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.