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CA NTIL EV&S
'iw
c b—4 aL I' L -
Wx2
WaMmax 7
W
cwrved—raight I—Frjp Izçjg° W(&+/8a2b+/2ab3b3)
dmaxfJ('24E1
+Ja) _______________________2W
___ I/A J/Q
a b— c— ____ ____L. L
M
N ______RAW
4.— curved .4strai'ht f_-i ajdnaxjx c/C /5(15bumax. ,(/#
Bending moment, shear and deflection 1077
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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eria
l is
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his
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men
t is
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ect t
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e te
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cond
ition
s of
the
Ste
elbi
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cenc
e A
gree
men
t
CA NT/LEVERS
9 w I 2W
________ ________ ________ bL -I L 'I
M _T()_2] Mmax = w( 4)
i-.--— curved —H straight k— IIillWp—j1°'C 60E1 W(2#SOo2b,4Oab2-H/b)
/SbI dmox 60E1dmax Jf(i+ i7)
IA a b— aL L s.j
Mx IX Mx = MCM,,,x— P a
A [I
No shears
1N. B. For ant/—clockwise moments
the deflect/on is upwards.
—curved k— fCU/hLkSEX C 2E1
d=E2'(, Sb'
1078 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
Discuss me ...C
reat
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y 20
07T
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eria
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right
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ll rig
hts
rese
rved
. Use
of t
his
docu
men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
SIMPLY SUPPORTED BEAMS
w/z
L
//— T
N
_ 0122°I'ZL/
when x1—
'RB
When xa
djmt2n(2n)d#n2(242m]When xod4
where mx/L and naa/L
IA BT L
RB
R8W
RA=RB T
= S WL'"max 384 El
94 R5
Wa fSL2— c2jdà,cx.= 96(1
2 /ien .;—
RAfl\_______ RB
dmax j.j (8?_4Lb2#b3)
Bending moment, shear and deflection 1079
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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eria
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ll rig
hts
rese
rved
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of t
his
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men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
MA4nqx. —O•/28WL
when x,=0•5774L
L 84p8
Mx = Wx( 2)—WL/6
II .x;. I= WIS
R8
RB 2W/S
4,dmcx
0.0/304 wi!dinax
when x =05/PJL51AA BAT.—a 'F b -F a
L"A B
I'"mar. 4 ('- 3J
RA=RB =
Jdmax
WL560(1
Ij x 2X2Mx L"YLJ
Mmax'Wi/i2
RR8 wft
j dc,s
A5
d, ur=ëi(85,7aLL4c2L4I)
i,dmax.
—3iffk1
1080 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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hts
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his
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
SIMPLY SUPPORT(O BEAMS
,l. ti.
£k"\M -'"cx. 6
N'A=B= W/2
NRa
lOab# 5b2)
6
_____ RB
RA[— R8—w/2
4/max.
dmxJi(/5a2#2Oabi'.5b2)
N yzW/a-
Lb
Mmcx _(_m# 'j)iW,en x—
WmR8 —
z_when x= a/i—
_______IA
B .7
Bending moment, shear and deflection 1081
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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eria
l is
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right
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ll rig
hts
rese
rved
. Use
of t
his
docu
men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
SIMPLY SUPPORTED BEAMS
P p
Lpa
VA4nc P
I I
a>c
,,Ppib*Zc)"C LPc(b#2a)
MD— L
I I
L HP(b *)
LPot central deflection
odd the values for each Pderived from the formulain the adjacent diagram.
1
M,,,— T
I I
I IB'4—R8— .
J/nax.
PL3°;nax.
RA-R8-P
J/nax.
-
— Pb/L Po/L
always occurs within00774 L of tfie centre of the beo,nWhen ba,
d PL3rsa Ia i7centre 48E1L L (LII
This value is a/ways with/nS % of the maximum value.
