2C9
Design for seismic
and climate changes
Jiří Máca
List of lectures
1.
2.
3.
4.
5.
6.
7.
8.
9.
Elements of seismology and seismicity I
Elements of seismology and seismicity II
Dynamic analysis of single-degree-of-freedom systems I
Dynamic analysis of single-degree-of-freedom systems II
Dynamic analysis of single-degree-of-freedom systems III
Dynamic analysis of multi-degree-of-freedom systems
Finite element method in structural dynamics I
Finite element method in structural dynamics II
Earthquake analysis I
2
Finite element method in structural dynamics II
Numerical evaluation of dynamic response
1. Dynamic response analysis
2. Time-stepping procedure
3. Central difference method
4. Newmark’s 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
Dynamic response analysis
For systems that behave in a linear elastic fashion…
-
If N is small → it is appropriate to solve numerically the equations
  cu  ku  p(t )
mu
-
If N is large → it may be advantageous to use modal analysis
  Cq  Kq  P(t)
Mq
-
M and K are diagonal matrices
For systems with classical damping C is a diagonal matrix →
J uncoupled differential equations (see resolution of SDOF systems)
M n qn (t )  Cn qn (t )  K n qn (t )  Pn (t )
-
n = 1, J
For systems with nonclassical damping C is not diagonal and
the J equations are coupled → use numerical methods to
solve the equations
Dynamic response analysis
-
The direct solution (using numerical methods) of the NxN system of equations
  cu  ku  p(t )
mu
is adopted in the following situations:
-
-
For systems with few degrees of freedom
-
For systems and excitations where most of the modes
contribute 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
2. TIME-STEPPING PROCEDURE
-
Equations of motion for linear MDOF system excited by force vector p(t) or
earthquake-induced ground motion ug (t )
  cu  ku  p(t ) or  mιug (t )
mu
initial conditions
u(0)  u 0
and u (0)  u 0
time scale is divided into a series of time steps, usually of constant duration
t  ti 1  ti
i  u
(ti )
pi  p(ti ) ui  u(ti ) u i  u (ti ) u
i  cu i  kui  pi
mu
unknown vectors
i 1  cu i 1  kui 1  pi 1
 mu
i 1
ui 1 u i 1 u
Explicit methods - equations of motion are used at time instant i
Implicit methods - equations of motion are used at time instant i+1
Time-stepping procedure
-
Modal equations for linear MDOF system excited by force vector p(t)
u t 
J
  q  t   Φq(t )
n 1
n
n
  Cq  Kq  P(t)
Mq
M  ΦTmΦ C  ΦTcΦ K  ΦTkΦ P(t)  ΦTp(t)
i  Cq i  Kqi  Pi  Mq
i1  Cq i1  Kqi1  Pi1
Mq
unknown vectors
i 1
qi 1 q i 1 q
3. CENTRAL DIFFERENCE METHOD
explicit integration method
dynamic equilibrium condition
for direct solution
xn → ui
fn → pi
M, C, K → m, c, k
for modal analysis
fn → Pi
M, C, K → M, C, K
xn1  xn1
2t
x  2 xn  xn1
xn  n1
t 2
x n 
xn → qi
M 
xn  Cxn  Kxn  f n
 x  2 xn  xn1 
 xn1  xn1 
M  n1
C

 Kxn  f n
2



t
t


2




1

1  
2
1 


 1
 1
xn1   2 M 
C   f n   K  2 M  xn   2 M 
C  xn1 
t
2t  
2t 


 t
 t

K̂
b
a
Central difference method
for diagonal M and C, finding an inverse of
is trivial (uncoupled equations)
1 
 1
M

C
 t 2
2t 
1
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)
T
t  M

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
TJ
t
20
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.
Central difference method
(initial conditions)
(effective stiffness matrix)
Central difference method
(effective load vector)
for direct solution:
delete steps 1.1, 1.2, 2.1 and 2.5
replace (1) q by u, (2) M, C, K and P by m, c, k and p
4. NEWMARK’S METHOD
implicit integration method
dynamic equilibrium condition
for direct solution
xn → ui
fn → pi
M, C, K → m, c, k
for modal analysis
fn → Pi
M, C, K → M, C, K
xn → qi
xn 1  xn  txn  0.5   t 2 xn  xn 1t 2
xn 1  xn  1   txn  txn 1
(Newmark’s equations)
Mxn 1  Cx n 1  Kxn 1  f n 1
(“discrete” equations of motion)
 M  t C   t 2 K  
xn 1  f n 1  C  xn  1    t 
xn   K  xn  t xn   0.5    t 2 
xn 
K̂
fˆn 1
Newmark’s method
effective stiffness matrix
Kˆ  [ M  tC  t 2 K ]
effective load vector
fˆn 1  f n 1  C  xn  1    txn   K  xn  txn   0.5    t 2 
xn 
system of coupled equations
Kˆ 
xn 1  fˆn 1  
xn 1
unconditionally stable method
for γ = 1/2 and β = 1/4 (average acceleration method)
recommended time step depends on the shortest period of interest
t
TM
10
Newmark’s method
Newmark’s method
for direct solution:
delete steps 1.1, 1.2, 2.1 and 2.7
replace (1) q by u, (2) M, C, K and P by m, c, k and p
5. STABILITY AND COMPUTATIONAL ERROR OF TIME INTEGRATION
SCHEMES
TJ 

 t  
 

more
accurate
than
(t  0.551TJ )
TJ 

 t  
 

Stability and computational error of time integration schemes
Free vibration problem: mu  ku  0
u (0)  1 and u (0)  0  u (t )  cos nt
(t  0.1Tn )
Errors: amplitude decay (AD)
period elongation (PE)
numerical
damping?
6. EXAMPLE – DIRECT INTEGRATION – CENTRAL DIFFERENCE METHOD
bh3 0.3 * 0.33
E  3.43*10 kN / m , I 

