ornenclature E= Modulus ofElasticityofsteel at 29,000 ksi. I= MomentofInertia ofbeam,in.4. L= Total length ofbeam betweenreactionpoints ft. M,, = Maximum moment, kipin. M1 = Maximum momentin leftsection ofbeam,kip-in. M2 = Maximum momentin rightsection ofbeam,kip-in. M3 = Maximumpositivemomentinbeamwithcombinedendmomentcondi- tions, kip-in. M, = Momentat distancex fromend ofbeam,kip-in. P= Concentratedload, kips PI =Concentratedload nearestleftreaction,kips. P, =Concentratedloadnearestrightreaction,and ofdifferentmagnitudethan PI, kips. R= End beamreactionfor any conditionofsymmetrical loading,kips. R1 = Leftend beamreaction,kips. R2 = Right end or intermediate beam reaction,kips. R3 = Right end beamreaction,kips. V= Maximum vertical shear forany conditionofsymmetrical loading, kips. V, = Maximum vertical shear in leftsection ofbeam,kips. V2 = Verticalshearat rightreactionpoint,ortoleftofintermediatereaction pointofbeam,kips. V3 = Vertical shearat rightreactionpoint,or to rightofintermediatereaction pointofbeam,kips. V, = Verticalshearat distance x from end ofbeam,kips. W= Total load on beam,kips. a= Measured distancealong beam, in. b= Measureddistancealong beamwhich may begreater or less than a, in. I= Totallength ofbeambetweenreactionpoints, in. w= Uniformlydistributed load per unit oflength, kipstin. wl = Uniformlydistributed load per unitoflength nearest left reaction,kipslin. w, = Uniformlydistributed load per unitoflength nearestrightreactionand of differentmagnitude thanwI, kipslin. x= Any distancemeasuredalong beamfrom leftreaction,in. xl = Any distance measured along overhang section ofbeam from nearest reac- tionpoint, in. A, , = Maximumdeflection,in. A = Deflectionat pointofload,in. A,= Deflectionat any point x distance from left reaction,in. AXI = Deflectionofoverhang section ofbeamat any distance from nearestreac- tion point, in. The formulas givenbeloware frequentlyrequiredinstructuraldesigning.They are included hereinfor the convenience ofthose engineers who have infrequent use for suchformulas andhence mayfindreferencenecessary.Vanationfromthestandard nomenclature on page2 - 2931snoted. Flexural stress at extreme fiber: f= Mc/I= MIS Flexural stress at any fiber: f= My/I y= distance from neutral axis to fiber. Average vertical shear (for maximum see below) : v= V/A= V/dt(for beams and girders) Horizontal shearing stress at any section A-A: v= VQ/IbQ= staticalmomentaboutthe neutral axis ofthe entire section ofthat portion ofthe cross-section lying out- side ofsection A-A, b= width atsection A-A (Intensity ofverticalshear is equal to that ofhorizontal shear acting normal to i tatthe same pointand both are usually amaximum atmid-height ofbeam.) Slope and deflection at any point: E I e= Mx and yare abscissa and ordinate respectively ofapoint dxa ontheneutralaxis,referredtoaxesofrectangularco- ordinates