finite element and numerical methods in structural dynamics · ref: dynamics of structures, theory...
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Finite Element and Numerical Finite Element and Numerical
Methods in Structural Methods in Structural
Dynamics Dynamics
S M ShiyekarS M ShiyekarProfessor Civil Engineering Department
Fundamentals of Structural Dynamics and Applications to Earthquake Engineering
Sanjay Ghodawat Group of Institutes, Faculty of Engineering
7 to 12 Dec 2015
S M Shiyekar, Professor Civil Department
Degrees of Freedom
Independent displacements, rotations
that specify completely the displaced or
deformed position and orientation of the
node in a body or system.
S M Shiyekar, Professor Civil Department
Dynamics of Single Degree of
Freedom (SDOF) System
Static Force
P ku=
( )P t ku≠
Dynamic Force
S M Shiyekar, Professor Civil Department
Governing Differential Equation
2
( )
( ) N et force
( ) m ass x acceleration
P t ku
P t ku
P t ku
≠
− =
− =
2
2
d u( )
d t
(
)
)
(
P t ku
P t ku
u ku P t
mu
m
− =
−
+ =
= &
&&
&
S M Shiyekar, Professor Civil Department
Free Vibration
0mu ku+ =&&
S M Shiyekar, Professor Civil Department
Natural FrequencyEvery periodic motion is governed by
2 2
( ) cos( ) sin( )
sin( ) cos( )
cos( ) sin( )
( ) cos( ) sin( ) (Fourier)
n n
n n n n
n n n n
u t A t B t
u A t B t
u A t B t
u t A B
ω ω
ω ω ω ω
ω ω ω ω
θ θ
= +
= − +
= − −
= +
&
&&
2 2
2
2
2
2
cos( ) sin( )
( ) 0
( )
,
n n n n
n
n
n
n n
u A t B t
u u
m u ku
k m
k k
m m
ω ω ω ω
ω
ω
ω
ω ω
= − +
= −
− + =
=
= =
&&
&&
S M Shiyekar, Professor Civil Department
2 2
2 2
2
( ) cos( ) sin( )
( ) cos( ) sin( )
sin( ) cos( )
cos( ) sin( )
cos( ) sin( )
n n
n n n n
n n n n
n n n n
u t A B
u t A t B t
u A t B t
u A t B t
u A t B t
u u
θ θ
ω ω
ω ω ω ω
ω ω ω ω
ω ω ω ω
ω
= +
= +
= − +
= − −
= − +
= −
&
&&
&&
2
2
2
2
( ) 0
( )
,
n
n
n
n n
u u
m u ku
k m
k k
m m
ω
ω
ω
ω ω
= −
− + =
=
= =
&&
S M Shiyekar, Professor Civil Department
SDOF System Natural Frequency
Rayleigh-Ritz Method
{ }
{ }
0
0
( ) sin( ) : displacement
( ) cos( ) : velocity
n
n n
u t u t
u t u t
ω
ω ω
=
=&
S M Shiyekar, Professor Civil Department
SDOF System Natural Frequency
Rayleigh-Ritz Method
{ }
{ }
0
0
( ) sin( ) : displacement
( ) cos( ) : velocity
n
n n
u t u t
u t u t
ω
ω ω
=
=&
Strain energy stored in spring =
Kinetic energy of system =
SOE
E
2 2 2
0 0
2
Kinetic energy of system =
maximum maximum
1 1
2 2
;
KO
SO KO
n
n n
