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,
V..............................................................
.............................................................
M,., (81 CMte~ ...............................................
wl
='2
=w(i-xJ
w12
=a
.............................................................. ='!!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
R, =
v,.............................................................
R,- V,a
V,
=T
v_ .................................................. .
.............................................................
2W
3
=~ - wx'
3
12
(at x- ~ = 0.5571) ............................. = !j}• 0.128 Wr
M,
.............................................................. = !!!=..(/ a12
x")
X= IJ1-Jfi s0.5191) .................... • 0.0130 w
:,
.............................................................. =, w; ~x' - 1orx" .. 1r')
(at
80 112
3. SIMPLE BEAM- LOAD INCREASING UNIFORMLY TO CENTER
Total Equiv. Uniform Load
........................... - 34W
R = V .............................................................. =~
v.
2
(when X<~) ....... .................................. = ~
M""" (at oenter)...............................................
M,
e-4x")
wr
=s
(when x<~ ) ......................................... =
wx(~- ~~ )
(at center) ............................................... = wP
~
60 1
(when
X< ~) ..........................................
2
•
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
'+a'Jw;/12 ~12 - 4x")
DESIGN OF FLEXURAL MEMBERS
3-214
Table 3-23 {continued)
Shears, Moments and Deflections
4. SIMPLE BEAM- UNIFORM LOAD PARTIALLY DISTRIBUTED
R, = V, (max. when
R2 R,=
l
v,
(max. when
8
<c) ................
8>
= T,C2c+ o)
c) ............... =¥,(2a+O)
(when x> a and< (a+l>)) .... = R, - w(x - a)
v.
(atx~ a•~) ············· ··· ·
k;;;,::;'r'-'"W-DII:d(2 M,..
M,
=R, (a•;:)
(when X < a) .................•....... = R,x
(whenx> aand <(8+b)) .... =R,x- ~(x -a'f
(when X> (8+b)) ................... =
~
(1- x)
5. SIMPLE BEAM- UNIFORM LOAD PARTIALLY DISTRIBUTED AT ONE END
R,= V,=
v.,... .......................................
R,= v,..................................................
R, V,
= !fTC21-a)
waZ
=2~
(when X< a) .............................. = R, - wx
R,2
................................
T V M,.. (at X= ~)
= 2w
lfti:r:o;Jo:a:r:j l 2
Shear
Mx (W hen X< a ) """""""""""""""" """"" = R,x- w.l
2
6. SIMPLE BEAM -
M,
(when X> a) .............................. = R2(I - x)
a,.
(when X< 8) .............................. = :; ~ (21-af - 2ax2 (21 - a)+ix3)
6x
(when X> a) .............................. =
2 11
wa:4~~x)0xl-2x2 - a2 )
UNIFORM LOAD PARTIALLY DISTRIBUTED AT EACH END
w1a (21 - a)+ w2c 2
2/
w2c(21-c)+w,a2
2/
(when x <. a) ......................... = R, - w,x
(when a < X< (a+b)) ............
(when
=R, - w1a
X > (8+b)) .................. = ~ -w2 (1-x)
R,
( al X • ;;·when R 1 < w1s
( atx=l-!'t.when
w2
J .... =!!{_
~< w2c)
2w,
:!
2
=
2
w, .l
(when X< a) ......................... = R1x - - 2
(When a< X< (a+b)) ............
M,
8
= R,x- "'' (2x-a)
2
w2(i- xf
2
(when x>(B+b)) .................. "R2 (t- x) - - --
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
BEAM DIAGRAMS AND FORMULAS
3-215
Table 3-23 (continued)
Shears, Moments and Deflections
7. SIMPLE BEAM- CONCENTRATED LOAD AT CENTER
Total Equiv. Unaorm Load ............................ .... = 2 P
R=V .................................................................. =~
:fl.
M,..... (at point of load) ..................................
M,
4
(when X< I
2 ) ...........................................