through aselected point ofsupport. (Firstintegrationgivesslopes;secondintegrationgivesdeflections.Constants ofintegration must be determined.) Uniform load:~aA + 2Mb I1 Concentrated loads: Considering any two consecutive spans in any continuous structure: Ma. Mb, MC=momentsatleft,center,andrightsupportsres~ctivelv.ofanv -- - pair ofadjacent spans. &and k =length ofleft and rightspans respectively, ofthe pair. 11 and IZ=moment ofinertia ofleft and rightspans respectively. wland wz =load per unit oflength on left and rightspans respectively. PIand PZ =concentratedloads on left and rightsDans res~ectivelv. a;and a; =distanceofconcentratedloadsfrom iefts u ~ b r t inleftandrieht * - spans respectively. - bland bz=distance ofconcentratedloads fromrightsupport inleft and right spans respectively. The above equations are for beams withmoment ofinertia constant in each span butdifferingindifferent spans,continuous overthreeormore supports.Bywriting such an equationforeachsuccessive pairofspansandintroducingtheknownvalues (usually zero) ofend moments, all other moments can befound. - Coeff. SimpleBeam BeamEixedOneEnd Supported atOther 0.0703 0.1250 0.3750 0.6250 0.0054 1.OQOO0.4151 0.1563 0.1875 0.3125 0.6875 0.0093 1.5000 0.4770 0.2222 0.3333 0.6667 1.3333 0.0152 2.6667 0.4381 P P P Pgative moment (kip-ft.): Deflection coeff. forequivalent simple 0.3600 0.6000 1.4000 2.6000 0.0265 4.8000 0.4238 simple span uni - for BeamFixed Both Enc 0.2000 0.4000 - 2.0000 0.0130 3.2000 0.3120 m load (kips): ic loading conditions Formeaning ofsymbols, see page2 - 293 4.SIMPLEBEAM-UNIFORMLOADPARTIALLYDISTRIBUTED wb RI= V,.(max.when a < c). .= ~ ( 2 c f b)wb R2= Va(max.when a > c)..= ; i ~ - (2a+b) VX( whenx >aand a and < (a + b)).=RIX -( x- a) l2 M~ (whenx> (a+b)). . . .=RI (1-x) Moment 5.SIMPLEBEAM-UNIFORMLOADPARTIALLYDISTRIBUTEDATONEEND Rl= V lmax. wa . . . . . ...=(21-a) wait R z = V a . . . . . . . . . .=- 21 ~1V, ( whenx( a+b) ) . =Ra(1-x)--------2 7.SIMPLEBEAM-CONCENTRATEDLOADATCENTER . . . . . Total Equiv. Uni form Load=2P P . . . . . . . . . . . R = V =-2 . . . . .P6= - 4 1Px . . . . Mx( w h e n x < ? . ) . =-2 . . . . .PI' Amax.(at poi ntof load = - 48EI . . . . .Px Moment AX( when x < 1.= w( 3 / 1 - 4 ~ q8. SIMPLEBEAMCENTRATEDLOADATANYPOINT Ik . 4 Total Equiv. Uni form Load RI = VI max.when a < b ( R2 = ~a(rnax.when a > b Mmax.(atpoi ntof load ) ) atpoi ntofload AX(when x< a .. when 8 Pab =- 12 Pb =- I - ' Pa -- I Pa b =- I Pbx =- I - Pab (a + 2b)2/3a(a + 2b) 27ElI Paabz =- 3EI I = s ( / 2 - b * - x a )EQUALCONCENTRATEDLOADS RIGALLYPLACED 8 Pa . . . . . . Total Equiv. Uni form Load- I R = V . . . . . . . . . . .= PMmax.betweenloads). . . . . .Pa ( Mx(whenx < a). . . . . = Px Pa . . . . . . . . . amax.(atcenter)-(31"4a) 24EI . . . . . . . Ax(whenx< a)~ ( 3 l a - 3 a . - x a )6EI Ax( wh e n x > a a n d < ( l - a ) ) . .