Conjugate Beam METHOD

Method (2)

Method (2)

 Conjugate beam Method ....

 Conjugate beam Method ....

Deflection (  Deflection ( yy₎₎ )) Deflection (  Deflection ( yy₎₎  Conjugate beam:  Conjugate beam: (Conjugate beam) (Conjugate beam) (Elastic load ) (Elastic load ) (  ( -M _{EI  EI  )}-M ) ( M ) ( M )  Rotation or Slope( 

 Rotation or Slope( θθ_{)or (y')= )or (y')= Shear force Shear force of elastic of elastic load =load =} Q Q_{elasticelastic}

Deflection ( 

Deflection ( δ δ _{)or (y)= Moment of elastic load =)or (y)= Moment of elastic load =} M M_{elasticelastic}

 Real(Original) beam

 Real(Original) beam  Conjugate beam: Conjugate beam:

 Load =  Load =  d²M(x) d²M(x)_dx²dx²  = w  = w  Shear = Shear =  dM(x) dM(x)_dxdx  = = Q Q Moment = Moment = MM_(x)(x)  Elastic load =

 Elastic load =  d²(y) d²(y)_dx²dx²  =- =- _EI  EI MM  Rotation =

 Rotation =  d(y) d(y)_dxdx  = = Q Q_{elasticelastic}

Deflection

Deflection = = y y == MM_{elasticelastic}

wt/m' wt/m' Deflection) Deflection) Slope(  Slope( θθ₎₎ Slope(  Slope( θθ     - M/EI  - M/EI   Rotation)  Rotation)

θ

δ = 0 , θ ≠ 0 M = 0 , Q ≠ 0

Q

Free end Fixed support 

δ ≠ 0 , θ ≠ 0 M ≠ 0 , Q ≠ 0

Fixed support  Free end

δ = 0 , θ = 0

Q M

δ = 0 , θ ≠ 0 _{M = 0 , Q ≠ 0}

Interior support (Roller or Hinge) _{Internal hinge}

θ_L

θ_R

 (Q_L=Q_R ) Internal hinge

δ ≠ 0 , θ_L ≠ θ_R ≠ 0

Interior support (Roller or Hinge) θ_L θR

M ≠ 0 , Q_L ≠ Q_R

(Original beam)

(Conjugate beam)

M = 0 , Q = 0

End support (Roller or Hinge) End support (Roller or Hinge)

Conjugate Support 

 Real Support 

 Examples

 Real Beam Conjugate Beam

Indetreminate Beam

(θ_R = θ_L )

Steps of Solution :

( y₎

.Conjugate beam method 

 Reactions

 FOR Real beam

Bending Moment Diagram _( B.M.D)

 Examples 4 t/m` 2 t/m` 15 t  _{21 t}  14 t  WL²/8=32m.t  WL²/8=9m.t  8 m.t  12 m.t  C 3t/m`

B.M.D

WL²  8 = 24m.t  6t  12t.m

=

WL²  8= 24m.t  12t.m (1) (2)

B.M.D

WL²/8=32m.t  WL²/8=9m.t  8 m.t  12 m.t 

B.M.D

6t   ₍ θ₎

4  (  (  C 12mt  8t  8t 

B.M.D

Modified B.M.D

12mt  6mt  12mt    12mt  12mt  24mt 

Or 

6mt  12mt  6mt  3mt  6mt 

 FOR Conjugate beam

(Conjugate Beam) (Conjugate Beam) (Elastic loads) .(Conjugate Beam) (C.g) .( B.M.D) .( B.M.D) ( B.M.D) ( Elastic loads) 1 2ML Datum ML Datum  nerti )  nerti ) (Elastic loads) (Given) .(Modified B.M.D) 8t  2 2   R  R

2 3ML Datum Datum M₁ M₂ 1 2M2L 1 2M1L Datum M₁ M₂ Datum M₁ M₂ = 1 2M1L 1 2M2L Datum M₁ M₂ = 2 3ML Datum Datum M₁ M₂ + 1 2M2L 1 2M1L Datum M₁ M₂ = + 2 3ML Datum Datum M₁ M₂ 1 2M2L 1 2M1L

(Deflection ) (Rotation)

 Rotation (  θ _{)= Deflection (y) =}

Q_elastic M_elastic Q Q Q Q M M M M - v e  + v e +v e - v e  Positive )  Shear force Negtive )  Positive ) Moment  Negtive ) ( θ _{)= + ve (  θ )= - ve ( y )= + ve (} y _{)= - ve} Q_{elastic M} elastic + -θ θ

Solved Examples

Using the Conjugate beam method, determine the rotation at points (a,b,c and d) and deflection at points(c,d and e).