= Px
2
t..... (at point of load) ....................................... = ;;:,
t.,
(when x <
~ ) ............................................. = 4=~~~12 - 4 x2)
8. SIMPLE BEAM- CONCENTRATED LOAD AT ANY POINT
Total Equiv. unaorm Load
8Pab
... aT
R ,• V, ( • V.,..when a< b) .............................. •
R,=
Ef-
v, ( = v..... when a> b) .............................. =o/-
M,..., (at point of load) ....................................... = P~
M,
A,...
(when X< a) . .............................................
=~X
(atx-r
=Pa~(a+ 2:;~
8 2
; o)_wllena>o) ..................
(at point of load) ................................... ....
= Pa'lo'
£
3 11
(when X< a) ............................................ = :~Q2
- o2 - x')
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) ..............................................
t...... (at center) ..................................................
A,.. (when a= I
3
(when
X<
= Px
=
2~1 (31
2
-4 a2)
)........................................... =.E!_
El
a) ..
28
= Px(3ta- 3a? - x2)
SEI
(when a< K< (/-a)) ............................... "';; (stx - 3x' - a2)
1
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
DESIGN OF FLEXURAL MEMBERS
3-216
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)
R,=
v, (= 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<b) ......... ............................ =A,I>
M,
(when
M,
(when a< X < ( 1-b )) .................................... = R,x- P(x - s)
X<
a) ................................................... = R,x
11. SIMPLE BEAM- TWO UNEQUAL CONCENTRATED LOADS UNSYMMETRICALLY PLACED
R,= V, ....................................................................... =
P1 (1-a)+P2 b
1
R,• v, ...
V,
(when a< x< ( 1-b)) ................................... = R, - P,
M,
(
M,
( = M...,when R, < P,) .................................. = R,t>
M,
(when
M,
(when a< X< ( 1- b )) ................................... = R,x-P,(x - a)
= M.,.,when R, <
P,) ..................................
= R,a
x <a) ................................................... •
R1 x
12. BEAM FIXED AT ONE END, SUPPORTED AT OTHER- UNIFORMLY DISTRUSTED LOAD
Total Equiv. U niform Load .................................... = wl
R,= V,
.................................................................. = 3:1
R7 = V: = Vhllllt ····························~·········
=Swl
8
v,
M,
(at X =
3
a')
............
.................................................................. =R1 x(at
T
X=fs~
+
w.-2
2
./33} 0.422 I) ..................... =,::EI
................................................................. = ;~ Q• -3tx2 .2x")
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
4 1
3-217
BEAM DIAGRAMS AND FORMULAS
Table 3-23 (continued)
Shears, Moments and Deflections
13. BEAM FIXED AT ONE END, SUPPORTED AT OTHER- CONCENTRATED LOAD AT CENTER
Total Equiv. unnorm Load ...................... ~~
2
R,
R,
R,·
v,·····················~··································
R, =
v, = V""" ........................................... = \ 1:
•
~=
=~~~
M..., (at fixed end) ...................................
M,
(at point of load) ............
M,
(at x < ~ ) ...... ................................... = 5 ';;
M,
(when X >
1
~ ) ..................................
•
P(~- \1:
)
II""" (at x = ..!..... o.447/) ........................ = ...!f_. 0.00932
J5
48E/J5
.
(at poont of load) ..............................
PP
El
7P13
=768 E/
(at x< ~ ) ......................... .......... •
96~1 ~r2 -sx"-)
(at X> ~ ) ............................. .......... = : E (x- rf (t t x - 21)
1
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 )
R,
M,
(at point or load) ............................. = R1 s
M,
(atfixedend) ................................... . Pab(a+ l)
M.
(at x< a) .......................................... = R,x
M.
(when
2r2
R,
x >a) . ...................................
v.~~en a<0.41-41 at h
[
=
R, x - P(x -a)
2
.a2) J.. " ~ ~ -if)'
0r2 -a2) 3EI 0r2 t~j
12
i(
2
Pab
) ..... •
( v.l\ena>0.4141 at x=l,g
, 21+8
Ll,
T
(at point of load) .............................