=*(3lx-3xr-aa) 6EI Formeaning ofsymbol s,seepage2 - 293 10.PEEBEAM-T UNS . . . . .( P RZ= Vt max.whena < b)-I( l - a f b ). . . .) = I ( [ - b + a ) 1 Vx( wh e n x > a a n d < ( l - b ) ) , .=;(b-a). . . .MI (max.whena > b ) =Rl a . . . . . Ma x . w e a< b)Rab Mx (whenx a a n d < ( l - b ) ) . .=Rt x- P( x- a)CONCENTRATEDLOADS YPLACED - Ra=Vz- Pi(I-a)+ Pzb . . . . . . . . . . .1 - R==V=- Pl a + Pa( 1 - b) . . . . . . . . . . .1 Vx (whenx > a and< (1-b))..=RI - P i). . . = Ri a . . . Mz(max.whenR2 < Pa)=Rab . . . . . . Mx(whenx < a)=Ri x Moment Mx (whenx> aand < (I-b))..=Rlx-PI(x-a) TONEEND,SUPPORTEDATOTHER- LYDl STRlBUTEDLOAD . . . . .Total Equiv. Uni form Load =wl3wl . . . . . . . . .R i = V a . . =- 8 5wl . . . . . . . . .R, Rt= Vzmax. = - 8 . . . . . . . . . . . .Vx=RI-wx w1= Wlmax. =-. . . . . . . . . . .8 3 MI( h t ~ = ~ l )9 =wla . . . . . . .wxz Mx. . . . . . . . . . . .=R I X - ~A( a = 1+= 4215) .=185EI WX Ax. . . . . . . . . . . . .= ---- (13 -31x2+2x 48EI Formeaning ofsymbols,seepage2 .. 293 END,ORTED AT OTHER- TEDAT CENTER 3P Total Equiv. Uni form Load. . . . = - 2 58 Ra = Va . . . . . . . . . .=-16 11P Rs = Vamax.. . . . . . . . = - 16 3Platfixed end). . . . . = - 16 5Platpoi ntof load). . . . = - 32 . . . . . .5Px =- 16 1l l x. . . . Mxwhen^>^).=P(;--) PI3PI= = .44721)..= -= .009317- 48EI 21%El7PIJ . . . d ) . =- 768El Px . . . . .=- 96EI(3ts - 5x21 PPORTED AT OTHER- LOAD AT ANY POINT MI(atpoi ntof load. . . . =Rza . . . . .Pab (atfixed end)= -- ( a f l )211 . . . . . MX(whenxa) =Rax-?@-a) Amax.(whena < ~ 1 4 1 atx = Amax.(whena > .4141atx=I . . . .Paaba ' A(atpoint,load= - 12Etls(31+a) Pb2x . . . . . . Ax(whenx < a)(3alZ-21x1-axI; BEA Forvariousstaati Formeanmg ofsymbols,seepage2 - 293 BOTHENDS-UNIFORMLYDISTRIBUTED LOADS 2wI Total Equiv. Uni form Load. . . .= 7 wIR = V . . . . . . . . . . =- 2 Vx. . . . . . . . . . .=w(g-x) w1= Mmax.(atends). . . . . .= -12 wl' (atcenter). . . . . . = -24 . . . . . . . . . . . Mx=( 61~- 12-6x2) 12 WI*Amax.(atcenter). . . . . . = - 384El wxz . . . . . . . . . . .Ax =-(I-x)Z 24EI 16.BEAMFI XEDATBOTHNDS-CONCENTRATEDLOADAT CENTER . . . Total Equiw. Uni form Load R = V . . . . . . . . .Mmax. (atcenter and ends.. 1 . . . . Mx(whenx a ) . . . . . = P( x - a )Amax.(at f reeend). . . . .= z ( 3 1 - b )) . . . .Pba Aa (atpoi nt of load= - 3EI AX(whenx a)=- 6EI( 3b- I +x) 22.CANTILEVERBEAM-CONCENTRATEDLOADATFREEEND . . . . Total Equiv.Uni form Load= 8P IP . . . . . . . . . . . Mx=Px Pi3 . . . .amax.( at f r eeend) = -3EI 111.. . . . . . . . . . . .ax= L ( 2 i 3 - 3 / 2 x + x 3 ) 6EI 23.BEAMFIXEDATONEEND,FREETODEFLECTVERTICALLYBUT NOTROTATE ATOTHER-CONCENTRATEDLOADATDEFLECTEDEND Total Equiv. Uni form Load. . . . =4P . . . . . . . . . aR = V = PM max.(atboth ends Pi) . . . . = - 2 . . . . . . . . . . .MX=P( i - X)PI3 Amax.