1082 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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hts
rese
rved
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of t
his
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men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
PLMmax rPL
MC Mf —SPL
MD
RAI I
I JB
dp
d 23PL3mczx. 648(1
/9 P1!dmax = 304(1
— — 2P
4/PL3768(1
= P
dinax.
d SJPLin ax. —
MC=ME
A*C
Bending moment, shear and deflection 1083
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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ll rig
hts
rese
rved
. Use
of t
his
docu
men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
SiMPLY SUPPOATED BEAMS
pPppPP P
LA (n—I) forces BA
'9
pp p pb F AB
MCflMDME!fL
A4 —A8 "2P
;,2'max.
When n is odd.(nL /) p1
Mmax —When n is even.
Mmcx. — n. PL/8
R8'4B("')'%
When n is oddPL3 r i7r it /
When n /5 even
dmaxiuu,p4,. . nft_ :# )J
d 6JPLmax. 1000E%
TOTAL LOAD
When n >10, consider the load uniformly distributedThe reaction at the supports = W/2, but the maximum SE
at the ends of the beam — W(n;/)11,AW
The value of the maximum bending moment — C. WL
The value of the deflection at the centre of the span — k.
Value otn A C k2 0 250O 0.1250 O• 0/05S 0.3333 0/I/I 001184 03750 0.1250 00/245 Q.4 01200 0.0/266 04/67 O•IZSO OO/277 O4286 0/224 0•0/288 0•4375 0•1250 00/289 04444 0/236 00l29/0 O•4500 0.1250 0•0/29
1084 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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hts
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men
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
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cenc
e A
gree
men
t
SIMPLY SUPPORTED BEAMS
L '.1
A CI BA MA(,a + b—I ___________
1 L MA ®M4>M5 'M8
MA Al8
_ Al4 1jt'—M8Al. GIL MC8 — Al . bIL (Al8 antic/ockw,M8
_________________ Shear diagram when MA "M8
PAl Va R4( Vs
A4 A5 M/L Al4 — MB
As shown
M.cb 'a b' When M4M8.dc-31 (zz) ML2For anti-clockwise moments dmcxaii —ythe deections are reversed ____________________________2nd degree_parabola. W Complement of parabola.KiI '°1
L L
Mx -i (m4-2m#m) Mx (mJm2#4m1_2m4)— Mmq .LfL
/6RAfl ______J RA4—R5—W/2 ,IA_R8_W/z
dmqx
a' 61WL3 28W1maz — dmax —
Bending moment, shear and deflection 1085
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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his
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
SIMPLY SUPPORTED BEAMS
w unit /oao'
C A 0 B
-
AAI
_
dc=o_4(jn3,Ln2_ i)(s-)
Where
/W.unitboo
CiA BAD-HNI L '1QH-
___ W(L#N)(LN)
m.x/L ,,1.N/L
/W.unFt /oaoçCADBflL NHA5
MA=MB=-
—4..',,rTA
RDL,
RA=RBWNd4f(fC
wL2 N/6(1
w— unit
_4J4
—j O•5774L0 BEL IQH
AMax. upward deflection is at 0.
NIA
1086 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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men
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
8(J/LT—/N BEAMS
1p1
WL
WLMC
4B WI?
—.10g/L .—O s8L—.12/L p—
WI!=384(1
V4—a-4..—b- "+ c—4
MA
MA,b [e(4L-Je)- c 31'4L —Jc)]
B,bLY4t--7°" (4L —sajRA
When r is the sinp/e support reactionM-M M-MA4 L _r3# L
/W wI2 W12
-- b -H L -4
ENMA=MB
= = w/2
____ I /dflfl
-J Wa / I
vinax=4f% (L—OJ
'-jvl'A
k—- L— -in
MA=-.in (Jnr"—em+o)M8_ jrn2 (4-3m) ,'Mmax'L,,2( 3si,)/2
k—x- When x=2(n,3—2m42)2
RA= W(m-2) 8 = Wm(2-m)2 2ri,
dna.
uWhen a=4/2 and x1=O-445L
WI!3JJ(%
WI!c/C = 384(1
When a = c.