,g
6EI ,21+8
=
Pa2 b3
12
EtP (31 +a)
(when X <8) . ................................... = Ptlx
•2Eir
(when x > a) .................................... =
:/J
12
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
~at" -2Jx2 -ax2)
(1-xf ~~· x-a2 x-2a?1)
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) ..........................•.............................. =
2
w1
= 24"
M,
(,at center) .......................................................
M,
........................................................................ = * ~lx -12 -sx2)
A.,.. (,at center) ...................................................... =
T
w/2
12
A,
........................................................................
w/4
El
384
= ~·;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<i) .................. .............................. =~(4<-/)
(at center) ....................................................... =
P/3
l~EI
(when x< ~) ................................................. = :;; (3t-4K)
1
17. BEAM FIXED AT BOTH ENDS- CONCENTRATED LOAD AT ANY POINT
h 'Tl
R ..
x --j
p
R,= v,(
= v_
R,= V, (=
..". R, M,
wh&n a< b) ....................................
82
v.... whena>b) .................................... =P
3
( :M"""whenil<b) .....................................
t 'l'·
M,
T
p~
=-;3"(3a•
o)
(a•3b)
:~81:2
M,
(• M,_when a> b) ..................................... • Pa b
M.
.
2~~
(at pomt ol load) ............................................. = 13
M,
(when
/2
x< a)
................................................. =
Amon (when a> b al x
Pab2
R,x- T
3
= ..1.!!L) ........................... = 2 Pa t?
3a+b
3Et(3a+of
Ax
.
(at pornt of load) ................................... ......... •
!J.,
PIJ2,(l
(when X< a) ................................................. = 6"£ir3(3a/- 3ax - ox)
~~
Ell'
3
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
BEAM DIAGRAMS AND FORMULAS
3-219
Table 3-23 (continued)
Shears, Moments and Deflections
18. CANTILEVERED BEAM -
LOAD INCREASING UNIFORMLY TO FIXED END
Total Equiv. Uniform Load ....................................... =
R
3w
R=V ...............................................................•......... =W
v,
......................................................................... ;
wx"
,z
M,_ (at fixed end) .................................................. = ~
................ ................ .. ...................... -
Wl<3
(at free end) ................................................... =
~~~
~3;2
......................................................................... =
;
00 112
(!' - st' x +4 t 5)
19. CANTILEVERED BEAM- UNIFORMLY DISTRIBUTED LOAD
Total Equiv. Uniform Load ..................................... = 4wt
R
= V.................................................. ..................... = wl
Vx
........................................................................
~ WX
M,_ (at fixed end) .................................................. =
wt2
2
........................................................................ = .,.2
2
w/4
a,..,
(at free end) ................................................... ; SEt
.1-x
........................................................................
= 24wEI~4 -4/3 x +- 3t4)
20. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATE AT
OTHER - UNIFORMLY DISTRIBUTED LOAD
Total Equlv. Uniform Load ..................................... =
3wl
R= V ........................................................................ =wt
V..:
...................... ......................................... ......... =w;lf
M,
(at deflected end) ............................................ -- w/2
6
- w/2
M.,.. (at fixed end) .................................................. -
M,
3
........................................................................ =~Q' - 3x")
!J..,.. (at deflected end) .......................................... =
........................................................................ =
A MERICAN INS11TUTE OF STEEL CONSTRUcnON
w/ 4
Et
24
~224fl
-·2)
3-220
DESIGN OF FLEXURAL MEMBERS
Table 3-23 {continued)
Shears, Moments and Deflections
21 . CANTILEVERED BEAM- CONCENTRATED LOAD AT ANY POINT
flr'
a
R
b
Total Equlv. Unlfonn load ........•............................
= 8~
R =V ........................................................................
