CONJUGATE BEAM METHOD

Conjugate beam is an imaginary beam of same

lenge as original beam

, but is supported in

such a manner that , when loaded with diagram

If real beam , the sheal force and Bending moment

at a section in conjugate beau gives slope and

election at corresponding section of original beam.

Real beam support Conjugate beamSupport

hinged/rolleshinged/roller ⑭X T

fixed free end.

⑭ –

fixed end free end #·

InteriorSupport internal hinge —

⑳

InteriorSupport internal hinge — ⑳

Load on Conjugate Beam- > diagram Area of original

Theorems :- Beam.

&Slope at any port on original beam-there force at

that point in conjugate beam.

② Deflection at any port on original beam-Bending moment at

that point in conjugate beam.

SIGN CONVENTIONS :-

–

1) If M/ diagram is positive , loading outs downwards

of Mez diagram is negative , loading

acts upward .

2) If sheal force -> tre -> slockwise slope

– re -> anti-clockwise slope.

if sheal fork ->

3) if BM ->re -> deflection – downward .

If BM -re -> deflection -> upward .

Problems to point load at free end.

1) Cantileve bear subjected

I w

A B – Real Beam

–

wal -> Bending

moment diagram.

– > diagram

|23

B slope at Bow real beau ,

OB = sheal force at B' of conjugate Beam.

SFi' = RB'= xxL=C

deflection at B of real beam = Oc = Bending moment at B'

of conjugate beam = BMB '

= Cavea of loading x Centroidal distance was)

= (* XL)xzx

= 2) Cantilever Beam subjected to uniformly

distributed load

Wim

mmm – Real beam –

G -> Bending

moment diagram (B . M . D)

w -> diagram

↑

-conjugate beamI

slope at Bow real beau , OB = sheal force at B' of

conjugate Beam.

OB = SFi' = RB'=xx =3

deflection at B of real beam = Oc = Bending moment at B'

of conjugate beam = BMB '

& B = BRB' = =Xxx(x)=

–

③Simply supported beam AB spanh subjected to point loud W at midepan . find slopes at 1 and B and

deflection at midspar , C .

—

W

Y2 X

42 ->

real beam .

B Do T

↑ RA ↑RB

WL F

id -> B . M .D

F —

diagram 42 42

↓ B1 -> conjugate bear

↑ Rig' Ral

Hope at a o real beam-shear force at Alie

RAY of

conjugate beam.

< Fy = 0 => RA'= RB = = Y -xw

↑ 4E1

equilibrium Ra + R= ( Equations

44MA = 0 = L** E) – RBX) = 0

RB'= : Ral=-WTEI//

Ther OA = RA= (2)

OB = RB = E G

= Banding moment at s of conjugate

beam.

= Rsx E – * xEx W * (5 +E)

=- 5r = (

⑦ find Hope and deflection at support and midspan

of simply supported beam subjected to udl w/m.

DA= RAI *b

-> w/m ↳ A = 2munmum Sh

↑ g – Tk L-

↑A+ Rp = EXLX ↑

-> BMD Ra'+ Rs=

we ?

– EMA' = O

k 1- [Rbx4) +) = 0

↑

#) -> digram

R

=-O

At 1- B Ral =We

& c = BM,

↑

= Oc = Ent/

Q) find maximum slope and deflection of following beam using conjugate

beam method?

50 kN may slope at supports A & B

A 2m am " – real

beam man deflection

at

& * ↑

midspan , i. e -> C.

50X4 = 50 kNm

i& -> B-M .D

idiagram

·I i – conjugate beam –

&

&

c &↑ – ↑R C

OA = RA or SEAl and OB = SFil = RB'

RA' + RB = Ex4x = EMAl = 0

LEXivx-4xRp =ER==/

Ra = E – : Slope at A =A= (2)

OB=(

&c = BMC = (2) -x)=

2 find slopes at A & B and deflection at C and D

of following beam using conjugate beau method ?

↓SOKNCork B

A " i F ↑m am am

To analyse the beam , loads will be considered separately.

and the surre gives slope and deflection.

30KN 2 KN Real

Real t beam 2

Y B beam 1 / "4m = *

Om D Do

A T ↑m

L

B 2m T

Wob =2xx – = 30kNm

gram * I T

↑

I

' ↓ – A 6m .

Y 2m B

Mdia . –

~El

↑

I· – 6m- –

= –

there M/c values at D andC are calculated being

Interpolation]

conjugate beam of 30K Conjugatbeam of 20 .

KN

Q

S ~ I

②

"30A M B E 11 1 ↑ –

=>>4m ->↑ 4m

/ D' I "Atra"< 8m- RB RA

p

to conjugate beam theory , sumoy shear forces at

According A and Al will give

the shope at A of real beam. To

find that , calculate RA , RA" , Ris' and RB" .

① RA' + RB) = = X8X

=> RA' + Ri = 2 EMAl = 0

Ex *Xx(E) – 8XR = 0 = Ris =1

due to symmetry RA =

240- A " B & ⑳

k G7b 7

& RA" + Ris"=X8X Distance of C-G

from A =>

EMA" = o (Ea

& x8Xx (A)SXRD"=O R =E

Ra" =

10- –

Now slope at A of Real beam ,

is OA =

shear force at A of conjugate beam = Ra' + Ra"

–

Slope at B of real beam , in OB = shear force at B

of conjugate beam = RB' + Rs

=>2+

DEFLECTION at C

-6 ! I El El

. B/ #Im ↑ A,-

– Ris' RB 1(

Defection at C of real beamie Oc = Bending moment at

of the conjugate beam

= [(RA'x 4) -(e) + ((2" x4) – (tx4x)) um

Total S. F Total S .F

A- c A"->

I

=(x)-()+ 4) – 5 &

= DEFLECTIONAT D

30

& El 30 -43 -X-

k- I El –" ! &

2m B D'

-At R 8m- RB

RB

deflection at D of real beam , is Op = Bending moment

at c of the conjugate beam-

= ((Risx2) – (tx2xx] + ((Ri"x2) -(xx2xx)] = ((x2) – ( ***) + (C +2) – (Ex2xx)

=

Our website has a team of professional writers who can help you write any of your homework. They will write your papers from scratch. We also have a team of editors just to make sure all papers are of HIGH QUALITY & PLAGIARISM FREE. To make an Order you only need to click Ask A Question and we will direct you to our Order Page at WriteDemy. Then fill Our Order Form with all your assignment instructions. Select your deadline and pay for your paper. You will get it few hours before your set deadline.