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CIVL3310 STRUCTURAL ANALYSIS
Professor CC Chang
Chapter 8: Chapter 8: DeflectionsDeflections
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
EI
M curvature
+M-Mtensile side
tensile side
Example 8.1Draw 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
All members are axially inextensible!
M=0No curvature
M≠0
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
)x(y
)x(y1
)x(y2
)x(y)x(y 21 =
Direct Integration
Bernoulli-Euler beam
EI
M
dx
yd
2
2
EI
M
dx
dθ
dxdxEI
My dx
EI
Mθ
apply boundary conditions
+ disp
Example 8.3The cantilevered beam is subjected to a couple moment Mo at its end. Determine the eqn of the elastic curve. EI is constant.
x
21
2
1
2
2
2 CxC
xMvEI
CxMdx
dvEI
Mdx
vdEI
o
o
o
EI
xMv
EI
xM oo
2 ;
2
EI
LMv
EI
LM oA
oA 2
;2
at x = 0dv/dx = 0 & v = 0C1 = C2 =0.
Max
Direct Integration
20m 20m 50m 20m 20m
5 kN/m3 kN/m
ACB
FD E
0
50 kN 30149241
50
-150-59
91 90
-30V
M1 M2
M3
M
dxdxEI
My
+ _ +
y
Moment-Area Theorems
dxEI
Md
EI
M
dx
yd
2
2
EI
M
dx
dθ
dxEI
MBAAB
AB / 1st moment-area theorem+
BA
Moment-Area Theorems
2nd moment-area theorem
dxEI
Mxdx
EI
Mxt
B
A
B
ABA /
dxEI
Mxdxdt
Horizontal dis btw A & centroid of M/EIA-B
M/EI area btw A & B
d
EI
M
dx
dθ
Moment-Area Theorems
dxEI
Mxdx
EI
Mxt B
ABABA /
dxEI
Mxdx
EI
Mxt A
BABAB ' /
'x
Moment-Area Theorems
+
dxEI
MBAAB /
1st moment-area theorem 2nd moment-area theorem
dxEI
Mxt BABA /
Example 8.5
Determine the slope at points B & C of the beam. Take E = 200GPa, I = 360(106)mm4
ACCABB // ;
radEI
kNm
mEI
kNm
EI
kNmm
EI
kNmABB
00521.0375
)5(50100
2
1)5(
50
2
/
dxEI
MBAAB /
radACC 00694.0/
Conjugate-Beam Method
Mathematical analogy
-slope-deflection Load-shear-momentEI
M
EI
M
dx
dθ
θdx
dy
EI
M
dx
yd
2
2
wdx
dV
Vdx
dM
wdx
Md
2
2
Conjugate-Beam Method
Mathematical equivalence
-slope-deflection Load-shear-momentEI
M
EI
M
θ
y
w
V
M
Actual beam Conjugate beam
Conjugate-Beam Method
P
EI
MEI
PL
Actual beam Conjugate beam
0
0
y
θ
0
0
y
θ
0
0
M
V
0
0
M
V
EI
PL
w
V
L
x
EI
PL1
L
xx
EI
PL
2
2
L
xx
EI
PL
62
32
P
My
θ
?
or
Conjugate-Beam Method
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
OKmxmx
xEI
x
EI
)90( 71.6
02
2
145
Conjugate beam: Max Moment 0Vdx
dMNote:
when maxMM
xEI
2 0')71.6(3
171.6
)71.6(2
2
1)71.6(
45
MEIEI
6.71
mmm
mmmmmmkN
kNmEI
kNmM
8.160168.0
)])10/(1()10(60][/)10(200[
2.201
2.201'
44344626
3
3
max
Conjugate-Beam
20m 20m 50m 20m 20m
5 kN/m3 kN/m
ACB
FD E
MMEI
Conjugatebeam
CBF
D E
MEI
V=
M= y
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?
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