E
E E
ku mu
k k
m m
ω
ω ω
=
=
= =
S M Shiyekar, Professor Civil Department
Continuum System: Cantilever Beam by Rayleigh-Ritz Method
Cantilever Beam: Vibrating in SHM
[ ]
( )
0
2
2
L 2
0
2 2
( , ) sin( ) ( )
''( )
MStrin Energy in Bending =
2
[ ''( )]
n
L L
u x t z t x
d yM EI EIu x
dx
dx
EI
EI z x EI
ω ψ
ψ
=
= =
∫
( )( )
( )
2
2
2 0
2
2
20 2
0
0 0
22
0
2 2
0
0
[ ''( )][ ''( )]
2 2
1 1( ) [ ( )]
2 2
( )[ ''( )]
( )[ ( )]
L L
SO
L
KO n
SO KO
L
n L
EI z x EIE z x
E mv m x z x
E
EI x x dx
m x x dx
E
ψ
ω
ω
ψ
ψ
ψ
ψ
= =
= =
== →
∫
∫
∫ ∫
∫S M Shiyekar, Professor Civil Department
2
2 0
2
0
( )[ ''( )]
: Rayleigh Quotient
( )[ ( )]
L
n L
EI x x dx
m x x dx
ψ
ω
ψ
=
∫
∫
Continuum System: Cantilever Beam by Rayleigh-Ritz Method
( ) ( )N
ju x Z xψ=∑1
( ) ( )
Gen. Co-ord., ( ) Shape Function
Selection of Shape Function
1. Must be Continuous
2. Also derivative must be continuous
j
j
j
u x Z x
Z x
ψ
ψ
=
=
= =
∑
S M Shiyekar, Professor Civil Department
Find Natural Frequencies of Cantilever Beam by Rayleigh-Ritz Method
1
1( ) 1 cos
2
xx
L
πψ
= −
2
3( ) 1 cos
2
xx
L
πψ
= −
S M Shiyekar, Professor Civil Department
1 2
" " " "
11 1 1 12 1 2
0 0
" " " "
1 3( ) 1 cos ; ( ) 1 cos
2 2
( )[ ( ) ( )] ; ( )[ ( ) ( )] ;
( )[ ( ) ( )] ; ( )[ ( ) ( )] .
L L
L L
x xx x
L L
K EI x x x dx K EI x x x dx
K EI x x x dx K EI x x x dx
π πψ ψ
ψ ψ ψ ψ
ψ ψ ψ ψ
= − = −
= =
= =
∫ ∫
∫ ∫
Find Natural Frequencies of Cantilever Beam by Rayleigh-Ritz Method
" " " "
21 2 1 22 2 2
0 0
11 1 1 12 1
0
( )[ ( ) ( )] ; ( )[ ( ) ( )] .
( )[ ( ) ( )] ; ( )[ (
L
K EI x x x dx K EI x x x dx
m m x x x dx m m x x
ψ ψ ψ ψ
ψ ψ ψ
= =
= =
∫ ∫
∫ 2
0
21 2 1 22 2 2
0 0
) ( )] ;
( )[ ( ) ( )] ; ( )[ ( ) ( )] .
L
L L
x dx
m m x x x dx m m x x x dx
ψ
ψ ψ ψ ψ= =
∫
∫ ∫S M Shiyekar, Professor Civil Department
Find Natural Frequencies of Cantilever Beam by Rayleigh-Ritz Method
{ }{ }
4
3
2
1 0 0.2267 0.5755;
0 81 0.5755 1.924432
0. Solving Eigen Value Problem
24.0059 3.5235
n
EIK m mL
L
K m
EI EI
π
ω φ
ω ω
= =
− =
= =1 22 2
1 22 2
24.0059 3.5235;
:
22.0345 3.5160;
EI EI
L m L m
Exact
EI EI
L m L m
ω ω
ω ω
= =
= =
S M Shiyekar, Professor Civil Department
Disadvantages of Rayleigh-Ritz Method
• Difficult selection of Shape Function
• Suitable for Specific boundary conditions
• Difficult to select additional functions for
accuracyaccuracy
• Difficult to integrate
• Full flooded [k] and [m] matrices
• Not suitable for computer aided numerical
methods.
