( upwrds) Fy = = RY RY w = R Y = w / ( upwrds) Step : Drw FD of bem cut just before the RHS (Section -). F..D. (Section -) w R w / o M() V() R Y =w/ Step 3: Equilibrium for FD of bem cut just before the RHS (Section -). s fr s V() nd M() re concerned the UD cn be temporrily replced by its resultnt R w (=w) pplied t the centroid of the UD distribution in the moment equilibrium eqution. So if we tke moments bout the RHS of the bem we get: O = = R Y Rw M = M w w M ( ) = nd differentiting w.r.t. '', s Eq. (5.4), gives : dm ( ) ( ) w V = = w d Step 4: lotting the Sher Force nd ending Moment Digrms ccording to M() nd V() to depict the digrms ( ) = ( w / ) ( w ) M ( ) ecture Notes of Mechnics of Solids, Chpter 5 7

8 w oding Digrm w/ V() w/ w/ Sher Force Digrm -w/ M() w /8 rbol ending Moment Digrm t is worth pointing out tht one should not completely replce such UD by its corresponding resultnt concentrted force R w (=w) in the beginning of the solution. There is significnt difference of the Sher Force nd ending Moment Digrms between concentrted force (Emple 5.) nd UD (Emple 5.). t is lso interesting to note tht the UD corresponds to n inclined line in the Sher Force Digrm nd qudrtic curve (prbol) in the ending Moment Digrms. ecture Notes of Mechnics of Solids, Chpter 5 8

9 Emple 5.: Determine the sher force nd bending moment equtions nd plot them for bem loded with UD between nd nd two concentrted forces t C nd E. F..D. (globl equilibrium) R Y UD=w=kN/m kn m 5m 5m m R DY C D kn E Step : Determine the ground rections; M = = 5 5 RDY 3 = R DY = 3. 5kN Fy = = R Y RDY = R Y = 7. 5kN Step : Drw FD of bem cut just before the RHS (Section -). Note: The only problem with Mculey's Nottion is tht it does not work when UD stops. t however does work for UD which strts nywhere long bem nd continues to the end. The problem cn be corrected by pplying UD of equl mgnitude but opposite sense where the first UD ends. F..D. (Section -) nd ppliction of equivlent UD w=kn/m kn o M() 7.5kN 3.5kN V() Step 3: Equilibrium for FD of bem cut just before the RHS (Section -). Tke moments bout RHS: M O = = M ( ) = nd differentiting w.r.t. '', s Eq. (5.4), gives the sher force eqution s: dm V ( ) = = d ( ) M ( ) ecture Notes of Mechnics of Solids, Chpter 5 9

10 Step 4: lotting the Sher Force nd ending Moment Digrms oding Digrm UD=w=kN/m 7.5kN kn m 5m 5m m 3.5kN C D kn E Sher Force Digrm V() kn ending Moment Digrm M() knm qudrtic gin, the UD segment corresponds to n inclined line in SFD nd qudrtic curve in MD. Emple 5.3: Determine the sher force nd bending moment equtions nd plot them for cntilever bem loded with moment M = 4kNm nd force F= kn. Globl F..D. M M =4kNm kn C R Y 4m.5m Step : Determine the ground rections; The cntilever bem is fully clmped in the left hnd end s shown. The ground rection for this point should hve rection force R Y nd rection moment M.. So the globl equilibrium is given s F y = = R Y = R Y = kn (- downwrds) M = = M = M = 95kNm (- clockwise) ecture Notes of Mechnics of Solids, Chpter 5

11 Step : Drw FD of bem cut just before the RHS (Section -). F..D. (Section -) 95kNm M =4kNm O M() kn 4m V() Step 3: Equilibrium for FD of bem cut just before the RHS (Section -). Tke moments bout RHS: O 4 M = = 95 4 ( ) M ( ) M = nd differentiting w.r.t. '', s Eq. (5.4), gives the sher force eqution s: dm ( ) ( ) V = = = d Step 4: lotting the Sher Force nd ending Moment Digrms 95kNm oding Digrm M =4kNm kn C kn Sher Force Digrm V() kn - M() knm 95 ending Moment Digrm 5 55 t is interesting to observe tht due to concentrted bending moments M (rection moment) nd M (eternl moment), there is respectively sudden lep nd drop in the ending Moment Digrm. n ddition, such concentrted moments do not ffect the Sher Force Digrm (Note tht the drop t point (the left end) in SFD is due to the concentrted rection force R Y. n fct, both M nd M do not pper in sher force eqution V() t ll). ecture Notes of Mechnics of Solids, Chpter 5