Steel Designers Manual (6th Edition) - Bending Moment, Shear and Deflection Tables
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Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
1077
CA NTIL EV&S
'iw L
c b—4
a
I'
-
L
Wx2 Wa
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
Mmax 7 W
Frjp Izçjg cwrved—raight I—
°
W(&+/8a2b+/2ab3b3)
dmaxfJ(' +Ja) ___ a
b— c— L.
24E1
_______________________ 2W
I/A J/Q
____ ____ L
M
N RAW
i aj ______
4.— curved
dnaxjx
.4strai'ht f_-
c/C /5(1
umax. ,(/#
5b
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1078
Bending moment, shear and deflection CA NT/LEVERS
w
9
2W
I
________L________ -I
________L
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
M _T()_2]
1
/SbI
dmax Jf(i+ i7)
IA
a
b—
L
'I
Mmax = w( 4)
i-.--— curved —H straight k—
llWp—j °'C 60E1
b
Mx IX
IIi
dmox
W(2#SOo2b,4Oab2-H/b) 60E1
a L
s.j
Mx = MC
M,,,x— P a
No shears
A[
I
1
N. B. For ant/—clockwise moments
the deflect/on is upwards.
k—
—curved
SEX
d=E2'(,
Sb'
fCU/hLk C 2E1
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
SIMPLY SUPPORTED BEAMS
w/z
IA
T
B
L
L
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
RB
//
—
N
T
R8 W
RA=RB T
= S WL' "max 384 El
94 R5
dà,cx.=
Wa fSL2— c2j
_
ien
2/
96(1
012
2°I'ZL/
when x1—
.;—
RAfl\
'RB
_______ RB
When xa
djmt2n(2n)d#n2(242m] When xo
dmax j.j (8?_4Lb2#b3)
d4where mx/L and naa/L
1079
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1080
Bending moment, shear and deflection
L
M Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
x,=0•5774L
II
.x;.
— WL/6
I
R8
= WIS RB 2W/S
RA=RB =
Jdmax
4,dmcx dinax
2)
Mx = Wx(
A4nqx. —O•/28WL when
84 p8
0.0/304 wi!
WL5
60(1
51
when x =05/PJL
AA'F T.—a
"A
b L
BA
-F a
B
I' 4 ('- 3J "mar.
Mx
Ij x 2X2
L"YLJ
Mmax' Wi/i2 A5
RR8 wft j dc,s d, ur=ëi(85,7aLL4c2L4I)
i,dmax. —3iffk1
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
SIMPLY SUPPORT(O BEAMS
-
N yzW/a ,l.
ti.
£k"\ MCreated on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
'"cx. 6
N
Lb
Mmcx _(_m# 'j) iW,en x—
NRa
'A=B= W/2
R8 —
Wm
lOab# 5b2)
6
_____
RB
z_ RA[
when x= a/i—
_______IA
— R8—w/2
4/max. B
dmxJi(/5a2#2Oabi'.5b2)
.7
1081
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1082
Bending moment, shear and deflection
SIMPLY SUPPORTED BEAMS
p
P
1 L
pa
V
A4nc P
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
M,,,— T I
I
I
I
IB
I
RA-R8-P
'4—R8— .
J/nax.
J/nax. °;nax.
-
PL3
a>c
"C ,,Ppib*Zc) L MD—
L
I
I
— Pb/L
Pc(b#2a)
Po/L
P(b *)
LH
L
always occurs within
00774 L of tfie centre of the beo,n
When ba, d
PL3rsa
Ia i7
centre 48E1L L (LII This value is a/ways with/n
S % of the maximum value.
Pot central deflection odd the values for each P derived from the formula in the adjacent diagram.