.Ey(L3#2Lba #4Lc?—8a)
Bending moment, shear and deflection 1087
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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his
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
B/JILT-/N BEAMS
LA L8 A C ___
M
MA ___________M8 MA V NMB- WL 7/Qx 9x ) M - SWL
#Mmax WL/233 N5Qn x=OS5L MWL//6MA = —WI//S M8 =— WL/iO
-
R8 RB
R=O.SW R8=O7W RA=RB=W/2
—4J•22LF—OS6L —Ø22L
WI! — 1.4W!!max. - 384E1
when x, = 0• 525L
W/2 2W/L HI'a
La- b—+_aJ ____CL—4 F I.
MA M8 (SL44aL_402) MA = M8 -WL//6
M=WL/48
JRBR8 W/2 A =R3 W/Z
. L/2 —H ¼
O-WL3364(1
1088 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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eria
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his
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men
t is
subj
ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
B(//LT-/N BEAMS
fwTw/a____L
MAV NMaMA
AB W/2
____ I /d________4
Wa /dmax =
2W/a
bL
MA =-J'2(Jo/obL)M3
mAC. Mx =P.X+M3-2W(x.-bP
InCB.Mx RB.X#M5
A = (/OL—sLa42a)2, prO,, —
W/2 W/a W/2-Y HJ—a I b -f
L
1NMBMA Maz_R(4L_3a)
= = W/2
dmax = (/SL—Ma)
W
bI. L .1
MAMB
MA — _!10L_15#502)
M3 _lOL2(5L4t
= (/oL-/sLa'#8a)
R = '/5L —ec)
Bending moment, shear and deflection 1089
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
Discuss me ...C
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
BUILT—/N BEAMSparabolic total /oao W
JRa'!4 W/2
F4axJ1.3 WL3
d,,, 384E!
Any symmetrical load WrA ______
symmetrical diagram
NMBMA = M8 —A,JL
where A5 is the area of the 'free'bending moment diagram
ARB
A =R8 = W/2
A,- - I The fic.re shown
is ha/f the bending* + moment diagram
C 1* and +7fr- X, -H-X--- LareC.GI
A5x —AIX1lmaxatC ZETWhere A' is the area of the fixing
moment diagram
RA R8 W/2
rA
O•4WL3dma 384 (2'
a rb2br
2 -2br----—-
aab r&_
CM. f(Ja—L)M8c -M(3b-L)When
'4I=P8 = slope of moment diagramM#M M#M
EabWhen '2/L = in,
M. L2m2(/_m)2(1_2m)2E%
For ant/clockwise momentsreverse the deflect/ons
con,c*mentparabo/p
JM8A -M5=-WL//O
MAV NMBAq rM8i.._WL/2O
1090 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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his
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
BUILT—/N BEAMSp p
L/2 L/2 I-. b 8JJF. L
MA L—' Al5 MA LZ EEE3 M8- MA = - M5 = M = PL/8 MA=- M5 = —
_______ Mc2Pa2b
__________ 1!A1_____ _____ A__________
=P (iT) (I#2 *)= = /,,21Q8 = z
4mcx __I.— Xp9 d — Po3b3dinax = /92 El C —
d _ZPa2b3 WñQnX="3EI(3L2a)2 JL-Za
p p p p
Pa MA/ \MB- __Pa(L-q) MA=MB=_JPLh6MA—MB— L
MC MD = P02,'L MC =MD =
RAfl AI ILJ I IPA=P'P
PLdmax = /30(a)7
Bending moment, shear and deflection 1091
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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men
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o th
e te
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and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
BUILT—/N BEAMSP p p p______ IrgfA C D
'.—L/y —1--L/j —.—L/j.