 6.75*104
12
12
7
2
3.10m
ag (t ) / g
m
k
3.10m
0.2
3.10m
1.0
1.2
m
1.7
3.10m
0.30m / 0.30m
k
12 EI
12 * 3.43 *107 * 6.75 *104
 18640kN / m
k  2* 3  2*
L
3.14
m  60kN sec2 / m
 k  k   18640  18640
K 


 k 2k   18640 37280 
T1  0.58 sec
T2  0.22 sec
m 0  60 0 
M 


 0 m  0 60
2.5 3.4
 0.35
T sec
Example – direct integration – central difference method
Explicit time stepping scheme
General forcing function
 1

xn1   2 M 
 t

1
1

2
 1
 


 1

xn 1   2 M   f n   K  2 M  xn   2 M  xn 1 
t
 t
 


 t


 f n  Kxn   2 xn  xn1
(undamped system)
 1

0

  f1   K (1,1) K (1,2)   x1    x1   x1 
 x1 
2 M (1,1)
    
 x   t 
  x    2 x    x 
1
f
(
2
,
1
)
(
2
,
2
)
K
K

 0
  2  n 
  2  n   2  n  2  n 1
 2  n 1

M (2,2) 
t 2
1
xn1 
f1  K (1,1) * x1n  K (1,2) * xn2  2 * x1n  x1n 1
M (1,1)

x
2
n 1

t 2

f 2  K (2,1) * x1n  K (2,2) * xn2  2 * xn2  xn21
M (2,2)


Example – direct integration – central difference method
for the base acceleration case a (t )
x
1
n 1
t 2

 M (1,1) * an  K (1,1) * x1n  K (1,2) * xn2  2 * x1n  x1n 1
M (1,1)
x
2
n 1
t 2

 M (2,2) * a n  K (2,1) * x1n  K (2,2) * xn2  2 * xn2  xn21
M (2,2)




for the problem considered
x
1
n1
xn21
t 2

 60 * an  18640 * x1n  18640 * xn2  2 * x1n  x1n1
60
t 2

 60 * an  18640 * x1n  37280 * xn2  2 * xn2  xn21
60




Example – direct integration – central difference method
t  0.01sec
Example – direct integration – central difference method
t  0.05sec
Example – direct integration – central difference method
t  0.08 sec  tcrit 
Tmin


0.22

 0.07
inaccurate solution
Example – direct integration – central difference method
t  0.09 sec  tcrit
unstable solution
(unstable solution)
7. ANALYSIS OF NONLINEAR RESPONSE – AVERAGE ACCELERATION
METHOD
Incremental equilibrium condition
i  cu i   f S i  pi
mu
Incremental resisting force
 f S i   k i sec ui  k i T ui
(assumption)
secant stiffness:
 k i sec
(unknown)
tangent stiffness:
(known)
 k i T
Need for an iterative process
within each time step…
Analysis of nonlinear response – average acceleration method
1
1

  4 ;   2 


(effective load vector)
(effective stiffness matrix)
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
Analysis of nonlinear response – average acceleration method
Modified Newton-Raphson (M N-R) iteration
To reduce computational effort,
the tangent stiffness matrix is
not updated for each iteration
8. HILBER-HUGHES-TAYLOR (HHT) METHOD (α – method)
implicit integration method – generalization of Newmark’s method
numerical damping of higher frequencies (elimination of high frequency
oscillations) + stable and second order convergent
dynamic equilibrium condition
for direct solution
xn → ui
fn → pi
M, C, K → m, c, k
for modal analysis
fn → Pi
M, C, K → M, C, K
xn → qi
xn 1  xn  txn   0.5    t 2 
xn   
xn 1t 2
xn 1  xn  1    txn  txn 1
(Newmark’s equations)
Mxn 1  1     Cxn 1  Kxn 1     Cxn  Kxn   f  tn 1 
(“discrete” equations of motion)
 
xn 1
(unknowns)
tn 1  tn  1    t
HHT method
3 parameters – values for unconditional stability and second-order accuracy:
  1    4
2

1

2
 0.3    0
the smaller value of α – more numerical damping is introduced to the system
for α = 0 – Newmark’s method with no numerical damping
HHT method for transient analysis of nonlinear problems
Mxn 1  fint  xn  , xn    f  tn  , xn  
xn 1  xn  txn   0.5    t 2 
xn   
xn 1t 2
system of algebraic equations
xn 1  xn  1    txn  txn 1
f int  xn  , xn   vector of internal forces
(depends non-linearly on displacements and velocities)
  n   1     n 1     n
variables computed by convex combination