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DEFLECTIONS AND
SLOPES OF BEAMS
Table G-1 Deflections and slopes of cantilever beams
IV Fi„
V =^ deflection in the y direction (positive upward)
v' =^ dv/dx —^ slope of the deflection curve
|a_^^^£ ^ 8b^ — —^ v{L)^ —^ deflection^ at^ end^ B^ of^ the^ beam^ (downward) 0g = —^ v'(L)^ =^ angle^ of^ rotation^ at^ end^ B^ of^ the^ beam^ (clockwise)
L
r \ (^) EI = (^) constant
1
1
" =^ -^^^' -ALx + x^) v' =^ -^(3Z.2 -^ 3Lr + x^) 1 M 1 1 i i 24EI 6EI 1 ql' qO ^ (^) SEI ^ 6EI
R qx^ v=-y—{6a--4ax + x-) (0 <^ jc <^ a)
v' =^ --^(3fl' -^ 3fl.r + .v^) (^) (0 <^ X <^ a) 6EI
j 1 1 1 1 1
qa? qa^
qa"* (^) , qa^
At X =^ a: V^ —^ v^ =
%EI 6EI
qa^ qii' 8a^ =^ ——(4L -a) d„ = -— 24EI^ '^ *^ 6£/ (Continued)
qbx^ " =^ ~T;I7(3L litI +^ 3a-2x) (0 <^ ;c <^ a)
qbx v' = (^) -TT^a + a-x) (0 < (^) .X < (^) a) 2EI
V =^ -^rj—ix^ -^ ALx^ + 6L-X- -^ Aah + a^) (^) {a <^ x <^ L)
v' =^ --r^Ax' -^ 3Lx2 + ?,Lh -a') {a<x< (^) L) 6EI qa-b qabL Atx =^ a: v=———(3L + a) (^) v' = --^ \2EI (^) 2EI
V =^ -^{3L^ Px^ -^ X) v' 6EI
Px
5« =^
PL^
3EI
PL"
2EI
V = -^(3a -X) (^) v' = (^) -~:(2a - (^) x) (0 < (^) x < (^) a) bEl (^) 2E
-a —A^b- (^) V = PaK^ (^) , , Pa" (^) , -TTryOx oEI -a)^ v'^ =^ --—2EI^ {a^ <^ x^ <^ L)
Atx =^ a: V =^ Pa^^ v ,^ = Pa- 3EI (^) 2EI
_ Pa-^ Pa^
2EI
2EI
MqX
EI
EI
(Continued)
Table G-2 Deflections and slopes of simple beams
'
'\ /"•^ B
/
EI = constant
V =^ deflection in the y direction (positive upward)
v' =^ dv/dx =^ slope of the deflection curve
Sc =^ -v{LI2)^ =^ deflection^ at^ midpoint^ C^ of^ the^ beam^ (downward)
x^ =^ distance from support A to point of maximum deflection
^max ~^ ^I'max ~^ maximum^ deflection^ (downward)
6^ = —^ v'(0)^ =^ angle^ of^ rotation^ at^ left-hand^ end^ of^ the^ beam
(clockwise)
dg =^ v'{L) =^ angle of rotation at right-hand end of the beam (coun-
terclockwise)
TTTTTTT ^ (^) iSSSr
V =^ ^(L3 -^ 2Lr2 + x^) 24 EI
v' =^ —{L^ -^ 6Lx- + 4x^) 24EI 5qV (^) _ qO
TTT
-^^(9L? -^ 24Lx~ + 16;r-') (^) ( <^ a: <^ - 384£/ (^) V 2
v' =^ ^—(9L3 -^ 12Lx- + 64x-^) (^) ( <^ x <^ - 384£/ I 2
V =^ (8;r3 -^ 24Lx- + 17Z,lv -^ L^) ( - <^ .v <^ L 384£/ \
v' =^ 384£:/ {24x--4%Lx+\lL-)^ (-<jc<L
5qL^ _3qlJ (^) _ IqO 768£/ 128£/ 384£/
TT a:
qx 24LEI
24LEI
qa
(a* -^ 4a^L + 4a'^L- + 2a-x'^ -^ 4a Lx' + Lx^) (^) {Q <^ x <^ a)
(a-* -^ 4a^L + 4a^L- + baV~ -^ \2aLx- + 4Lx^) (0 <^ Jt <^ a)
24L£/
qa-^ •, 24LEI
i-a^L + 4Lh + ah -^ 6Lx^ + 2x^) (a <^ x <^ L)
(4L- + a^ -^ MLx + 6x^) {a<x<L)
e, =^ -^^(2L -^ aY Ob =^ -;^—(20^ -^ a'^) 24LEI 24LEI (Continued)
APPENDIX G DEFLECTIONS AND SLOPES OF (^) BEAMS 885
J^
^-7^^7-^
Px 48EI
8c =^ 8„
(3L- -^ Ax-)
PO
16EI
(L^ -^ 4x-) 0<x<-
48£'/
PL'
\6EI
a:
*—^ a^ ><—^ b^ —*j
Pbx 6LEI
Pab{L + b)
{C- -b'-- x'-)^ Pb
6LEI
(L- -b- -^ 3,x-) (0 <^ ;c <^ a)
Pab{L + a)
lia>b, 8c
lfa>b, x.
6LEI "^ 6LEI
PbOV- -^ 4fc2) 4SEI
lia<b, 8c
PaOL- -^ Aa-)
4SEI
L'-b'- and 8„^ PbjL'^ -^ b'-f
9\3LEI
r i'
..A
V =^ -—(3aL -^ 3a' -^ x') tEI 2EI
{aL -^ a"- -^ x') (0 <^ jc <^ a)
Pn Pn V =^ --!-^(3Lx -^ 3x' -^ a') v' =^ -(L -^ 2x) {a<x<L- a) 6EI 2EI Pn c m« (^) 24EI
Pa{L -^ a)
lEI
Mn
r>
SB
V =^ —rrzr{2L' -^ 3Lx + x')
M^
8c =
6LEI
16EI
X, =L\
f)
3EI
and 6„„ =
V — 6LEI
* 6EI
M,L'
(2L- -^ 6Lx + 3x')
9V3EI
Mn
A ^
I. L^.
L
L J
24LEI
(L' -^ 4x')
5c =^ e^^ =^ 24EI
Mc. 24LEI MqL 24EI
{V-- \2x') 0<x<|
(Continued)