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
PL
Mmax r Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
RAI
I I
=P
MC Mf —
PL
MD
JB
dp
dinax.
dmczx. 23PL3 648(1
din ax. — SJPL
A*C
MC=ME
— — 2P
/9 P1!
dmax = 304(1
4/PL3
768(1
SPL
1083
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1084
Bending moment, shear and deflection
SiMPLY SUPPOATED BEAMS
pp p p A
F
b
pPppPP
AB
LA
P
(n—I) forces B
'9
MDME!fL Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
MCfl
When n is odd. (nL /) p1 Mmax — When n is even. Mmcx. — n. PL/8
'4 B("')'%
A4 —A8 "2P
;,2'max.
it /
When n is odd
6JPL dmax. 1000E%
PL3 r
R8
i7r
When n /5 even
dmaxiuu,p4,. . nft_ :# )J TOTAL LOAD
When n >10, consider the load uniformly distributed The 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
2 S 4 5
0 250O 0.3333
7 8 9
O4286
6
/0
03750
C
k
0.1250
O• 0/05
0/I/I
00118 00/24 0.0/26
Q.4 04/67
0.1250
0•4375
0•1250
04444
0/236
OO/27 0•0/28 00/28 00l29
O•4500
0.1250
0•0/29
01200
O•IZSO
0/224
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
SIMPLY SUPPORTED BEAMS L
CI BA
A
a 1
L
+ b—I
'.1
MA(, MA
___________'M8 ®M4>M5
MA
Al8
_ Al. GIL MC8 — Al . bIL Al4 1jt'—M8
(Al8 antic/ockw,M8
_________________
Va R4(
PAl Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
Shear diagram when MA "M8
Vs Al4 — MB
A4 A5 M/L As shown
M.cb 'a b' dc-31 (zz)
When M4M8. ML2
dmcxaii —y
For anti-clockwise moments the deections are reversed 2nd degree_parabola. W
____________________________ Complement of parabola.
L
L
KiI
Mx -i (m4-2m#m) —
RAflJ
'°1
Mx Mmq
R
A4—R5—W/2
(mJm2#4m1_2m4) .LfL
/6
______ ,IA_R8_W/z
dmqx
a'maz — 61WL3
dmax —
28W1
1085
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1086
Bending moment, shear and deflection
SIMPLY SUPPORTED BEAMS
w unit /oao' C
A
0
B
/W.unFt /oaoç
CADBfl L NH A5
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
-
MA=MB=RD L,
AAI
—4..',,
rTA
_
dc=o_4(jn3,Ln2_ i) Where
(s-)
/W.unit
CiA
-HNI
d4f( RA=RBWN
L
boo
BAD
'1QH-
fC
wL2 N
/6(1 w— unit
0 BE IQH
—j O•5774L
L _4J4 A Max. upward deflection is at 0.
NIA ___
W(L#N)(LN)
m.x/L ,,1.N/L
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection 8(J/LT—/N BEAMS
wI2
1p1/W
L
-4
EN
WL
MA=MB
WL
MC Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
b-
--
H
4B —.10g/L
W12
= = w/2
WI?
.—O s8L—.12/L p—
I /dflfl
____ = WI!
Wa
-J
384(1
/
I
vinax=4f% (L—OJ
V4—a-4..—b- "+ c—4
'-jvl
'Ak—-
-
L— in
MA
MA,b [e(4L-Je)- c 31'4L —Jc)]
MA=-.in (Jnr"—em+o)
B,bLY4t--7°" (4L —saj M8_ jrn2 (4-3m) ,'Mmax
'L,,2( 3si,) 2 /2
RA
When r is the
A4
sinp/e
M-M L
k—x- When x=2(n,3—2m42)
support reaction
M-M
_r3# L
RA =
W(m-2) 8 = Wm(2-m) 2
2ri,
dna.
u When a = c.
When
a=4/2 and x1=O-445L WI! 3JJ(%
.Ey(L3#2Lba #4Lc?—8a) c/C =
WI!