:p
M- (at fixed end) .................................................. = PC
M,
(when x > 8) ...................... .............................
= P(x - e)
- Pb.
.
(at free end) ................................................... - 6E/(~/ - b)
(at point of load) ............................................. =
Pb3
W
PC2
(when)(< 8) ...................................................
= m(3/- 3x- b)
(when x> 8) ...................................................
P(!- xf
= ---;a-(lb- l•x)
22. CANTILEVERED BEAM- CONCENTRATED LOAD AT FRIEE END
Total Equiv. Unifonn Load .....................................
R
R= V ...........
= 8P
. ........................ : p
M_, (at fixed end) ................................................. = PI
........................................................................ = Px
- p(J
. .......... .. . -w
t....,
(at free end) .
t.,
........................................................................ = ~ ~13 -312 x•><')
61
23. BEAM FIXED AT ONE END, FREE TO DEFLECT VERTICALLY BUT NOT ROTATE AT OTHERCONCENTRATED LOAD AT DEFLECTED END
Total Equiv. Unifonn Load .....................................
= 4P
R=V
.................................................................. = P
l
M...,
(at both ends) ............. .............................
1
I
M,
....................... . .................. =
=~
v
t.....
I
M_ t.,
pli-·)
P/3
(at deflected end) .....................................
= 12 El
..................................................................
- ~(!+2 x)
!
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
_ P(!-xf
3-221
BEAM DIAGRAMS AND FORMULAS
Table 3-23 (continued)
Shears, Moments and Deflections
24. BEAM OVERHANGING ONE SUPPORT- UNIFORMLY DISTRIBUTED LOAD
..... = fi~:z -a1)
R, = V, .................
R~=
=Ti(1•4
V,+V3 .••• . •• .. ... ••
v,
v,
.........,...........
VK
(between supports) ..........
v••
(forovemang) ..................
M,
.............. =.!!..(l+a'/(1-a'/
(alx=l[•-L])
2
12
812
M,
(atR,) ............................... -
M,
(between supports) .......... = ~e
v,
- - --·· ··· - --~ · -···-·
= wa
...................................•....... =
Ti(!2 + a2)
= R1 - wx
= w(a - x1 )
- .....
(for ovemang) ..................
2
= !fC•-x,f
(between supports) ......•... = ; ;
2 11
" ••
.... ~
(lof ovamang)
-a2 -xi)
(!' - 212 x> +lx3 - 2a2fl •2a2l)
wx, ( 2
3
,}
2
3)
E \.4s J- 1 +6 x1 - 4ax1 +x1
24 1
NOTE: For a negative value of ' •· deflecnon is upward.
25. BEAM OVERHANGING O NE SUPPORT- UNIFORMLY DISTRIBUTED LOAD ONI OVERHANG
R, -- V, .......................................... -R~ = V,+V~ ..........
wal
21
................. = T,C21••)
V,
........................................... = wa
V,,
(for overhang) ..................
M,.., (at R,) ...............................
= w~
- wtJ1
··---v
M,
(between supports)
M,,
(for overhang) ........ .•.......
A.... (for overhang at
= w(a- x 1 )
X
= ~(•- xd
3
x,= a) .... = w
a
(41•3•}
24EI
2
4,
(between supports) ...... ... = waE xt
\;z
12 11
A,,
(toroverhang) ..................
_,2)
= ;; 0e21•6a2 x1 -4ax~··:)
1
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
3-222
DES IGN OF FLEXURAL MEMBERS
Table 3-23 {continued)
Shears, Moments and Deflections
26. BEAM OVERHANGING ONE SUPPORT -
R,=
CONCENTRATED LOAD AT END OF OVERHANG
V, .............................................
I
................. = T(!+a)
R, = V,• V, ....
v,
:E!!.
............................................................. = p
M..., (at
R,l .................................................... = Pa
M,
(between supportS) ............................. = P~x
M,,
(for ovemang) ......................................
......