(atdeflectedend) . . . . = -----12EI P (1-x)Z . . . . . . . . . . . Ax=-12EI(1 +ex) 2 -304 For meaning ofsymbols,seepage2 - 293 (betweensupports) (foroverhang ) (betweensupports) (foroverhang.. ) (betweenSupports) (foroverhang).. 25.BEAMOVERHANGINGONESUPPORT-UNIFORMLY DISTRIBUTEDLOADONOVERHANG waa RI=VIL. . . . . . . . + =- 21 wa Ra=vl +va. . . . . . . ==( 2l +a). . . . . . . . . . . va=wa . . . . Vx,(foroverhang)=w ( a- xd waa M max.(atR ~ ) . . . . . . -- 2 wazx Mx(betweensupports)..= -21 ) . . . . Mxl (foroverhang-- - ( a - ~ a ) ~between supportsat x= Amax.(foroverhangatxs= a).=(41 + 3a) ) waax ax(betweensupports..-12EI 1(12-xz) . . . .Ax,(foroverhang)= 524EI(4azl +6a~xr4axi z+xr Formeanlng ofsymbols, seepage2 - 293 26.BEAMOVERHANGINGONES UPPORT-CONCENTRATED LOADATENDOFOVERHANG Pa Rt = Va . . . . . . . . .=- 1 P Ra=Vz+Va. . . . . . .=- ( [ +a)1 Va. . . . . . . . . . .=P . . . . . . M max.(atR. ) =Pa Pax Mx(betweensupports)..= -1 . . . . Mx,(foroverhang)=P (a-XI) I dH)= -$& = .tJ6415! %! ?Amax.(betweensu pports atx =- ElAmax.(foroverhang atxn= a).= -!??3EI( 1+a) Pax Ax(betweensupports)..= -(12 - x. )6El lPxi.A (tor overhang). . . . - -6EI(2al-k 3axr -x12) 27.BEAMOVERHANGINGONESUPPORT-UNIFORMLY DISTRIBUTEDLOADBETWEENSUPPORTS . . . Total Equiv. Uni form Load=wlwlR = V . . . . . . . . . .=- 2 . . . . . . . . . . . VX=w(f- X)( . . . . .w1= Mmax.atcenter) = - 8 WX x. . . . . . . . . . . =-(I-X) 2 5wl r . . . . . Amax.(atcenter)= - 38481 WX . . . . . . . . . . .Ax = -(la-21x2+ x') 24El wl'xl Axl. . . . . . . . . .=- Moment24EI 28.BEAMOVERHANGlNGONESUPPORT-CONCENTRATED LOADATANYPOINTTWEENSUPPORTS 8Pab Total Equiv. Uni form Load . . . = --12 Moment . . . RI = Va (max.when a < b = . . .) - Ra=~a( max.when a > b - - . . . Mrnax.(atpoi ntof load ) - Aaj atpoi ntof load ) . . .= ~b - 1 Pa - I Pa b - 1 Pbx - 1 Pab (a + 2b)d 3a(a + 2b) 27EI 1 Pa%= - 3EI 1 Pbx (12-bz-xa) Pa (1-x) (21x-xr-a2) 6EI 1 Pabxr rn ( l +=) For meaning of symbols,seepage2 - 293 29.CONTINUOUSBEAM-TWOEQUALS ONONESPAN Total Equiv. Uniform hod=49w[64 7 R%=Va. . . . . . . . - w [16 . . . . . .5 R3Ra=V&Va- w I8 Ro=Vs. . . . . . .=- .J- wl716 V YI. . . . . . . - w l 9 16 7 16 ; 2 wl'a t x =- 1 ) ..=- atsupportRr).=&wl rWX when x < l ) ..=- (71 -8x) 16 AMax.(0.4721 fromRI)=0.0092wPI EI30.CONTINUOUSCONCENTRATED LO 13 Total Equiv. Uniform Load.=-g-P 13 . . . . . . . . R,Ra=\lr- P32 11 . . . . . . Ra=Va+Vs- P16 3 V,Ro=Vs. . . . . . . =-- 32 19 V1. . . . . . . . = - P32 13 Rllrnax.(atpoint ofload).=PIM~1(atsupportR*).=& P IA Max.(0.4801 fromR1)=0.015PPI EI31.CONTINUOUSBENS-CONCENTRATED Formeaning ofsymbols,see page2 - 293 32.BEAM-UNIFORMLYDISTRIBUTEDLOADANDVARIABLEENDMOMENTS 33.BEAM-CONCENTRATEDLOADATCENTERANDVARIABLEENDMOMENTS I For variousstaticloa Formeanlng ofsymbols,see page2 - 293 34.CONTINUOUSBEAM-THREEEQUALSPANS-ONEENDSPANUNLOADED wlwl AMax. (0.4301 from A)= 0.0059wl4/El 35.CONTINUOUSBEAM-THREEEQUALSPANS-ENDSPANSLOADED AMax.