BEAM DIAGRAMS AND FORMULAS
BEAM DIAGRAMS AND FORMULAS
3-2 13
Table 3-23
Shears, Moments and Deflections
1. SIMPLE BEAM- UNIFORMLY DISTRIBUTED LOAD
Total Equiv. Unlform Load ........................... = wl
R ~ V.............................................................. ='w2l
=w(i-xJ V, .............................................................
=a M,., (81 CMte~ ............................................... w12
.............................................................. ='!!f-Q - N) (at oente~ ............................................... ~ swr~'I
384 .............................................................. =~3 -21x".x")
2. SIMPLE BEAM- LOAD INCREASING UNIFORMLY TO ONE END
Tolal Equiv. Uniform load ........................... ~ '6 ~ . 1.0Jw
9v3
=T R,= v,.............................................................
R,- V,a v_
.................................................. .
2W
3
V, ............................................................. =~ - wx' 3 12
(at x- ~ = 0.5571) ............................. = !j}? 0.128 Wr
M, .............................................................. = !!!=..(/ - x")
a12
X= IJ1-Jfi w:, (at
s0.5191) .................... ? 0.0130
.............................................................. =, w; ~x' - 1orx" ..1r') 80 112
3. SIMPLE BEAM- LOAD INCREASING UNIFORMLY TO CENTER
Total Equiv. Uniform Load ........................... - 34W
R = V .............................................................. =~
v.
2 e-4 x") (when X a and< (a+l>)) .... = R, - w(x - a)
k;;;,::;'r'-'"W-DII:d(2 M,.. (atx~ a?~) ????????????? ??? ? =R, (a?;:)
M,
(when X < a) .................?....... = R,x
(whenx> aand b) ..............................
M,..., (at point of load) ....................................... = P~
M, (when X< a) . ............................................. =~X
(atx-r A,...
8; 2o)_wllena>o) .................. =Pa~(a+2:;~
(at point of load) ................................... .... = Pa?'lo'
3 11
- x') (when X< a) ............................................ = :~Q2 - o2
9. SIMPLE BEAM- TWO EQUAL CONCENTRATE D LOADS SYMMETRICALLY PLACED
Total Equiv. Uniform Load ................................ = s~a
R:V ................................................................... = P M,_ (between loads) ....................................... = Pa
M, (when X < a) .............................................. = Px
2 1 a2) t...... (at center) .................................................. = ~ (312 -4
A,..
(when a=
I
3
)...........................................
=.2E8!E_l
(when X < a) ..
= Px(3ta- 3a? - x2) SEI
(when a< K< (/-a)) ............................... "';; (stx - 3x' - a2)
1
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
3-216
DESIGN OF FLEXURAL MEMBERS
Table 3-23 {continued)
Shears, Moments and Deflections
10. SIMPLE BEAM- TWO EQUAL CONCENTRATED LOADS UNSYMMETRICALLY PLACED
R,= V, ( = V.,.. when a< b) .................................... = !j-Q - a.b)
v, R,= (= v.... when a> b) .................................... =!f-(! - b+a)
V, (when a< x< ( 1- b )) .................................... = -'j-(o - s) M, ( = M,.., when a> b) ......... ............................ = R,a M, (=M,_whena M, (when X < a) ................................................... = R,x M, (when a< X < ( 1-b )) .................................... = R,x- P(x - s)
11. SIMPLE BEAM- TWO UNEQUAL CONCENTRATED LOADS UNSYMMETRICALLY PLACED
R,=
V, .......................................................................
= P1 (1-a)+P2b
1
R,? v, ...
V, (when a< x< ( 1-b)) ................................... = R, - P,
M, ( = M.,.,when R, < P,) .................................. = R,a
M, ( = M...,when R, < P,) .................................. = R,t>
M, (when x ~ ) .................................. ? P(~- 1: )
II""" (at x = ..!..... o.447/) ........................ = ...!f_. 0.00932 PP
J5
48E/J5
El
= (at po.ont of load) ..............................
7P13 E/
768
96 1 (at x< ~ )......................... .......... ? ~ ~r2 -sx"-)
(at X> ~ )............................. .......... = : E1(x- rf (t t x - 21)
14. BEAM FIXED AT ONE END, SUPPORTED AT THE OTHER- CONCENTRATED LOAD AT AIIIY POINT
R,= v,.............. ......................................... = ptJ2 (a+21) 2P
R, = V, ................. ..................................... = ;~ ~12 _ ,.2 )
M, (at point or load) ............................. = R1s
M, (atfixedend) ................................... . Pab(a+ l)
2r2
R,
R, M. (at x< a) .......................................... = R,x
M. (when x >a) . ................................... = R, x - P(x -a)
[v.~~en a a) .................................... =
(1-xf ~~? x-a2 x-2a?1)
12
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
3-218
DESIGN OF FLEXURAL MEMBERS
Table 3-23 (continued)
Shears, Moments and Deflections
15. BEAM FIXED AT BOTH ENDS - UNIFORMLY DISTRIBUTED LOADS Total Equlv. Uniform Load ..................................... = 2;1
R=V ........................................................................ =?
R V, ........................................................................ =w(~- x)
M,.,..
(at ends) ..........................?..............................
=
w/2
12
24" M, (,at center) ....................................................... = w12
M, ........................................................................ =* ~lx -12 -sx2)
A.,.. (,at center) ...................................................... = w/4El 384
T A,
........................................................................ = ~?;1(1-xf
16. B EAM FIXED AT BOTH ENDS- CONCENTRATED LOAD AT CENTER
Tolal Equlv. Uniform load ..................................... = P
R R= V........................................................................ = ~
R M,... (at center and ends) ...................................... = ~
M, (whenx
8) ...................................................
=
P(!- xf ---;a-(lb-
l?x)
22. CANTILEVERED BEAM- CONCENTRATED LOAD AT FRIEE END
Total Equiv. Unifonn Load ..................................... = 8P
R R= V ...........
. ........................ :p
M_, (at fixed end) ................................................. = PI
........................................................................ = Px
t...., (at free end) .
. .......... .. . -w - p(J
6 1 t., ........................................................................ = ~ ~13 -312x?> ................
................
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