~
( be1ween supports at
=P(a- x1)
x ~ J3 ................ -_
(for overhang at x, =
I )
Pa1 2
J3El =0.0642 Pal'
Ti
9
a)........................ = ~~ (!••)
(between supportS) ............................. = :~
~~.
(for overtlang) ......................................
e- X' )
Px1 (.
2)
=SEI
~a/+3ax, - x,
27. BEAM OVERIHANGING ONE SUPPORT- UNIFORMLY DISTRIBUTED LOAD BETWEEN SUPPORTS
Total Equiv. Uniform Load ........................... = wl
R =V ................................ ......................... ... =
v~
..............................................................
!Yf
=w(t-·)
M,_ (at center) ............................................. =
w/2
8
M,
.............................................................. =!!!f-(1- x)
~.....
(at center) ............................................. -
_ Swl•
Et
384
.............................................................. = ~~,e -21, ·xl)
2
_ wt3 x 1
.............................................................. - 24E/
28. BEAM OVERHANGING ONE SUPPORT- CONCENTRATED LOAD AT ANY POINT BETWEEN SUPPORTS
Total Equlv. Un~orm Load ...........................
,.
= !!f!!!.
R, = v, (= V,...when a< b) .......................... =
T
R 1 = v, (= v...,.when
T
a> b) ......................... =
M'""' (at point of load) ..................................
M,
= P~b
(when x< a) ......................................... = P~x
/•<••2b) wltena>bJ ·····
( atx=, - 3-
= Pab(a+2b)~
27E/I
~.
(at point of loac) ...........................
2 2
= Pa b
3EII
~.
(when x < a) .........................................
= :~~2 -b' -X')
(when x> a) ........................................
= Pa (~~,X)~lx-x2 - a2 )
6
.......................... = Pabx; (l•s)
6EII
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
3-223
BEAM DIAGRAMS AND FORMULAS
Table 3-23 (continued)
Shears, Moments and Deflections
29. CONTINUOUS BEAM -TWO EQUAL SPANS - UNIFORM LOAO ON ONE SPAN
Tolal Equiv. Uniform Load ................. = ~wl
R, =V,...................................................
= {swl
R, = V, +V, ........................................... = ~wl
R, = V, ................................
= _ _!_wl
16
v,
= l..wJ
16
M,.,., (at x= ~l ) .............................. = ;,~w/2
=
iiwP
M,
(at support R,) ...........................
M,
(when x < f) ............................... = *(71-Bx)
6 - (at 0.472 /from R ,).................... =
0.009~wt'
30. CONTINUOUS BEAM-· TWO EQUAL SPANS- CONCENTRATED LOAD AT CENTER OF ONE
SPAN
Total Equiv. Uniform Load ................. = ~ P
R=V.................................................... = ~~P
R=V+V ........................................... : 1l p
16
R =V ................................. ................. =-332p
v.
.................................................... = 19 p
32
M,.,., (at point of load) ....................... = ~ PI
M
(at s upport R,) .......................... = fiPI
.::.""' (at 0.480 I from R,) ....
= 0015PI,
E:l
31. CONTINUOUS BEAM - TWO EQUAL SPANS - CONCENTRATED LOAD AT ANY POINT
R, = V, .......................... ............... = :~
R,
v,++:11 ~Mli~~Jrn~rrrr~lTv,
01
2
-a(!+a))
R, = V, +V, ............................................
-
= .!:.!.~12 + b(!+a))
R,= V, ...................................................
= _Psb(l+a)
V,
=: ; (4P+b(l+a))
....................................................
