lecture 1 : work and energy methods
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Lecture 1 : Work and Energy methods. Hans Welleman. Content. Meeting 1Work and Energy Meeting 2Castigliano Meeting 3Potential Energy. Lecture 1. Essentials Work, virtual work, theorem of Betti and Maxwell Deformation or Strain Energy Work methods and solving techniques Virtual work - PowerPoint PPT PresentationTRANSCRIPT
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Lecture 1 : Work and Energy
methodsHans Welleman
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Content
Meeting 1Work and Energy Meeting 2Castigliano Meeting 3Potential Energy
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Lecture 1
Essentials– Work, virtual work, theorem of Betti and Maxwell– Deformation or Strain Energy
Work methods and solving techniques– Virtual work– Strain Energy versus Work– Work method with unity load– Rayleigh
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Work
F
u
uF
FuFA
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Deformation or Strain Energy
loaded situation
u
F
unloaded situation
F=0
spring characteristics
u
force
221 ukEV
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Virtual Work : Particle
x
y
z
Particle
zzyyxx FuFuFuA
For a kinematical admissible displacement Virtual Work is generated by the forces
Equilibrium : Virtual Work is zero
Equilibrium conditions of a particle in 3D
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VW : Rigid Body (in x-y plane)
Same approach, with additional rotational degree of freedom (see CM1, chapter 15)
x xi y yi zi( )i i i
A u F u F T In plane equilibirum conditions
for a rigid body
Equilibirum : Virtual Work is zero
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MECHANISMS
Kinematically indeterminate Possibilities for mechanisms ?
Hinge, N, V no M
Shear force hinge, N, M no V
Telescope, V, M no N
Interaction Forces (at the interface) do not generate Work !
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RESULT
For mechanisms holds:
The total amount of virtual work is generated only by external forces
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MECHANISMS ?????
Not a sensible structure Correct, but …….
=
M
work = 0
Monly M generates work !
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With Loading …....
Total (virtual) work is zero !
F
=
FM M
total work = 0 !
results in value of M
M FM
u
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Example : M at the position of F
l
F
a b
M
F
M
u
1
u
a
2
u
b
x-axis
z-axis
1 2 0
0
A F u M M
u uA F u M M
a b
1 10F M
a b
F abM
a b
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Standard Approach Generate Virtual Work for the chosen generalised
force (forces or moments) Only possible if the constrained degree of freedom
which belongs to the generalised force is released and is given a virtual displacement or virtual rotation
In case of a statically determinate structure this approach will result in a mechanism. Only the external load and the requested generalised force will generate Virtual Work (no structural deformation).
The total amount of Virtual Work is zero.
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Example : AV
AV
F
u
l
a bz-as
F
0V
V
u bA A u F
lF b
Al
ub
l
AV
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“TASTE” FOR BEAMS Support Reactions - remove the support Shear force - shear hinge Moment - hinge Normal force - telescope
u u
V
V
u
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Force in bar DE ?
Step 1: release the elongation degree of freedom of this bar with a telescope mechanism and generate virtual work with the normal force N
Step 2: Determine the virtual Work
Step 3 : Solve N
Example : TrussHorizontal displacement =
Rotation Vertical Distance to Rotational Centre (RC)
Compute the amount of Work…
14
12
4
2
D
E
wu a w
aw
u a wa
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Assignment : Virtual Workmoment at the support and support reaction at the roller
2,5 m 3,5 m
50 kN5 kN/m
x-axis
z-axis
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A B
Work and the reciprocal theorem
ababbb
aaa
uFuF
uFA
21
21
babaaa
bbb
uFuF
uFA
21
21
Fa
ubauaa
Fb
ubb
uab
2 : first Fb than Fa
1 : first Fa than Fb
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Work must be the same
Order of loading is not important
This results in:
theorem of BETTI
bababa uFuF
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Reciprocal theorem of Maxwell
displacement = influencefactor x force
bbbbbababa
bababaaaaa
FcuFcu
FcuFcu
baab
ababbaba
cc
FcFFcF
bababa uFuF
Rewrite BETTI in to:
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Result : Betti – Maxwell reciprocal theorem
bbbababbbab
babaaaabaaa
FcFcuuu
FcFcuuu
b
a
bbab
abaa
b
a
F
F
cc
cc
u
u
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Strain Energy
Extension (tension or compression)
Shear Torsion Bending Normal- and shear stresses
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Extension
oppervlak
221*
2*
2EAE
EA
NE VC
dx
dx
N
NN
d
strain
force
work
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Strain Energy
In terms of the
generalised stresses EC
In terms of the
generalised displacements EV
See lecture notes for standard cases
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SUMMARY
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Work methods
Work by external loads is stored in the deformable elements as strain energy (Clapeyron)
Aext = EV
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Example 2 :Work and Energy
0,5 l 0,5 l
F
wmax
EI
x-axis
z-axis
BA
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Work = Energy ?
