2C09-08 Design for Seismic and Climate Changes

7/30/2019 2C09-08 Design for Seismic and Climate Changes

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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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5/31

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


8/31

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


10/31

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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(initial conditions)

(effective stiffness matrix)


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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


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


16/31

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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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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sec01.0t


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0.05sect


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07.022.0

sec08.0 min

Ttt crit

inaccurate solution


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(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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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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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


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

2C09-08 Design for Seismic and Climate Changes

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