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2C9
Design for seismic
and climate changes
Ji Mca
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List of lectures
1. Elements of seismology and seismicity I
2. Elements of seismology and seismicity II
3. Dynamic analysis of single-degree-of-freedom systems I
4. Dynamic analysis of single-degree-of-freedom systems II
5. Dynamic analysis of single-degree-of-freedom systems III
6. Dynamic analysis of multi-degree-of-freedom systems7. Finite element method in structural dynamics I
8. Finite element method in structural dynamics II
9. Earthquake analysis I
2
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1. Dynamic response analysis
2. Time-stepping procedure
3. Central difference method
4. Newmarks method
5. Stability and computational error of time integration schemes
6. Example direct integration central difference method
7. Analysis of nonlinear response average acceleration method
8. HHT method
Finite element method in structural dynamics II
Numerical evaluation of dynamic response
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Dynamic response analysis
For systems that behave in a linear elastic fashion
- IfNis small it is appropriate to solve numerically the equations
( )t mu cu ku p
- IfNis large it may be advantageous to use modal analysis
( )t Mq Cq Kq P
- For systems with classical damping C is a diagonal matrix
Juncoupled differential equations (see resolution of SDOF systems)
( ) ( ) ( ) ( )n n n n n n nq t C q t K q t P t
- For systems with nonclassical damping C is not diagonal and
the Jequations are coupled use numerical methods to
solve the equations
M and K are diagonal matrices
n = 1, J
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Dynamic response analysis
- The direct solution (using numerical methods) of the NxNsystem of equations
is adopted in the following situations:
( )t mu cu ku p
- For systems with few degrees of freedom
- For systems and excitations where most of the modescontribute significantly to the response
- For systems that respond into the non-linear range
- Numerical methods can also be used within the modal analysis for system with
nonclassical damping
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2. TIME-STEPPING PROCEDURE
- Equations of motion for linear MDOF system excited by force vectorp(t) or
earthquake-induced ground motion
( ) or ( )gt u t mu cu ku p m
( )gu t
initial conditions 0 0(0) and (0) u u u u
time scale is divided into a series of time steps, usually of constant duration
1i it t t ( ) ( ) ( ) ( )
i i i i i i i it t t t p p u u u u u u
1 1 1 1i i i i i i i i mu cu ku p mu cu ku p
Explicit methods - equations of motion are used at time instant i
Implicit methods - equations of motion are used at time instant i+1
1 1 1i i i u u u unknown vectors
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Time-stepping procedure
- Modal equations for linear MDOF system excited by force vectorp(t)
( )t Mq Cq Kq P
1
( )J
n n
n
t q t t
u q
( ) ( )T T T T t t M m C c K k P p
1 1 1 1i i i i i i i i Mq Cq Kq P Mq Cq Kq P
unknown vectors 1 1 1i i i q q q
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3. CENTRAL DIFFERENCE METHOD
explicit integration method
dynamic equilibrium conditionfor direct solution xnui fnpi M, C, Km, c, k
for modal analysis xnqi fnPi M, C, KM, C, K
2
11
11
2
2
t
xxxx
t
xxx
nnnn
nnn
1 1 1 1
2
2
2
n n n n
n n n n nn n
M x Cx Kx fx x x x x
C Kx f t t
122
1
21 2
112
2
11nnnn xC
tM
txM
tKfC
tM
tx
aK b
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Central difference method
fordiagonalMand C, finding an inverse of
is trivial (uncoupled equations)
1
2 2
11
C
tM
t
MT
t
time step
conditionally stable method - stability is controlled by the highest frequency
or shortest period TM (function of the element size used in the FEM model)
the time step should be small enough to resolve the motion of the structure.
For modal analysis it is controlled by the highest mode with the period TJ
20J
Tt
the time step should be small enough to follow the loading function
acceleration records are typically given at constant time increments (e.g.
every 0.02 seconds) which may also influence the time step.
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Central difference method
(initial conditions)
(effective stiffness matrix)
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Central difference method
for direct solution: delete steps 1.1, 1.2, 2.1 and 2.5
replace (1) q by u, (2) M, C, Kand P by m, c, kand p
(effective load vector)
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4. NEWMARKS METHOD
implicit integration method
dynamic equilibrium conditionfor direct solution xnui fnpi M, C, Km, c, k
for modal analysis xnqi fnPi M, C, KM, C, K
11
2121
15.0
nnnn
nnnnn
xtxtxx
txxtxtxx
1111 nnnn fKxxCxM
2 21 1 1 0.5n n n n n n nt C t K x f C x t x K x t x t x
(discrete equations of motion)(Newmarks equations)
1
nf
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Newmarks method
unconditionally stable methodfor = 1/2 and = 1/4 (average acceleration method)
recommended time step depends on the shortest period of interest
][ 2KttCMK
21 1 1 0.5n n n n n n nf f C x tx K x tx t x
1 1 1
n n nK x f x
effective stiffness matrix
effective load vector
system of coupled equations
10MTt
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Newmarks method
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Newmarks method
for direct solution: delete steps 1.1, 1.2, 2.1 and 2.7
replace (1) q by u, (2) M, C, Kand P by m, c, kand p
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5. STABILITY AND COMPUTATIONAL ERROR OF TIME INTEGRATION
SCHEMES
JT
t
( 0.551 )Jt T
JTt
more
accurate
than
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Stability and computational error of time integration schemes
Free vibration problem: 0 (0) 1 and (0) 0 ( ) cos nmu ku u u u t t
Errors: amplitude decay (AD)
period elongation (PE)
( 0.1 )n
t T
numerical
damping?