S M Shiyekar, Professor Civil Department
Finite Element Method
Beam Element
1 1
2 2
1:
1:
u
u
ψ
ψ
=
=2 2
3 3
4 4
1:
1:
1:
u
u
u
ψ
ψ
ψ
=
=
=
S M Shiyekar, Professor Civil Department
Polynomial Function for DOF2 3
1 2 3 4
2
2 3 4
1 1
2 2
2 3
3 1 2 3 4
0 2 3
( 0) 0 0 0
( 0) 0 0 0
( )
i
i
j
u a a x a x a x
duu a a x a x
dx
u x a
u x a
u x L a a L a L a L
= + + +
= = + + +
= = + + +
= = + + +
= = + + +3 1 2 3 4
23
4 2 3 4
( )
( ) 0 2 3
u x L a a L a L a L
duu x L a a L a L
dx
= = + + +
= = = + + +
1 1
2 2
2 3
3 3
2
4 4
1 0 0 0
0 1 0 0
1
0 1 2 3
u a
u a
u aL L L
L Lu a
=
S M Shiyekar, Professor Civil Department
Polynomial Function for DOF
2 3
1 2 3 4
2
2 3 40 2 3
i
i
j
u a a x a x a x
duu a a x a x
dx
= + + +
= = + + +
1 0 0 0 1 1
2 2
2 3
3 3
2
4 4
1 0 0 0
0 1 0 0
1
0 1 2 3
u a
u a
u aL L L
L Lu a
=
1 1
2 2
2 23 3
4 4
3 2 3 2
1 0 0 0
0 1 0 0
3 2 3 1
2 1 2 1
a u
a u
a uL L L L
a u
L L L L
= − − −
S M Shiyekar, Professor Civil Department
{ }
{ }
1
22 3
3
4
1
22 3
1
1 0 0 0
0 1 0 0
3 2 3 1
i
a
au x x x
a
a
u
u
=
=
Shape Function
{ } 22 3
2 23
4
3 2 3 2
2 2 2 3
1 22 3 2
2 3 2 3
3 42 3 2 2
3 2 3 11
2 1 2 1
3 2 21 ; ;
32 ;
ux x x
uL L L L
u
L L L L
x x x xx
L L L L
x x x x
L L L L
ψ ψ
ψ ψ
= − − −
= − + = − +
= − = − +
S M Shiyekar, Professor Civil Department
Element Stiffness and Mass Matrices
2 2 2 3
1 22 3 2
2 3 2 3
3 42 3 2 2
3 2 21 ; ;
32 ;
x x x xx
L L L L
x x x x
L L L L
ψ ψ
ψ ψ
= − + = − +
= − = − +
" "
0
0
( )[ ( ) ( )] ,
( )[ ( ) ( )]
L
ij i j
L
ij i j
K EI x x x dx
m m x x x dx
ψ ψ
ψ ψ
=
=
∫
∫
L L L L
S M Shiyekar, Professor Civil Department
Element Stiffness and Mass Matrices
2 2
3
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
e
L L
L L L LEIK
L LL
L L L L
− − = − − −
−
2 2
2 2
156 22 54 13
22 4 13 3
54 13 156 22420
13 3 22 4
e
L L
L L L LmLm
L L
L L L L
− − = − − − −
S M Shiyekar, Professor Civil Department
Cantilever Beam: Finite Element Method
( )
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )
2 2
1,2
3
2 2
12 6 12 62 2
6 4 6 22 2 2 2
12 6 12 62 22
6 2 6 42 2 2 2
e
L L
L L L LEI
K
L LL
L L L L
−
− =
− − −
− S M Shiyekar, Professor Civil Department
Cantilever Beam: Finite Element Method
S M Shiyekar, Professor Civil Department
Cantilever Beam: Finite Element Method
[ ] [ ]{ }{ }20
nK m ω φ − =
Solution of Eigen Value ProblemSolution of Eigen Value Problem
• Power Method
• Stodola Method
S M Shiyekar, Professor Civil Department
Cantilever Beam: Finite Element Method, Results
No. of Finite Elements
Mode 1 2 3 4 5 Exact
1 3.5327 3.51772 3.51637 3.51613 3.51606 3.51602
2 34.8069 22.2215 22.1069 22.0602 22.0455 22.03452
3 75.1571 62.4659 62.1749 61.9188 61.6972
4 218.138 140.671 122.657 122.320 120.902
5 264.743 228.137 203.020 199.860
6 527.796 366.390 337.273 298.556
Ref: Dynamics of Structures, Theory and Applications to Earthquake Engineering, II nd Edition,
Anil K Chopra, PHI Publications, New Delhi 2005 S M Shiyekar, Professor Civil Department
Thank You
S M Shiyekar, Professor Civil Department
Thank You