doc NLIDABLKIN
Documentation of the NLIDABLKIN function.
helpFun('NLIDABLKIN')
Non Linear Implicit Dynamic Analysis of a BiLinear KINematic hardening
hysteretic structure with elastic damping
[U,UT,UTT,FS,EY,ES,ED,JITER] = NLIDABLKIN(DT,XGTT,M,K_HI,K_LO,UY,...
KSI,ALGID,U0,UT0,RINF,MAXTOL,JMAX,DAK)
Description
General linear implicit direct time integration of second order
differential equations of a bilinear elastoplastic hysteretic SDOF
dynamic system with elastic damping, with lumped mass.
The General Single Step Single Solve (GSSSS) family of algorithms
published by X.Zhou & K.K.Tamma (2004) is employed for direct time
integration of the general linear or nonlinear structural Single
Degree of Freedom (SDOF) dynamic problem. Selection among 9
algorithms, all designed according to the above journal article, can
be made in this routine. These algorithms encompass the scope of
Linear Multi-Step (LMS) methods and are limited by the Dahlquist
barrier theorem (Dahlquist,1963).
Input parameters
DT [double(1 x 1)] is the time step of the integration
XGTT [double(1:NumSteps x 1)] is the acceleration time history which
is imposed at the lumped mass of the SDOF structure.
M [double(1 x 1)] is the lumped masses of the structure. Define the
lumped masses from the top to the bottom, excluding the fixed dof
at the base
K_HI [double(1 x 1)] is the initial stiffness of the system before
its first yield, i.e. the high stiffness. Give the stiffness of
each storey from top to bottom.
K_LO [double(1 x 1)] is the post-yield stiffness of the system,
i.e. the low stiffness. Give the stiffness of each storey from
top to bottom.
UY [double(1 x 1)] is the yield limit of the stiffness elements of
the structure. The element is considered to yield, if the
interstorey drift between degrees of freedom i and i+1 exceeds
UY(i). Give the yield limit of each storey from top to bottom.
KSI [double(1 x 1)] is the ratio of critical viscous damping of the
system, assumed to be unique for all damping elements of the
structure.
ALGID [char(1 x :inf)] is the algorithm to be used for the time
integration. It can be one of the following strings for superior
optimally designed algorithms:
'generalized a-method': The generalized a-method (Chung &
Hulbert, 1993)
'HHT a-method': The Hilber-Hughes-Taylor method (Hilber,
Hughes & Taylor, 1977)
'WBZ': The Wood–Bossak–Zienkiewicz method (Wood, Bossak &
Zienkiewicz, 1980)
'U0-V0-Opt': Optimal numerical dissipation and dispersion
zero order displacement zero order velocity algorithm
'U0-V0-CA': Continuous acceleration (zero spurious root at
the low frequency limit) zero order displacement zero order
velocity algorithm
'U0-V0-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) zero order displacement zero order
velocity algorithm
'U0-V1-Opt': Optimal numerical dissipation and dispersion
zero order displacement first order velocity algorithm
'U0-V1-CA': Continuous acceleration (zero spurious root at
the low frequency limit) zero order displacement first order
velocity algorithm
'U0-V1-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) zero order displacement first order
velocity algorithm
'U1-V0-Opt': Optimal numerical dissipation and dispersion
first order displacement zero order velocity algorithm
'U1-V0-CA': Continuous acceleration (zero spurious root at
the low frequency limit) first order displacement zero order
velocity algorithm
'U1-V0-DA': Discontinuous acceleration (zero spurious root at
the high frequency limit) first order displacement zero order
velocity algorithm
'Newmark ACA': Newmark Average Constant Acceleration method
'Newmark LA': Newmark Linear Acceleration method
'Newmark BA': Newmark Backward Acceleration method
'Fox-Goodwin': Fox-Goodwin formula
U0 [double(1 x 1)] is the initial displacement.
UT0 [double(1 x 1)] is the initial velocity.
RINF [double(1 x 1)] is the minimum absolute value of the eigenvalues
of the amplification matrix. For the amplification matrix see
eq.(61) in Zhou & Tamma (2004).
MAXTOL [double(1 x 1)] is the maximum tolerance of convergence of the
Full Newton Raphson method for numerical computation of
acceleration.
JMAX [double(1 x 1)] is the maximum number of iterations per
increment. If JMAX=0 then iterations are not performed and the
MAXTOL parameter is not taken into account.
DAK [double(1 x 1)] is the infinitesimal acceleration for the
calculation of the derivetive required for the convergence of the
Newton-Raphson iteration.
Output parameters
U [double(1 x 1:NumSteps)] is the time-history of displacement
UT [double(1 x 1:NumSteps)] is the time-history of velocity
UTT [double(1 x 1:NumSteps)] is the time-history of acceleration
FS [double(1 x 1:NumSteps)] is the time-history of the internal
force of the structure analysed.
EY [double(1 x 1:NumSteps)] is the time history of the sum of the
energy dissipated by yielding during each time step and the
recoverable strain energy of the system (incremental).
cumsum(EY)-ES gives the time history of the total energy
dissipated by yielding from the start of the dynamic analysis.
ES [double(1 x 1:NumSteps)] is the time-history of the recoverable
strain energy of the system (total and not incremental).
ED [double(1 x 1:NumSteps)] is the time-history of the energy
dissipated by viscoelastic damping during each time step
(incremental). cumsum(ED) gives the time history of the total
energy dissipated from the start of the dynamic analysis.
JITER [double(1 x 1:NumSteps)] is the iterations per increment
Notation in the code
u=displacement
un=displacement after increment n
ut=velocity
utn=velocity after increment n
utt=acceleration
uttn=acceleration after increment n
__________________________________________________________________________
Copyright (c) 2018-2023
George Papazafeiropoulos
Major, Infrastructure Engineer, Hellenic Air Force
Civil Engineer, M.Sc., Ph.D.
Email: gpapazafeiropoulos@yahoo.gr
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