384(1
1087
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1088
Bending moment, shear and deflection B/JILT-/N BEAMS
LA
A
8
L
C
___
M NMB
MA ___________M8 MA V - WL 7/Qx 9x
#Mmax
WL/233 N5Qn
SWL MMWL//6
)
x=OS5L
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
MA = —WI//S M8 =— WL/iO -
RB
R8
RA=RB=W/2
R=O.SW R8=O7W
—4J•22LF—O S6L —Ø22L
WI! max. when x, = 0• 525L
1.4W!!
—
- 384E1
W/2 2W/L
La- b—+_aJ L—4 MA
M8
(SL44aL_402)
HI'a
____C F
I.
MA = M8 -WL//6
M=WL/48
JRB
R8
W/2
A = R3 W/Z
. L/2 —H ¼ O-WL3 364(1
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
B(//LT-/N BEAMS 2W/a
fwTw/a L
____
b L
NMa
MAV
MA =-J'2(Jo/obL)
MA
M3 mAC. Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
Mx =P.X+M3- 2W(x.-bP
InCB.Mx RB.X#M5
AB W/2 ____
A = (/OL—sLa42a)
I /d____ ____ 4
dmax =
2 , prO,,
—
Wa /
-Y HJ W/2
W/a
—a I
b
W
W/2
b
-f
L
I.
1NMB MA Maz_R(4L_3a)
L
MA
.1
MB
MA — _!10L_15#502)
M3 _lOL2(5L4t
= = W/2 =
(/oL-/sLa'#8a)
R = '/5L —ec) dmax =
(/SL—Ma)
1089
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1090
Bending moment, shear and deflection
BUILT—/N BEAMS parabolic total /oao W
JM8
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
NMB
MAV
A -M5=-WL//O
Aq rM8i.._WL/2O
JRa
RA R8 W/2
W/2
'!4
con,c*ment
parabo/p
F4axJ
O•4WL3
1.3 WL3
d,,, 384E!
dma 384 (2'
r
Any symmetrical load W
a rb 2br rA
______
A
symmetrical diagram
NMB MA = M8 —A,JL where A5 is the area of the 'free' bending moment diagram
2 -2br----—-
aab r&_
CM. f(Ja—L)M8c -M(3b-L) When
A RB
A =R8 = W/2 - - I The fic.re shown
A,
is ha/f the bending
* + moment diagram fr- X, -
H-X--lmaxatC
C 1* and +7 LareC.GI
A5x —AIX1 ZET
Where A' is the area of the fixing moment diagram
'4I = P8 = slope of moment diagram
M#M M#M
Eab
When '2/L = in,
M. L2m2(/_m)2(1_2m) 2E%
For ant/clockwise moments reverse the deflect/ons
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
BUILT—/N BEAMS
p
p
I-.
L/2
L/2
b
MA L—'
Al5
- MA = - M5 = M = PL/8
A1_____ _____ Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
MA LZ EEE3 M8 M5 = —
MA=Mc
_______ __________
= = /,,2
4mcx p9 dinax = /92 El
8JJ
L
F.
2Pa2b 1!
A__________ =P (iT) (I#2 *)
z
1Q8 =
I.— X
dC— —
__ Po3b3
d _ZPa2b3 WñQnX= JL-Za "3EI(3L2a)2
p
p
p
p
Pa MA/ \MB
- __Pa(L-q) L
MA=MB=_JPLh6
MA—MB—
MC MD = P02,'L
RAfl
LJ
dmax PL = /30(a)7
AI
MC =MD = I
II
PA=P'P
1091
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1092
Bending moment, shear and deflection
BUILT—/N BEAMS
p
______ fA C P
D '.—L/y —1--L/j —.—L/j.