MA MB MA LF- 5N M8MA =M = —/9PL/72
MD =1/
PArJ__I
LJARA = JP/2
p p p P P p pII + + 1I
MAVNMB MAL/L NEiNMBMA =M3=—I/PL/j2MD = Mf = SPL/32
max. = 96(1
l,L/3 P.-L/3L2
MA =M8 = — 2PL/9
MC =MD = PL/p
AL I
S PL3dmax 548?1
4/PL3
A C D £ BLL/4--L/4 L/4 L/4
34C D E PBL/L/4 +L/4 +L/4 4L14
MA=MB =r—SPL/,6= JPL/15
2P
— PL3dmax — 96(1
1092 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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men
t is
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
BUILT—/N BEAMS
HJ-
Y1F,ennisoø6 ,I / /Cimox 114-4c (,ji
Whennis even,
•:.:i•• O...o.&. ,. 4;.' •O4•4•.EfIiiiitiIIIn CONTINUOUS BEAM
}d L/n#-+-L/n4L/n+-L/-#L1-+-44L
L. - When n >10, consider the load .nitorrrdy oYstiibuted -The load on the outside stringers is carried c'/rect/y by the supportsThe continuous beam Is assumed to be horizontal at each supportThe reaction at the supports for each s,oan = W/2. but the maximum
shear force in any span of tM continuous beam = V/J.IAWThe value of the fixing moment at each support = — B. WL
The value of the maximum positive moment for each span = C. W4The value of the maximum deflection for each span —0'0O26
Value ofn A B C2 02500 00625 0•0 6253 0•3333 0074/ 003704 03750 0078/ 00469S 04000 0-0800 0•04006 04/67 Qc// 0•04397 04286 0O8/5 00408B Q•4375 00820 00 4309 04444 00823 004/310 04500 00825 00425
p pp pA C V t PL/5 +Lg +L,/s+L4
ppppppp4 (n-i) forces B
I I sôcev,naLJ,
MA LNMB MAMA M52PL/5MD wM — PL/5
AArLMA=MB=— P1,'/)
'A B=2'°
d = 13PL3max. 1000 El
COLUMN LOAD PER SA4NW
Bending moment, shear and deflection 1093
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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men
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ect t
o th
e te
rms
and
cond
ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
WL= — ---
— WL3dmax. - ____
PROPPED CAN T/L(VERS
/WA C B
WebL
BAb .f- c—.JL°'
MA7tSMA =— d2-c2)(29c2_c12)
-____L R8=r5—
Where and r3 are the simplesupport reactions for the beam
(MA being considered positive)
w/
MA
W
A C B
3L/8 -— 9W!.C —
___ Ca -f-b
MA=— f (2n)2wñere a/Ln
#MmaxRA -n (4-n)]
RAi4J[8n2(4n)]1
P3 Hn2( fl)
dC(/_I2n#7n2_n)
—--_J P8
P3 =fx/Lm
LA'max.
d=j(m —Jm3#2m)
MA
'P
MA=—
A I
J2 (o - n 2)
= f (n'-n-pi.e)
a, d=J42/n3(3n?-6b5When x a.
dj[2p4—p3n (n-6n,'.8)+_________ pn2(jn2_8n#6)]
1094 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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and
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ition
s of
the
Ste
elbi
z Li
cenc
e A
gree
men
t
PROPPED CANTILEVERS
W/2 W/21'
C U BA—f— a -H
L
MA
if = % then between B and U.Al,. = ftx4xa(4 -Sm+Zm9]+M,n0x. when x=I'4Jm#2rn2,
—20)
RB
MAZ_1027SLf
X/L = mx —HMx(2Qm'—27m#7)A 7WL —'''max.
'9:4=0 67L]
= (2i."#soi. 42)p8 = (2L2_30L#40')
Vx i(9_rn2)A PB =
- OO0/WL3El
When x=0•598L
W/? W/2
-L
MA
RB
3WLMA
#MmaxO0454 WL[When x 0 283L]
4W= T RB = * p_LA - /3WB - 32
- 0•0047WL'dmax (I
When x=O•447L—ii
When x=0404L
Bending moment, shear and deflection 1095
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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PROPPED CANT/LEVERS
a_____ C B A A _____a
L iR8 L A5___ ___ 0•577bI S
0/28 Wa
MAMA
_________________ JLBQt we en
CandA, Mx=R5.x-j!('x-b) M5=R5.x -(JaL/5aL#2OL2) =_ j9(si.2_j,2)
#Mmox when xb# fI/-
___ RAt NJR5A8 =('5L —a) = 9 (si!—,/)
W P5 A8 = (t+SaL2)
2W2W W
__________ ____________ -____ C ______ ____a " b—' 4—a .. b I
L R3 L
H0577aH 0/28 Waf—x —Wab H042Jb—
MA
When ,n=a/L 2 3m#2 I
MC=AB.b MA=_f/0L2-3b2)Between AandC RA I
=RAW52/a2 I_______________ Between C and B \
}.—x--H \ IRS Vx=RA_Wrc24Z N
A8 = '('/SL —4a) A3 = [L(//L -/S,i#(5L- a)]RA=W-RB RA=W-RB
1096 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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gree
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fAA 51 CD1. L
IWa2
M5=—2M— F
c-'3'L 8TN
IJA 8+ CD
'1. L
M'184 'C 0
1.