213 ~
413
M.,.. (atpolntof load) ........................ = Paor.12-a(!•a))
3
41 '
M,
(atsupport R ,) .......................... = ~(!•a)
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
412
3-224
DESIGN OF FLEXURAL MEMBERS
Table 3-23 {continued)
Shears, Moments and Deflections
32. BEAM- UNIFORMLY DISTRIBUTED LOAD AND VARIABLE END MOMENTS
.................................................... ... = T{l - x)•(M-' ~-M2
)x-M,
/2 - (-M1 •w- M2 ) • ( ~
M1 - M, )
b (to locate lnfiecUon points) ................... -_ 4
.2 + 12
Mt x<t/ 3 - 8- M11 - -4M2/]
tJ., -_ - wx [ x 3 - ( 21-t -4M1 - -4M2 ) ;r
24£1
wlwl
w
w
w
33. BEAM- CONCENTRATED LOAD AT CENTER AND VARIABLE END MOMENTS
-
R,-V1
w . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. . . . . . . . . . . . .. . .. . . . .. . .
-
-_p2 -t- -M, -M,
1
_p
R,-V, . ......................................... ............... -
2
-M,--M,
1
: r:!._M1 +M2
M,
(at center) ......................... .
M,
(when x <
~ ) ................................ =(~·M,~ M2 )x- M1
M,
(when x >
~ ) ................................ = ~{1- x)+ (M, - 1M2 )x -M,
tJ.,
(when x < -I )= - Px ( 31 2 2
48EI
4
,
4x-
2
x) (21 - x)•M,(I•x)J)
-8{1- -[M,
PI
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
2
3-225
BEAM DIAGRAMS AND FORM U LAS
Table 3-23 (continued)
Shears, Moments and Deflections
34. SIMPLE BEAM- LOAD INCREASING UNIFORMLY FROM CENTER
Total Equiv. Unifonn l oad ............ = 2 ~
~v
_w
............................................... -2
V,
{when x<
-
f)........................ = ~('-,2 •)
2
Mmu (at center) .............................. =
M,
(whenx<
f1
fJ ........................ =~(x- 2 f ·:;)
• 3W/3
El
t;..,.. (at center) .............................. -
t.,
(when x<
320
fJ ........................ = 1~EI(x•-~ ·~;: _31:•)
35. SIMPLE BEAM- CONCENTRATED MOMENT AT END
Total Equiv. Unifonn Load
=8M
R=V ...............................................
=7
R
I
M...,. ............................................... =M
M.
............................................... =
t..,.. (at x= 0.423
A,
M(1-f)
2
/) ...................... =0.0042 "::,
............................................... =
6~1 (3x2 - ~ - 2/x)
36. SIMPLE BEAM- CONCENTRATED MOMENT AT ANY POINT
Total EQulv. Uniform Load
R=V ...............................................
=7
M,
(when x < a) ..........................
=Rx
M,
(when x> a) .......................... =R (I- x)
t.,
(when x< a) ..........................
fl,
{when x >a) ......................... =
=
6~,[(sa- ~ -2}- ~ ]
3
6~1 [3(a'
A M ERICAN INSTITIITE OF STEEL CONSTRUCTION
+
x> )- ~ -( 21+ 3 ~ ) • ]
3-226
DESIGN OF FLEXURAL MEMBERS
Table 3-23 (continued)
Shears, Moments and Deflections
37. CONTINUOUS BEAM - THREE EQUAL SPANS- ONE END SPAN UNLOADED
wl
wl
B~
R, • O.Ja3w/
oj
c~
R,• 1.20wl
'
R. • 0.450wl
R, = .()_(}33Qwl
<~- {0.4301/rom A) • O.QOS9 wl'fEI
38. CONTINUOUS BEAM -THREE EQUAL SPANS- END SPANS LOADED
wl
w/
j
j
c,l,
ol
A
B
~·~------------~·~··---------------.-~
- -~------------~·'
R,=0.450wl
R, = 0.550wl
R, = 0.550wl
R, = 0.45Qw/
0.4SOwl
Sh<Jar
M~l
<~- (0. 4791/rom
A or D)
=0.0099 wi'IEI
39. CONTINUOUS BEAM -THREE EQUAL SPANS- ALL SPANS LOADED
wl
w/
w/
Al.~--------------s~j~,------~----~c~l
R. = 0.400wl
R, = 1.10wl
R. = 1.10wl
0.400 w/
<~- (0.4461 from A or D)
= 0.0069 wl'lr!J
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
3-227
BEAM DIAGRAMS AND FORMULAS
Table 3-23 (continued)
Shears, Moments and Deflections
40. CONTINUOUS BEAM- FOUR EQUAL SPANS- THIRD SPAN UNLOADED
wl
w/
w/
A~~--~------~e~L----~----~c-'~--~~--~o~l----~------•~!