p
Irg
l,L/3 P.-L/3L2
MB MA LF- 5N M8
MA
MA =M8 = — 2PL/9
MC =MD = PL/p
AL Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
p
MA =M = —/9PL/72 MD =1/
PArJ__LJA
I
I
RA =
S PL3
dmax 548?1
pp + + C D £ BL L/4--L/4 L/4 L/4 II A
p
JP/2
4/PL3
PPpp
1I 34C D E PB L/L/4 +L/4 +L/4 4L14
MAVNMB MAL/L NEiNMB MA=MB =r—SPL/,6 = JPL/15
MA =M3=—I/PL/j2 MD = Mf = SPL/32
2P
— PL3 dmax —
96(1
max. = 96(1
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
BUILT—/N BEAMS
p pp p A CVt L/5 +Lg +L,/s+L4
P
ppppppp
4 (n-i) forces I
I
B
sôcev,naLJ,
MAMALNMB MA M52PL/5
MA=MB=—
P1,'/)
MD wM — PL/5
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AArL
HJ
'A B=2'°
Y1F,ennisoø6 13PL3 dmax.=1000 El
Cimox
Whennis even,
/ / 114-4c (,ji
,I
EfIiiiitiIII •:.:i••
COLUMN
LOAD PER SA4NW
4;.' •O4•4•.
O...o.&. ,.
n CONTINUOUS BEAM
}d L/n#-+-L/n4L/n+-L/-#L1-+-44 L
L. - When n >10, consider the load .nitorrrdy oYstiibuted
-
The load on the outside stringers is carried c'/rect/y by the supports The continuous beam Is assumed to be horizontal at each support The reaction at the supports for each s,oan = W/2. but the maximum
shear force in any span of tM continuous beam = V/J.IAW The value of the fixing moment at each support = — B. WL The value of the maximum positive moment for each span = C. W4 The value of the maximum deflection for each span —0'0O26
Value ofn
A
2 3
02500
4 S
6 7 B 9 10
0•3333
03750 04000 04/67 04286 Q•4375 04444 04500
B
C 0•0 625
00625 0074/ 0078/
00370 00469
Qc//
0•0439
0-0800
0O8/5 00820 00823 00825
0•0400
00408 00 430 004/3 00425
1093
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1094
Bending moment, shear and deflection
PROPPED CAN T/L(VERS
/w A
W
___ a
B
C
C
-f-b
MA
3L/8 =
—WL ---
— 9W!.
C—
MA =—
f (2n)2wñere a/Ln
#Mmax -n (4-n)]
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RA
—--_J P8
P3 =f
1
RAi4J[8n2(4n)] fl)
P3 Hn2(
x/Lm LA'max.
d=j(m —Jm3#2m)
dC(/_I2n#7n2_n)
WL3 dmax.— - ____
/W A
C Web
B
L
MA 'P
MA7tS MA =—
MA=—
AI J2 (o - n 2)
= f (n'-n-pi.e)
a, d=J42/n3(3n?-6b5
When x a.
dj[2p4—p3n (n-6n,'.8)+ _________ pn2(jn2_8n #6)]
BA
.f- c—.J L°'
b
d2-c2)(29c2_c12)
-____ R8=r5— L Where and r3 are the simple support reactions for the beam (MA being considered positive)
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection PROPPED CANTILEVERS
W/2 1'
W/2
CU BA —f— a -H L
MA
if = % then between B and U.
Al,. = ftx4xa(4 -Sm+Zm9] +M,n0x. when x=I'4Jm#2rn2, Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
—20)
MAZ_ —H X/L = m Mx(2Qm'—27m#7) 1027SLf x
A 7WL '''max.—
=0 67L] '9:4
RB
Vx i(9_rn2)
= (2i."#soi. 42)
PB =
A
p8 = (2L2_30L#40') - OO0/WL3
El
When x=0•598L W/?
