Chapter 8:
Deflections
CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang
Deflection- Displacement & Rotation



Deflections maybe due to loads, temperature, fabrication
errors or settlement
Linear elastic behavior: linear stress-strain relationship
and small deflection
Need to:
 Plot
qualitative deflection shapes before/after the analysis
 Calculate deflections at any location of a structure
Deflection Diagrams/Shapes
M
 curvature
EI
+M
tensile side
tensile side
-M
Example 8.1
Draw the deflected shape of each of the beams.
Need to show 1st order (slope) and 2nd order (curvature) information
tensile side
tensile side
Inflection point
Straight line
(why?)
Straight line
(why?)
Example 8.2

negligible

M=0
No curvature
M≠0
All members are
axially inextensible!
Calculation of Deflections




Direct integration
Moment-area method
Conjugate-beam method
Energy methods (in Chapter 9)
Calculation of Deflections
Note: Superposition can be used for linear structures
=

y( x )
 y1 ( x )  y 2 ( x )
y1 ( x )
y2( x )
Direct Integration
+ disp
Bernoulli-Euler beam
2
d y
dx
2
y  

M
EI
M
dθ
EI
dx
dx dx
θ 

M
EI
M
dx
EI
apply boundary conditions
Example 8.3
The cantilevered beam is subjected to a couple moment Mo at its end.
Determine the eqn of the elastic curve. EI is constant.
x
2
EI
d v
dx
EI
2
dv
dx
EI v 
 Mo
 M o x  C1
M ox
2
at x = 0
dv/dx = 0 & v = 0
C1 = C2 =0.
2
 C1 x  C 2
Max
 
M ox
v
;
EI
A 
M oL
EI
M ox
2
2 EI
;
vA 
M oL
2 EI
2
Direct Integration
5 kN/m
3 kN/m
A
0
50
V
F
B
C
20m 20m
241
50 kN
91
D
50m
E
20m 20m
30
149
90
-150
y  
M
dx dx
-30
-59
EI
M
M1
+
y
M2
_
M3
+
Moment-Area Theorems
2
d y
dx
2

M
dθ
EI
dx

M
EI
M 
d  
 dx
 EI 
B
 B   A  A
B / A
M
dx
B
A
EI
+
1st moment-area theorem
Moment-Area Theorems

2nd moment-area theorem
dθ

dx
M
EI
d
Horizontal dis btw A & centroid of M/EIA-B
dt  x d   x
M
dx
EI
tA/ B 

B
x
A
M
EI
dx  x 
M/EI area btw A & B
B
A
M
EI
dx
Moment-Area Theorems
B
t A / B  A x
M
EI
B
dx  x  A
A
M
t B / A  B x
x'
dx
EI
M
EI
A
dx  x ' B
M
EI
dx
Moment-Area Theorems
1st moment-area theorem
2nd moment-area theorem
+
B/A 

B
A
M
EI
dx
tA/B 
B
x A
M
EI
dx
Example 8.5
Determine the slope at points B & C of the beam. Take E = 200GPa, I = 360(106)mm4
B/A 
B  B / A;
B  B / A
 50 kNm
 
 EI

375 kNm
1  100 kNm
50 kNm


 (5 m )  
2
EI
EI

2
  0 . 00521 rad
EI
 C   C / A   0 . 00694 rad

 (5 m )

C  C / A

B
A
M
EI
dx
Conjugate-Beam Method

Mathematical analogy
M
EI
-slope-deflection
dθ
M
dV
EI
dx
θ
dM

dx
dy
dx
dx
2
w
V
dx
2
d y
Load-shear-moment

2
M
d M
EI
2
dx
 w
Conjugate-Beam Method

M
EI
Mathematical equivalence
-slope-deflection
M
Load-shear-moment
w
EI
θ
V
y
M
Actual beam
Conjugate beam
Conjugate-Beam Method
Conjugate beam
Actual beam
θ0
θ0
y0
y0
P
PL
PL
EI
EI
V 0
V 0
M 0
M 0
M
EI
?
PL 
x
1 
EI 
L
w
2
x 
PL 

x
EI 
2L 


P
V
θ
y
2
3
PL  x
x 

EI  2
6L 


M
Conjugate-Beam Method

or
Conjugate-Beam Method
Conjugate beams are mathematical/imaginative beams
No need to worry about their stability
Just use EQ concept to obtain V and M from w
Example 8.5
Determine the max deflection of the steel beam. The reactions have been computed.
Take E = 200GPa, I = 60(106)mm4
Max disp
Max M
Solution
Conjugate beam: Max Moment
Note:
dM
V  0
dx
when

45

EI
M  M max
1  2x 

x  0
2  EI 
x  6 . 71 m ( 0  x  9 m ) OK
2
 1  2 ( 6 . 71 ) 
1
( 6 . 71 )   
6
.
71

 ( 6 . 71 )  M '  0
EI

 2  EI
3
45
x
EI
 max  M '  
6.71

6
201 . 2 kNm
3
EI
 201 . 2 kNm
2
6
3
4
4
3 4
4
[ 200 (10 ) kN / m ][ 60 (10 ) mm (1m /(10 ) mm )]
  0 . 0168 m   16 . 8 mm
Conjugate-Beam
5 kN/m
3 kN/m
A
F
B
20m 20m
C
D
50m
E
20m 20m
M
M
EI
Conjugate
beam
B
C
V
=
D
M
=y
E
F
Summary


Can you plot a qualitative deflection shape for a
structure under loads (need to show 1st & 2nd order
information)?
Can you calculate structural deformation
(displacement & rotation) using
 Direct
integration?
 Moment-area principle?
 Conjugate-beam method?