R.= f.22wl
R,.• 0.3ao.vl
Ro= 0.357wl
it" O..U2wl
R.:O.MBwf
d-(0..4151/romE)=-0.0()g4 wi''IEI
41. CONT INUOUS BEAM -
FOUR EQUAL SPANS -
LOAD FIRT AND THIRD SPANS
w/
wl
Al~----~--~a•!~--~----~c~!----~----~o~~----~----~
·1
R~• 0.446wl
~·0.464wl
R, •0.$7lwl
R~•O.S1~1
R, • 1.0540'WI
.4- (D.4n 1from A) • 0.0097wi•IEt
42. CONT INUOUS BEAM -
FOUR EQUAL SPANS - ALL SPANS LOADED
w/
w/
wl
wl
A~~--~--~8~~----~--~c~
~ --~----0~~----~--~
·1
R . .. 0.3i3w/
R, : O.f. J.twl
~ · O.I28wl
4- (0.«0 I from A f"l'ltJ e) •
Ro'" 1. 1-lwl
0.~ wltEI
AMERICAN INSTITUTE OF STEEL CONSTRUCTION
R,• 0.393wl
DESIGN OF FLEXURAL MEMBERS
3-228
Table 3-23 (continued)
Shears, Moments and Deflections
43. SIMPLE BEAM -ONE CONCENTRATED MOVING LOAD
··~··
R1""" = v,max(at X= O) .... .. ................ .................. = p
( .
Mmox at p01nt of load, when x =
I)
2 ........ ......... ...... ·.... = 4PI
44. SIMPLE BEAM - TWO EQUAL CONCENTRATED MOVING LOADS
l
R1ma.x = V1mox(at x = O) ........................................ = P(2 - 7)
··~··
when a< (2 - h)l = 0.5861
[ undltrload 1 atx =
1(1- •) ...................= ~ (1- i)2
2
2
when a> (2- /2)1 = 0.5661
]
[ wnh one load at center of span (Case 43) ..... • -. . =
PI
4
45. SIMPL E BEAM - TWO UNEQUAL CONCENTRATED MOVING LOADS
R 1 , . , - V,.,.,(atx - 0) .. ..................... ........ ......... a P1 +
··~··
[uoderP,, at x =
~ (1- P,;•111 ) ] ...............= (P, + !',)~
Mmax may occur with larger
l
load at center ol span aod other
[
1- a
I',/
................. ~
~
1
load off spen (Case 43)
GENERAL RULES FOR SIMPLE BEAMS CARRYING MOVING CONCENTRATED LOADS
The maximum shear due to moving concentrated loads occurs al one support
l-- 1 -
when one ol the loads is at that support. With several moving loads, lhe location that
r--a~
R,
lb
f--x-
~
z~
••
will produce maximum shear must be detennined by trial.
P,( )-j-C.G. DP2
>--~ ~
llllll'h
-
R2
The maximum oonding moment produced by moving concentrated loads occurs
under one of the loads when that load is as far from one support as the center of
gravity ol alllhe moving loads on the beam Is from the other support.
In the accompanying diagram, the maximum bending momenl occurs unOOr load
P, when x = b. 11 should also be noted that this concnlon occurs when the centerline of the span Is midWay between the center of gravity of loads aod the nearest
concentmted load.
AMERICAN INSTITUTE OF STEEL CONSTRUCTION