BEAM DIAGRAMS AND FORMULAS - University of Southern California

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

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) - - --



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



3-216


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

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
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")


4 1

3-217



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


~at" -2Jx2 -ax2) (1-xf ~~· x-a2 x-2a?1)

3-218



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
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
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



3-219


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



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)

!


_ P(!-xf

3-221



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


3-222

DES IGN OF FLEXURAL MEMBERS


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


3-223



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)


412

3-224



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
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


2

3-225

BEAM DIAGRAMS AND FORM U LAS


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



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
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


3-227



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


R,• 0.393wl


3-228


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.


BEAM DIAGRAMS AND FORMULAS - University of Southern California

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