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General recurrence form of the Generalized Second-

Order Accurate GSSSS Algorithm for SDOF system

 Amplification matrix

Load vector

Recurrence relation

© George Papazafeiropoulos, 22-Dec-2013

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General explicit form of the Generalized Second-

Order Accurate GSSSS Algorithm

Free parameters: Λ6W1, Λ5W2, Λ3W3, Λ4W1, Λ1W1, Λ2W2, W1, λ1, λ2, λ3, λ4, and λ5.

© George Papazafeiropoulos, 22-Dec-2013

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Free parameters of integration algorithm

Free parameters (12) : Λ6W1, Λ5W2, Λ3W3, Λ4W1, Λ1W1, Λ2W2, W1, λ1, λ2, λ3, λ4, and λ5.

Weighted residual equation :

Weighted time field:

where wp are constant coefficients and Γ = (t – tn)/ Δt.

Definition of Wi :

© George Papazafeiropoulos, 22-Dec-2013

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Selection of integration parameters for various

common algorithms

© George Papazafeiropoulos, 22-Dec-2013

Central

DifferenceMethod

 Average

constant

acceleration

method

Linear

 Acc elerat ionMethod

Fox-

Goodwinformula

Family of

NewmarkMethods

ρ∞ (interval)

w1 -15 -15 -15 -15 -15

w2 45 45 45 45 45

w3 -35 -35 -35 -35 -35

W1Λ1 1 1 1 1 1

W2Λ2 0 0.25 1/6 1/12 β

W3Λ3 0 0.25 1/6 1/12 β

W1Λ4 0.5 0.5 0.5 0.5 γ

W2Λ5 0.5 0.5 0.5 0.5 γ

W1Λ6 1 1 1 1 1

λ1 1 1 1 1 1

λ2 0.5 0.5 0.5 0.5 0.5

λ3 0 0.25 1/6 1/12 β

λ4 1 1 1 1 1

λ5 0.5 0.5 0.5 0.5 γ

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Selection of integration parameters for various

U0-V0 algorithms

© George Papazafeiropoulos, 22-Dec-2013

U0-V0-Opt U0-V0-CA U0-V0-DA

ρ∞ (interval) [0 1] [1/3 1] [0 1]

w1  -15(1-2ρ∞)/(1-4ρ∞) -15*(1-5*ρ∞)/(3-7*ρ∞) -15

w2  15(3-4ρ∞)/(1-4ρ∞) 15*(1-13*ρ∞)/(3-7*ρ∞) 45

w3  -35(1-ρ∞)/(1-4ρ∞) 140*ρ∞/(3-7*ρ∞) -35

W1Λ1  1/(1+ρ∞) (1+3*ρ∞)/2/(1+ρ∞) 1

W2Λ2  1/2/(1+ρ∞) (1+3*ρ∞)/4/(1+ρ∞)  1/2

W3Λ3  1/2/(1+ρ∞)^2 (1+3*ρ∞)/4/(1+ρ∞)^2 1/2/(1+ρ∞)

W1Λ4  1/(1+ρ∞) (1+3*ρ∞)/2/(1+ρ∞) 1

W2Λ5  1/(1+ρ∞)^2 (1+3*ρ∞)/2/(1+ρ∞)^2 1/(1+ρ∞)

W1Λ6  (3-ρ∞)/2/(1+ρ∞) 1  (3+ρ∞)/2/(1+ρ∞)

λ1 1 1 1

λ2 1/2 1/2 1/2

λ3  1/2/(1+ρ∞) 1/2/(1+ρ∞) 1/2/(1+ρ∞)

λ4 1 1 1

λ5  1/(1+ρ∞); 1/(1+ρ∞) 1/(1+ρ∞)

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Selection of integration parameters for various

U0-V1 algorithms

© George Papazafeiropoulos, 22-Dec-2013

U0-V1-Opt -

generalized a-method

(Chung & Hulbert,1993)

U0-V1-CA - Hilber-

Hughes-Taylor

method (Hilber,Hughes & Taylor,

1977)

U0-V1-DA -

Wood–Bossak–Zienki

ewicz method (Wood,Bossak & Zienkiewicz,

1980)

ρ∞ (interval) [0 1] [1/2 1] [0 1]

w1  -15*(1-2*ρ∞)/(1-4*ρ∞) -15*(1-2*ρ∞)/(2-3*ρ∞) -15

w2  15*(3-4*ρ∞)/(1-4*ρ∞) 15*(2-5*ρ∞)/(2-3*ρ∞) 45

w3  -35*(1-ρ∞)/(1-4*ρ∞) -35*(1-3*ρ∞)/2/(2-3*ρ∞) -35

W1Λ1  1/(1+ρ∞) 2*ρ∞/(1+ρ∞) 1

W2Λ2  1/2/(1+ρ∞) ρ∞/(1+ρ∞)  1/2

W3Λ3  1/(1+ρ∞)^3 2*ρ∞/(1+ρ∞)^3 1/(1+ρ∞)^2

W1Λ4  1/(1+ρ∞) 2*ρ∞/(1+ρ∞) 1W2Λ5  (3-ρ∞)/2/(1+ρ∞)^2 ρ∞*(3-ρ∞)/(1+ρ∞)^2 (3-ρ∞)/2/(1+ρ∞)