1max2extA Fw
ll
vv xEI
MxEE
0
2
0
* d2
d
Unknown is wmax
Determine the M-distribution and the strain energy (MAPLE)
Work = Strain Energy (Clapeyron)
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Moment Distribution ?
Basic mechanics (statics) ? Take half of the model due to symmetry
lxxFxM 21
21 0)(
EI
lFx
EI
Fxx
EI
Fx
EI
FxE
lll
v 964d
4d
22
32
0
331
2
0
22
0
221
212
121
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Solution
2 31
max2
3
max
96
48
ext V
F lA E Fw
EI
Flw
EI
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Distributedload ?
Work = displacement x load (how?)
Strain Energy from M-line (ok)
Average displacement or something like that ????
l
q
w(x) EIBA
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Alternative Approach:Work Method with Unity Load
Add a unitiy load at the position for which the displacement is asked for.
Displacement w and M-line M(x) due to actual loading
Displacement w en M-line m(x) due to unity load
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Approach Add Unity Load (0 .. 1,0) Add actual Load (0 .. F)
1,0 kN
m(x)
EI
l
F
M(x)
EI
l
Total Work ? Strain Energy ?
1 12 2
2
0
2 2
0 0 0
1,0 1,0
( ) ( )d
2
( ) 2 ( ) ( ) ( )d d d
2 2 2
ext
l
v
l l l
A w F w w
M x m xE x
EI
m x m x M x M xx x x
EI EI EI
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Result
xEI
xMxmw
l
d)()(
0
Integral is product of well known functions. In the “good old times” a standard table was used. Now use MAPLE
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Work Method with Unity Load
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Example with distributed load
0,5 l 0,5 l
q
wmax
EI
1,0 kN
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Approach
Determine M(x) due to load q
(see example 1) Determine m(x) due to unity load
(notes : example 2)
Elaborate…
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Application Work & EnergyBuckling
F EI, EA
l
u
uF
F
F
just before buckling only compression
after buckling compression and bending
CONCLUSION :
Increase in Work during buckling is stored as strain energy by bending only. (Compression is the same)
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Buckling (transition)
(almost) Constant Normal Force Deformation by compression remains
constant
T H U S Work done by normal force and
additional displacement is stored as strain energy by bending only
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Additional displacement
z, w
x, u
duF
dwdx
dx
ww 22 dd wx
xx
wx
x
wu d
d
dd
d
d11d
2
21
2
F
Taylor approximation
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Clapeyron : A = Ev
l
xx
wu
0
2
21
F dd
d2
12
0
dd
d
l wA F x
x
ll
v xx
wEIxEIE
0
2
2
2
21
0
221 d
d
dd
2 22 212 2 2
0 0k k-Rayleigh2 2
12
0 0
d dd d
d d
d dd d
d d
l l
l l
w wEI x EI x
x xF F
w wx x
x x
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Example
F
f
2
4 ( )fx l xw
l
l
Assume a kinematically admissible displacement field
Elaborate the integrals in the expression and compute the Buckling Load …
Kinematic boundary conditions are met
Exact Buckling load is always
smaller than the one found with
Rayleigh(UNSAFE)