(0.4791 from Aor5) = 0.0099wP/El ----36.CONTINUOUSBEAM-THREEEQUALSPANS-ALLSPANSLOADED 0.400 wl 0.400 wl SHEAR I AMax.(0.4461 from A orD) = 0.0069wP/El Formeaning ofsymbols, seepage2 - 293 37.CONTINUOUSBEAM-FOUREQUALSPANS-THIRDSPANUNLOADED w2WE wl A Max. (0.4751 fromE)= 0.00)4wl4/EI 38.CONTINUOUSBEAM-FOUREQUALSPANS-LOADFIRSTANDTHIRDSPANS 39.CONTINUOUSBEAM-FOUREQUALSPANS-ALLSPANSLOADED For variouscome Thevalues glven ~nthese formulas do not Include Impact wh~chvanes accord~ngto the requfrements of each caseFor meanlng of symbols,seepage2 - 293 40.SIMPLEBEAM--ONECONCENTRATEDMOVINGLOAD R2 PI).=7M max.(atpoi ntofload,when x = - 41.SIMPLEBEA-TWOEQUALCONCENTRATED LOADS PLEBEAM-TWOUNEQUALCONCENTRATEDMOVING LOADS ( I - aRImax.= Vlmax.atx = o. ...=PI+Pz[[underpl,atx=$(l - max.may occur wi t h larg ad atcanter of span a ad off span (case 40) GENERALRULESFORSIMPLEBEAMSCARRYINGMOVING CONCENTRATEDLOADS Themaxi mumshearduet o movingconcentratedloads occurs atonesupport when one of t he loads i s att hatsupport. Wi t h several moving loadst he location t hatwi l lproduce maxi- mum shear must be det er hned by tri al . The maxi mum bending moment produced by moving con- '' centratedloads occursunder one of t he loads whent hatload i s as f ar f romone supportast he centerofgravityofal lt he moving loads on t he beam is f rom t he other support. I nt heaccompanyingdiagram,t hemaxi mumbending momantoccurs under loadPI whenx= b.It shoujd alsobe not ed t hatt hi s condi ti on occurswhent he centerl ~ n e oft he Moment span1s m~dwaybetween t he center of gravlty ofloads and t he nearest concentrated load. Equal loads,equallyspaced Svstem MOMENTANDSHEAR CO-EFFICIENTS EQUALSPANS,EQUALLYLOADED Given the simple span length, the depth of a beam or girder and the design unit bendingstress, the center deflection in inches may be found by multi- plying the span length in feet by the tabulated coefficients given in the fol- lowing table. FoFthe unit stress values not tabulated,the deflection can be found by the equation0.00103448 ( ~ ~ f b l d ) whereLis the span in ft, fbis the fiber stress in kips persq. in.and d is the depth in inches. The maximum fiber stresses listed in this table correspond to the allow- able unitstressesas providedinSects. F1.landF1.3 ofthe AISC ASD Specification forsteels having yield pointsranging between36 ksi and 65 ksi when Fb= 0.664;and between36 ksiand100 ksi when Fb= 0.604. The table values,as given,assume a uniformly distributedload.Fora single load at center span, multiply these factors by 0.80; for two equal con- centrated loads at third points, multiply by1.02. Likewise, for three equal concentratedloads at quarter points multiply by0.95. Ratio of Depth toSpan 118 119 1/10 111 1 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 1120 1121 1/22 1/23 1/24 1/25 1/26 1/27 1/26 1/29 1/SO MaximumFiber Stressin Kips PerSq.In.