 Ex(1)

 3t 

10t 10t  

 EI = Constant 

Solution

 Reactions and Bending moment diagram (Original beam)

 3t  10t 10t   16.2t  18.8t  10t 10t   2t/m' 2t/m' 6.0t.m 46.8t.m 47.4t.m  2*3² 8  =2.25 t.m  2*3² 8  =2.25 t.m a c d b e a c d b e 

7 

 Elastic loads( Area of bending moment)

6.0 46.8  47.4  2.25 t.m a c d b e 6.0 12 93.6 71.1  71.1  _70.2 4.5  4.5 

Conjugate beam

a e a b e

 Elastic loads on Conjugate beam

a b e      6 .      0      1      2      9      3 .      6      7      1 .      1      7      0 .      2      7      1 .      1      4 .      5      4 .      5 a c d

 Elastic Reactions

a b e      6 .      0      1      2      9      3 .      6      7      1 .      1      7      0 .      2      7      1 .      1      4 .      5      4 .      5 a c d b 138.7  132.7  164.3   _269.4 b

Modified B.M.D( M

 _EI )

( M

 _EI )

 Required rotation and deflection

 Point (a)

 Rotation ( θ ₎₌ Qelastic _{Deflection (y) =} Melastic

θ_a = 1_{EI[164.3] = +} 164.3_EI aa 164.3   Point (b) b 138.7  θ_b = 1_{EI[-138.7] = -} 138.7_EI  Point (c) θ_c = 1_{EI[164.3-4.5-71.1] = +} 88.7_EI      7      1 .      1      4 .      5 c 164.3  y_c = 1_{EI[164.3(3)-4.5(1.5)-71.1(1)] = +} 415.05 _EI  Point (d) θ_d = 1_{EI[ -138.7 -12 +93.6] = -} 57.1_EI      1      2      9      3 .      6 d b      1      3      8 .      7  Point (e) e 132.7   269.4 θ_e = 1_{EI[ -132.7] = -} 132.7_EI y_e = 1_{EI[ -268.8] = -} 269.4_EI  Left Right 

+ve Sign summary

+ ve   Clockwise   Anti-Clockwise   Clockwise   Down   Anti-Clockwise   Anti-Clockwise   Upward

Using the Conjugate beam method, determine the rotation at points (a,b,c and d ) and deflection at points(c and d ).

 Ex(2)

4t  4t/m' I I 2I   b c a d

Solution

b c a d 4*6² 8  =18.0t.m 12t.m b c a d 12 6t.m 18t.m 9t.m 6  3  9  18  18  4.5  18   36  Elastic loads a d b       9 1     8      1      8      4 .      5      1      8      3       6  Elastic Reactions a d b       9 1     8      1      8      4 .      5      1      8      3       6 a 10.12 7.88  _14.6 12.36

 Required rotation and deflection

 Point (a) θ_a = 1_{EI[+7.88] = +} 7.88_EI  Point (b) θ_b = 1_{EI[-14.62] = -} 14.62_EI  Point (c) y_c = 1_{EI[14.62(3) + 4.5(1.0) -18(1.125)]} = + 28.11_EI θ_c = 1_{EI[18 - 14.62 - 4.5]} = - 1.12_EI c b      4 .      5      1      8 14.62 c  Point (d) y_d = 1_{EI[+ 12.36] = +} 12.36_EI θ_d = 1_{EI[-10.12] = -} 10.12_EI

B.M.D

Conj.beam

I  I   2I 

I  I I    Clockwise   Anti-Clockwise   Downward   Anti-Clockwise   Downward   Anti-Clockwise

 Ex(3)

_{Using the Conjugate beam method, determine :}

* the rotation at points (a ,b,d and e ),

* the relative (change in) rotation at point( f ), * the deflection at points(d ,e ,f and j).

