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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