W/2
-L RB MA MA
3WL
#MmaxO0454 WL [When x 0 283L]
=4W T
dmax
RB = *
p_L A-
- 0•0047WL'
(I
When x=O•447L
/3W
B - 32
—ii When
x=0404L
1095
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1096
Bending moment, shear and deflection PROPPED CANT/LEVERS
a C
_____ L
BA
iR8 ___S___
A
_____
a L
A5
0•577b
I
0/28 Wa MA
MA
JL
_________________
BQt we en
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
CandA, Mx=R5.x-j!('x-b)
M5=R5.x =_ j9(si.2_j,2)
(JaL/5aL#2OL2) #Mmox when xb# fI/-
___
RAt NJR5 = 9 (si!—,/)
A8 =('5L —a) W P5
A8 = 2W
__________ C
____ a
" b—'
R3
L
(t+SaL2) W
2W
____________ 4—a______ .. L b____I
H0577aH 0/28 Wa
f—x — Wab
H042Jb—
MA
When ,n=a/L
2
3m#2 I
MC=AB.b Between AandC
_______________ =RAW52/a2 IRS }.—x--H \
A8 = '('/SL —4a) RA=W-RB
MA=_f/0L2-3b2) RA I
Between C and B \ Vx=RA_Wrc24Z
I
N
A3 = [L(//L -/S,i#(5L- a)] RA=W-RB
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
PROPPED CANTILEVERS
W.iw4,
51 CD
fAA
1.
I
L
Wa2
M5=—2M— F
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
c-'3'L
8TN
.jwap
d0
WL4f2(ep#4?q#dp3(p#i)J
- Omax.
I
—
54(1 M '1 84 'C 0
8+ CD
JA
L
'1.
1.
—-Pa
M3 -2M4 =—M
=
Tfl
q
4I
- 8 =-
— dmax
I0'O
Ido
= j4p2,pq#Jp #Jq) d,n0x
27(1
a#b)#a2(2# VJ
'-dmax"
1097
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1098
Bending moment, shear and deflection
PROPPED CANT/LEVERS
W/2
+ AB
na/L *7/q
Wa
Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
MA _—4(2L_a)
MA=— f-(4L—3a)
W'?Qn X< a,
tl =(9,r?x —/2nx# /2x— 4xg2) +Mmax occurs whQn
NB
A r—\
A -(4L?#ZaL— az)
(4L2 #4aL—JoT
R8 W—
I
W
L kO.4/SLtq MA
32
/
Bj
A
a —+—b —+-—a —4
—L——-——--R8
MA7Z RAI\ I—
MArn (5L2+40L—4G2)
#MmaxOO948 WL
A4fl
w
R,
I '?&
A477(2/1!#4aL 4a2) A8
—
Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
1099
PROPPED CANT/LEVERS
!F051
coIr4oleraQnt of parabola
— SWL
SWL
M .('/Om-ZOm#7m) #My,OO888WL,when X Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
0 •3965L
B—
Mx =(-4om+8om-om4I7m) #Mmax*O0399WL,whQn x —
02343L
A6
7W
MA
MAW-' lz' ,nax.MA-O./93PL wMn baO577L
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Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
1100
Bending moment, shear and deflection
PROPPED CANT/LEVERS
Pp
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ACDE ¼+V4+L/4 +-/4
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MA MA = —
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Steel Designers Manual (6th Edition) - Bending moment, shear and deflection tables Discuss me ...
Bending moment, shear and deflection
PROPPED CANILEVEPS
pppp
P PPPP Pp
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MA=— Ad
forces
B
MAE7ZTJ PL(,,?l)
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Created on 24 July 2007 This material is copyright - all rights reserved. Use of this document is subject to the terms and conditions of the Steelbiz Licence Agreement
rT(— 5/2
P8
RAEL (5n2-4n-i)
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GO•OZZl
!or9e, dmax.
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Any symmetrical load W
JA
AreaR L AreoS'B MA
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a=L
® _r—
a>O423L
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M-
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P82 =!— L
M1 9(2_6n#sn') MCA (2— 6n# Pn?3ni -Sn
It'B dmax occurs at point corresponding
to Xon M diagram, the area A being equal to the area 0 Vmax = Area SXx
L In Case!, R — SM/ZL
CaseS, A= M/L
1101
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