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6. EXAMPLE DIRECT INTEGRATION CENTRAL DIFFERENCE METHOD
mm 30.0/30.0
m10.3
m10.3
m10.3
m10.3
k
k
m
m
sec22.0sec58.0
600
060
0
0
3728018640
1864018640
2
/sec60
/186401.3
10*75.6*10*43.3*12
*2
12
*2
21
2
4
47
3
TT
m
mM
kk
kkK
mkNm
mkNL
EI
k
433
27 10*75.612
3.0*3.0
12,/10*43.3
bhImkNE
2.0
35.0
2.1
0.1 secT
gtag /)(
7.1
5.2 4.3
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Example direct integration central difference method
Explicit time stepping scheme
General forcing function
2 122122
2
1
1
1
121
1
21
1
12
1
2
1
2
1
2
12
12
1
1
1
21
*2*)2,2(*)1,2()2,2(
*2*)2,1(*)1,1()1,1(
2)2,2()1,2()2,1()1,1(
)2,2(
10
0)1,1(
1
21
nnnnn
nnnnn
nnnnn
nnnnn
xxxKxKfM
tx
xxxKxKfM
tx
xx
xx
xx
KKKK
ff
M
Mtxx
xxKxfMt
x
1
1 12 2 2
1 2 1n n n nx M f K M x M x
t t t
(undamped system)
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Example direct integration central difference method
for the base acceleration case )(ta
2 12212
2
1
1
1
1212
1
1
*2*)2,2(*)1,2(*)2,2()2,2(
*2*)2,1(*)1,1(*)1,1()1,1(
nnnnnn
nnnnnn
xxxKxKaMM
tx
xxxKxKaMM
tx
2 1221
22
1
1
1
1212
1
1
*2*37280*18640*6060
*2*18640*18640*6060
nnnnnn
nnnnnn
xxxxat
x
xxxxa
t
x
for the problem considered
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Example direct integration central difference method
sec01.0t
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Example direct integration central difference method
0.05sect
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Example direct integration central difference method
07.022.0
sec08.0 min
Ttt crit
inaccurate solution
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Example direct integration central difference method
(unstable solution)
crittt sec09.0
unstable solution
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7. ANALYSIS OF NONLINEAR RESPONSE AVERAGE ACCELERATION
METHOD
i i S ii m u c u f p
Incremental equilibrium condition
Incremental resisting force
secS i i i ii T
f k u k u
seci
k
i Tk
secant stiffness:
tangent stiffness:
(unknown)
(assumption)
(known) Need for an iterative process
within each time step
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Analysis of nonlinear response average acceleration method
1 1;4 2
(effective load vector)
(effective stiffness matrix)
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Analysis of nonlinear response average acceleration method
cont.
M N-R method
- unconditionally stable method
- no numerical damping
- recommended time step depends on the shortest period of interest
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Analysis of nonlinear response average acceleration method
Modified Newton-Raphson (M N-R) iteration
To reduce computational effort,
the tangent stiffness matrix isnot updated for each iteration
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8. HILBER-HUGHES-TAYLOR (HHT) METHOD ( method)
implicit integration method generalization of Newmarks method
numerical damping of higher frequencies (elimination of high frequency
oscillations) + stable and second order convergent
dynamic equilibrium condition
for direct solution xnui fnpi M, C, Km, c, k
for modal analysis xnqi fnPi M, C, KM, C, K
2 21 1
1 1
0.5
1
n n n n n
n n n n
x x tx t x x t
x x tx tx
1 1 1 11n n n n n nx Cx Kx Cx Kx f t
(discrete equations of motion)
(Newmarks equations)
1 1n nt t t
1nx
(unknowns)
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HHT method
2 1
1 4 0.3 02
3 parameters values forunconditional stability and second-order accuracy:
the smaller value of more numerical damping is introduced to the system
for = 0 Newmarks method with no numerical damping
HHT method for transient analysis of nonlinear problems
1 int
2 2
1 1
1 1
, ,
0.5
1
n n n n n
n n n n n
n n n n
Mx f x x f t x
x x tx t x x t
x x tx tx
int ,n nf x x vector of internal forces(depends non-linearly on displacements and velocities)
1
1n n n
variables computed by convex combination
system of algebraic equations