Deflections Work Energy
Transcript of Deflections Work Energy
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ECIV 320 Structural Analysis I
Deflections
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Deflection Diagrams & Elastic Curve
ASSUMPTION
Linear Elastic Material ResponseA structure subjected to a load will return to its original
undeformed position after load is removed
Double Integration Method
Moment Area Theorems
Conjugate Beam
Energy Methods
OBJECTIVE
Calculate Slope and Deflection due to Bending at any point on a Beam
METHODS
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Last Time - Conjugate Beam Method
Same amount of Computations as Moment-Area Theorems
Use to find slopes and deflection due to
bending of beams
Method is based on
Principles of Statics Only
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Energy Methods
Double Integration Method
Moment Area TheoremsConjugate Beam
Energy Methods
Calculate Slope and Deflection
Beams
Simple Loadings
Complicated Loadings
Trusses
Frames
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External Work & Strain Energy
Ue=Ui
Work of External
Forces & Moments
Strain Energy
Internal Forces
Conservation of Energy(Elastic Material Behavior)
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External Work
Force
x
eFdxU
0
Gradually Applied
PUe
2
1
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External Work
x
e FdxU0
Due to Another Force
' PU
e
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External Work
Moment
0
MdUe
Gradually Applied
MUe
2
1
Due to Another Moment
'MU
e
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Strain Energy- Axial Force
Hookes Law:
Stress:
Strain:
Final Deflection:
AE
LNNU
i22
12
E
AN
L AE
NL
N
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Strain Energy - Bending
EI
M
dx
d
dxEI
MMddU
i
22
12
L
idx
EI
MU
0
2
2
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Example
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Principle of Virtual Work
Bernoulli 1717
External and Internal Loads are Related throughEquations of Equilibrium
External and Internal Displacements are Related through
Equations of Compatibility
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Principle of Virtual Work
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Principle of Virtual Work
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Virtual Work
udL1Virtual
Loads
Real
Displ.
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Virtual Work for Trusses - External Loads
AENLdL /
udL1Virtual
Loads
Real
Displ.
u=n
AE
nNL1
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Virtual Work for Trusses - Temperature
TLadL
udL1Virtual
Loads
Real
Displ.
u=n
LTna1
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Virtual Work for Trusses - Fabrication Errors
LdL
udL1Virtual
Loads
Real
Displ.
u=n
Ln1
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Virtual Work for Trusses - Combined Effects
LnLTnaAEnNL1
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Example
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Step 1
Remove ALL External Loads
Apply Virtual (Unit) Load in the Direction of
the Unknown Displacement
Calculate Internal Forcesn
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Step 2
Apply REAL Loads
Calculate Internal ForcesN
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Step 3
Virtual Work Equation
LnLTnaAE
nNL1
0
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Solution
0
00.75
-1.25
1
0
80
120
-100
80
0.009
6
86
10
8
58,000
58,00058,000
43,500
58,000
0.029
0.011
0.049
0
0
0
00
0
7.2e-4 7.2e-4
7.2e-4
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Virtual Work for Beams and Frames - BENDING
dx
EI
MddL
udL1Virtual
Loads
Real
Displ.
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Virtual Work for Beams and Frames - BENDING
udL1Virtual
Loads
Real
Displ.
u=m
L
dxxEI
xMxm
0 )(
)()(1
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Virtual Work for Beams and Frames - BENDING
dxEI
MddL udL1
Virtual
Loads
Real
Displ. u=m
L
dxxEIxMxm
0 )()()(1
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Virtual Work for Beams and Frames - BENDING
dxEI
MddL
dLu1Virtual
Loads
Real
Displ.
u=mq
L
dxxEI
xMxm
0 )(
)()(1
To Calculate Slope