Transcript LES 2 : Arbeidstheorema’s en energieprincipes

Lecture 2 :
Castigliano’s
Energy Theorems
Hans Welleman
Castigliano
Fa
ua
Fb
ub
Fc
uc
Fx
ux
Aext  12 Faua  12 Fbub  12 Fcuc  ...
 12 Fxux
Ir J.W. Welleman
Work and Energy methods
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Differentiate the Work to a
specific force, e.g. Fx:
Aext 1 ua 1 ub 1 uc
 2 Fa
 2 Fb
 2 Fc
 ...
Fx
Fx
Fx
Fx
ux
 ux  Fx
Fx
1
2
1
2
Use Maxwell’s notation :
u a  c aa Fa  c ab Fb  c ac Fc  ...  c ax Fx
u b  cba Fa  cbb Fb  cbc Fc  ...  cbx Fx
u c  cca Fa  ccb Fb  ccc Fc  ...  ccx Fx
u x  c xa Fa  c xb Fb  c xc Fc  ...  c xx Fx
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Work and Energy methods
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Elaborate …
u a  (c aa Fa  c ab Fb  c ac Fc  ...  c ax Fx )

 c ax
Fx
Fx
u b  (cba Fa  cbb Fb  cbc Fc  ...  cbx Fx )

 cbx
Fx
Fx
u c  (cca Fa  ccb Fb  ccc Fc  ...  ccx Fx )

 ccx
Fx
Fx
u x  (c xa Fa  c xb Fb  c xc Fc  ...  c xx Fx )

 c xx
Fx
Fx
Ir J.W. Welleman
Work and Energy methods
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Result:
Aext 1
 2 Fa cax  12 Fb cbx  12 Fc ccx  ...  12 Fx cxx  12 ux
Fx
Maxwell :
c ax  c xa
Aext 1
 2 cxa Fa  12 cxb Fb  12 cxc Fc  ...  12 cxx Fx  12 ux
Fx
1
2
cbx  c xb
ccx  c xc
Castigliano’s 2nd theorem
Ir J.W. Welleman
ux
Aext
 ux
Fx
Work and Energy methods
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Castigliano
Work done by external loads Aext is
stored in the strain energy Ev
 Differentiate the strain energy to a force
at location x to find the displacement u
at x

Ev
 ux
Fx
Ir J.W. Welleman
Work and Energy methods
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Example
F
EI
wmax
0,5 l
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0,5 l
Work and Energy methods
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Strain Energy in terms of the load
M ( x)  12 Fx 0  x  12 l
l
1l
2
2
1
4
2
2
2 3
F x
M ( x)
F l
Ev  
dx  2 
dx 
2 EI
2 EI
96 EI
o
0
3
3
dEv 2 Fl
Fl
w


dF 96EI 48EI
Ir J.W. Welleman
Work and Energy methods
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Steps of the solution strategy
Castigliano : differentiate EV to a
force.
 EV is integral of the square of the
moment distribution
 so
…. differentiate after integration …
IS THIS SMART TO DO ??

Ir J.W. Welleman
Work and Energy methods
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Smart help
l
2
dEv
d M ( x)
w

dx

dF l md( xF) 
2 EI
0 M ( x)
w
dx
EI
0 l
2
dE v
d  M ( x) 
w


 dx
dF
dF  2 EI 
0
l
l
2M ( x) dM ( x)
M ( x) dM ( x)
w

dx 

dx
2 EI
dF
EI
dF
0
0
Ir J.W. Welleman
Work and Energy methods
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Castigliano

Second theorem

First theorem
Ir J.W. Welleman
Ev
 ux
Fx
Ev
 Fx
u x
Work and Energy methods
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