2t/m' 6t  a d b f  e c

Solution

 EI = Constant 

2t/m' 6t  a d b f  e c  3t  4t  a d b f  e c 9t.m 10t.m 16t.m 1t.m  5t.m 40 85.33  1.33   27  10 a d b _f  e c  j 6.75t  16t  16.25t   j a d b f  e c 85.33  1.33   27  10 40 16  29.33   24.13  10.2 16  27  1.33  85.33  10 40

11 

 Required rotations and deflections

 Point (a) θ_a = 1_EI[+29.33] = + 29.33_EI  Point (b) θ_b =1_{EI[-16] =-}16_EI a  29.33  b 16 θ_f/L = 1_{EI[-16+10-1.33] = -} 7.33_EI y_f  = 1_{EI[-16(2)+10(1.33)-1.33(1)] = -} 20_EI

 Point (f)(Internal Hinge )

θ_f/R = 1_{EI[-16+10-1.33+24.13 ] = +} 16.8_EI θ_f/rel = [θ_{f/R -} θ_f/L ]₌ + 16.8_{EI -} -7.33_{EI  = +} 24.13_{EI  =} Rf _EI f        1  .       3       3       1        0 16  24.13   Point (d) y_d=1_{EI[+29.33(4)+10(1.33)-42.7(1.5)]} =+66.6_EI a d 16t.m  5t.m 42.7  10  29.33   Point (e) y_e=1_{EI[+10.2(3)-13.5(1)] = +} 17.1_EI e c 9t.m 13.5  10.2  Point (j) M  j= 6.75(3) - 6(1.5) = 11.25t.m 2t/m' a 6.75t   j 11.25t.m  29.33 

 Real beam Conj. beam

 2.25 t.m 16.88  4.5  y  j = 1EI[+29.33(3)-16.88(1) -4.5(1.5)] = + 64.36 EI θ_d = 1_EI θe = 1_EI   Clockwise)   Anti-Clockwise)   Anti-Clockwise)   Upward)   Clockwise)   Downward)   Downward)   Downward) [29.33+10-42.7] = - 3.37_EI [13.5-10.2 ] = + 3.3_EI 6t    Anti-Clockwise)   Clockwise)

Using the Conjugate beam method, determine : * the rotation at points (c and e ),

* the relative (change in) rotation at point( d ),

* the deflection at points(c ,e and d ). [take EI = 6000 t/cm²]

 Ex(4)

 2I   2I  I  I  4t  a d e b c 2t/m'

Solution

 Reactions(Original beam)  2I   2I  I  I  4t  a d e b c 2t/m' 7t  13t  7t  a d e b c 7t.m _4t.m 8t.m 16t.m

B.M.D(Original beam)

a d e b c 7t.m _{4t.m 4t.m} 16t.m  2t.m  3.5  8.0 4.0 8.0 _4.0 42.67   21.33  Conjugate beam  Elastic Reactions 16.29  a d e c 8.0 4.0 _8.0 4.0  21.33   3.5  42.67   24.2 12.29   27.25  16t  7.0t.m

 Required rotations and deflections

 Point (c) c 12.29   27.25  θ_c = _{6000[-12.29] = - 0.002rad}1 θ_d/L = _{6000[+ 3.5] = 5.833*10}1 -4 rad y_d = _{6000[3.5(0.67)] = + 0.039 cm}1 y_c = _{6000[-27.25] = - 0.454 cm}1

 Point (d)(Internal Hinge ) a d

      3  .       5  24.2 θ_d/R = _{6000[+ 3.5 + 24.2 ] = + 4.6167*10}1 -3 θ_d/rel = _│θ_d/R - θ_d/L│= Rd_{EI  = 6000  = 4.033*10}24.2 -3 rad  Point (e) θ_e = _{6000[-16.29-8-4+21.33] = - 0.00116 rad}1 y_e = _{6000[16.29*4 + 8*2.67+ 4*1.33-21.33*1.5] = + 0.997 cm}1 16.29  e 4.0 _8.0  21.33    Anti-Clockwise   Clockwise   Downward   Upward  Clockwise   Anti-Clockwise   Downward