—-Pa=
q
- 8 =-
= j4p2,pq#Jp #Jq)d,n0x 27(1
PROPPED CANTILEVERS
W.iw4,
.jwap
d0 WL4f2(ep#4?q#dp3(p#i)J-Omax. —
54(1
TflM3 -2M4 =—M
4I
— dmax
IdoI0'O
a#b)#a2(2# VJ'-dmax"
Bending moment, shear and deflection 1097
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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gree
men
t
PROPPED CANT/LEVERS
W/2
AB
Wa
MA _—4(2L_a)
NBA -(4L?#ZaL— az)
W
L kO.4/SLtq
MA
#MmaxOO948 WLA4flR,
A32 B j/
+
na/L *7/q
MA=— f-(4L—3a)W'?Qn X< a,tl =(9,r?x —/2nx# /2x— 4xg2)+Mmax occurs whQn
A r—\
(4L2 #4aL—JoT
R8 W—
wIa —+—b —+-—a —4
I— —L——-——--R8
MA7ZMArn (5L2+40L—4G2)
RAI\ I '?&
A477(2/1!#4aL 4a2)A8 —
1098 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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gree
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t
SWL
M .('/Om-ZOm#7m)#My,OO888WL,when X
0 •3965L
7PL7(%
dmax'OOOPJ2 ft3
coIr4oleraQnt of parabola!F051
— SWL
Mx =(-4om+8om-om4I7m)#Mmax*O0399WL,whQn x —
02343L
A6
MAW-' lz' ,nax.MA-O./93PLwMn baO577L
M=('Z-# p)maxA1-0./74PL
A3=f3b#2L) RAPR5
PROPPED CANT/LEVERS
7WB —
MA
A51•• —
F9
Bending moment, shear and deflection 1099
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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gree
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PROPPED CANT/LEVERS
AC D EB*L/L/3 Lb
Pp P
A C D E¼+V4+L/4 +-/4
ISPLMA = —
/7PL JJPLD "E7PAL
I
PAJ3 J2
dmax00209 zr
P PP PLj.4C DEFBA
0nczx. — O.OZdS
MA
'4
MA
4P 2P
dx 00152
P P P
MA MA
/9PLMA=— 48M -. M ....SJPL0 96 £ 288
dxOQ/69
1100 Bending moment, shear and deflection
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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gree
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PROPPED CANILEVEPS
p p p pD E F19
L/4tL/4tL/44L/MA
P PPPP Ppforces B
MAE7ZTJPL(,,?l)
MA= 8nJSPL
MA=— -—Ad /S7PLrT(— 5/2
P8
GO•OZZl
RAEL (5n2-4n-i)
RB = (34z_4fl#/)
whennis!or9e, dmax. /85(1
Any symmetrical load W
AreaR L AreoS'B
MAVAr000 X
If AArea of free B.MDiagramM-A8
P =!—82 L
JA ____I.
a=L
a>O423L® _r—cO423L
a< O423L
M1 9(2_6n#sn')MCA (2— 6n# Pn?3ni
-Sn
It'B
LIn Case!, R — SM/ZL
CaseS, A= M/L
dmax occurs at point correspondingto Xon M diagram, the area Abeing equal to the area 0
Area SXxVmax =
Bending moment, shear and deflection 1101
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables
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