W1Λ6  (2-ρ∞)/(1+ρ∞) 1  2/(1+ρ∞)

λ1 1 1 1

λ2  1/2 1/2 1/2

λ3  1/(1+ρ∞)^2 1/(1+ρ∞)^2 1/(1+ρ∞)^2

λ4 1 1 1

λ5  (3-ρ∞)/2/(1+ρ∞) (3-ρ∞)/2/(1+ρ∞) (3-ρ∞)/2/(1+ρ∞)

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Selection of integration parameters for various

U1-V0 algorithms

© George Papazafeiropoulos, 22-Dec-2013

U1-V0-Opt U1-V0-CA U1-V0-DA

ρ∞ (interval) [0 1] [1/2 1] [0 1]

w1 -30*(3-8*ρ∞+6*ρ∞^2)/(9-

22*ρ∞+19*ρ∞^2)

 -60*(2-

8*ρ∞+7*ρ∞^2)/(11-

48*ρ∞+41*ρ∞^2)

 -30*(3-4*ρ∞)/(9-11*ρ∞)

w2

 15*(25-

74*ρ∞+53*ρ∞^2)/2/(9-

22*ρ∞+19*ρ∞^2)

 15*(37-

140*ρ∞+127*ρ∞^2)/2/(1

1-48*ρ∞+41*ρ∞^2)

 15*(25-37*ρ∞)/2/(9-

11*ρ∞)

w3

 -35*(3-

10*ρ∞+7*ρ∞^2)/(9-

22*ρ∞+19*ρ∞^2)

 -35*(5-

18*ρ∞+17*ρ∞^2)/(11-

48*ρ∞+41*ρ∞^2)

 -35*(3-5*ρ∞)/(9-11*ρ∞)

W1Λ1   (3-ρ∞)/2/(1+ρ∞) (1+3*ρ∞)/2/(1+ρ∞) (3+ρ∞)/2/(1+ρ∞)

W2Λ2   1/(1+ρ∞)^2 2*ρ∞/(1+ρ∞)^2 1/(1+ρ∞)

W3Λ3   1/(1+ρ∞)^3 2*ρ∞/(1+ρ∞)^3 1/(1+ρ∞)^2

W1Λ4   (3-ρ∞)/2/(1+ρ∞) (1+3*ρ∞)/2/(1+ρ∞) (3+ρ∞)/2/(1+ρ∞)W2Λ5   2/(1+ρ∞)^3 4*ρ∞/(1+ρ∞)^3 2/(1+ρ∞)^2

W1Λ6   (2-ρ∞)/(1+ρ∞) 1   2/(1+ρ∞)

λ1 1 1 1

λ2  1/2 1/2 1/2

λ3   1/2/(1+ρ∞) 1/2/(1+ρ∞) 1/(1+ρ∞)^2

λ4 1 1 1

λ5   1/(1+ρ∞) 1/(1+ρ∞) (3-ρ∞)/2/(1+ρ∞)

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Linear elastic SDOF system without damping

© George Papazafeiropoulos, 22-Dec-2013

int   ( )mu F p t  + =Differential equation of motion:

Force due to stiffness and damping:

Tangent stiffness:

Tangent damping:

intF ku=

T K k =

0T 

C  =

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Linear viscoelastic SDOF system

© George Papazafeiropoulos, 22-Dec-2013

int   ( )mu F p t  + =Differential equation of motion:

Force due to stiffness and damping:

Tangent stiffness:

Tangent damping:

intF ku cu= +  

T K k =

T C c=

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Linear elastic SDOF system with constant hysteretic

damping

© George Papazafeiropoulos, 22-Dec-2013

int   ( )mu F p t  + =Differential equation of motion:

Force due to stiffness and damping:

Tangent stiffness:

Tangent damping:

int

2k F ku u

η 

ω = +  

T K k =

2T 

k C 

  η 

ω 

=

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Bilinear elastic SDOF system without damping (1)

© George Papazafeiropoulos, 22-Dec-2013

int   ( )mu F p t  + =Differential equation of motion:

Force due to stiffness and damping:

( )

)

( )

min, ,

min,

,

,

int

max,

,,

max, ,

, ,

, , 0

, 0,

, ,

neg y neg

neg

 y neg

 y neg

 pos

 y pos y pos

 pos y pos

 f u u

 f u u u

uF 

 f 

u u uu

 f u u

  ∈ −∞

  ∈

=

  ∈   ∈ ∞

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( ))

( )

,

min,

,

,

max,

,

,

,

0, ,

, , 0

, 0,

0, ,

 y neg

neg

 y neg

 y neg

T  pos

 y pos

 y pos

 y pos

u u

 f u u

u

K   f u u

u

u u

  ∈ −∞

  ∈

=   ∈   ∈ ∞

Bilinear elastic SDOF system without damping (2)

© George Papazafeiropoulos, 22-Dec-2013

Tangent stiffness:

Tangent damping: 0T 

C  =

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Linear elastic SDOF system with coulomb friction

damping

© George Papazafeiropoulos, 22-Dec-2013

int   ( )mu F p t  + =Differential equation of motion:

Force due to stiffness and damping:

Tangent stiffness:

Tangent damping:

int

uF F ku

u= +

T K k =

0, 0

, 0T 

uC 

u

  ≠=

+∞ =