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Journal of Theoretical and Applied Mechanics, Sofia, Vol. 48 No. 1 (2018) pp. 46-58
DOI: 10.2478/jtam-2018-0004
NUMERICAL SOLUTIONS FOR NONLINEAR HIGH DAMPING
RUBBER BEARING ISOLATORS: NEWMARK’S METHOD WITH
NETWON-RAPHSON ITERATION REVISITED
A.A. MAR KOU1∗, G.D. MANOLIS2
1Norwegian Geotechnical Institute, Sognsveien 72, Oslo 0806, Norway
2Department of Civil Engineering, Aristotle University Panepistimioupolis,
Thessaloniki 54124, Greece
[Received 10 October 2017. Accepted 04 December 2017]
ABS TRACT: Numerical methods for the solution of dynamical problems in
engineering go back to 1950. The most famous and widely-used time step-
ping algorithm was developed by Newmark in 1959. In the present study, for
the first time, the Newmark algorithm is developed for the case of the trilin-
ear hysteretic model, a model that was used to describe the shear behaviour of
high damping rubber bearings. This model is calibrated against free-vibration
field tests implemented on a hybrid base isolated building, namely the Solarino
project in Italy, as well as against laboratory experiments. A single-degree-of-
freedom system is used to describe the behaviour of a low-rise building isolated
with a hybrid system comprising high damping rubber bearings and low fric-
tion sliding bearings. The behaviour of the high damping rubber bearings is
simulated by the trilinear hysteretic model, while the description of the be-
haviour of the low friction sliding bearings is modeled by a linear Coulomb
friction model. In order to prove the effectiveness of the numerical method
we compare the analytically solved trilinear hysteretic model calibrated from
free-vibration field tests (Solarino project) against the same model solved with
the Newmark method with Netwon-Raphson iteration. Almost perfect agree-
ment is observed between the semi-analytical solution and the fully numerical
solution with Newmark’s time integration algorithm. This will allow for ex-
tension of the trilinear mechanical models to bidirectional horizontal motion,
to time-varying vertical loads, to multi-degree-of-freedom-systems, as well to
generalized models connected in parallel, where only numerical solutions are
possible.
KEY WORDS: High damping rubber bearings; mechanical models; trilinear
hysteretic model; shear behaviour; nonlinear response; base isolation; New-
mark’s method.
∗Corresponding author e-mail: athanasiosmarkou@gmail.com
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Numerical solutions for nonlinear HDRBs 47
1. INTRODUCTION
The first time stepping algorithm was constructed by Houbolt [1] within the context
of solving for the dynamic response of aircraft frames using matrix methods of anal-
ysis. It was basically nothing more than a simple central difference method, whereby
velocities and accelerations were written in terms of displacements computed in the
immediately previous time steps. Within the next few years, Newmark [2] published
his landmark paper on what become Newmark’s method for the transient response
of buildings to earthquake-induced loads. His method was versatile because there
were two artificial parameters (β, γ ) that could be ‘tweaked’ to produce time step-
ping algorithms with different properties. For instance, there was a range of values
that yielded an unconditionally stable numerical integration algorithm, not to mention
that the central difference algorithm could also be recovered as a special case. Since
then, the terminology ‘Newmark-beta’ has been used to demarcate what are explicit
algorithms, in contrast to the newer development of implicit algorithms that over-
shoot the immediately next time step and then move backwards in order to correct it.
A comprehensive review of the truly explosive amount of work done on deriving a
plethora of numerical integration algorithms in structural mechanics as early as the
1980’s can be found in the edited book by Belytschko and Hughes [3].
In here, we focus on the nonlinear response of a class of ‘soft’ base isolators
for earthquake protection of conventional buildings known as high damping rubber
bearings (HDRB). These are becoming widely popular in Europe and there are cur-
rently two large firms that mass produce them according to specifications set by the
client. The same holds true in Japan, while in the United States a different category
of base isolators known as friction pendulums are popular. Recently, the authors
have developed a generalized trilinear hysteretic model (THM) that best describes
the mechanical response of HDRB specimens under harmonic excitation of variable
frequency that leads to induced strains of up to 200% [4]. In contrast with the basic
THM and its predecessor, the bilinear hysteretic model (BHM), which can be solved
analytically, this is no longer the case with the generalized THM. Therefore, it is im-
perative to develop variants of the Newmark-beta method that are specially tailored
for highly nonlinear response of a single-degree-of freedom representation. This al-
gorithm needs to be calibrated against experimental results so that its accuracy can
be established.
More specifically, in the next section, we present the physical model developed
to describe the behaviour of a low-rise building, namely the Solarino building [5],
isolated with hybrid base isolation system (HBIS). Next, the mechanical models for
the isolation system are presented. In the following section, the Newmark method
with the Newton-Raphson iteration scheme is presented for the case of the THM,
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48 A.A. Markou, G.D. Manolis
while in the subsequent section, the comparison between the numerical and semi-
analytical solution is shown and finally in the last section the conclusions are drawn.
2. PHY SI CA L MO DE L
In 2004 in Solarino, Eastern Sicily two reinforced concrete buildings were retrofitted
by using a HBIS and subsequently tested under free vibration excitation, where the
accelerations of the structure were recorded, [5], see Fig. 1. The hybrid isolation
system comprises HDRB with low friction sliding bearings (LFSB), see Fig. 2. The
recorded accelerations were used in order the physical parameters of the system to be
identified, [6]. Specifically, the physical model of the structure is a single-degree-of-
freedom (SDOF) system describing the behaviour of the Solarino building is shown in
Fig. 3. THM describes the shear behaviour of HDRB (see Fig. 4), a linear Coulomb
friction model (LDFM) describes the shear behaviour of LFSB (see Fig. 5) and a
linear viscous damping (LVD) accounts for any additional source of damping.
Table 1. Physical parameters of the SDOF system of the Solarino building
HDRB LDSB LVD Mass
kekh1kh2fsucffkfc m
[kN/m] [kN/m] [kN/m] [kN] [m] [kN] [kN/m] [kNs/m] [kNs2/m]
14770 21920 13745 172 0.0285 34 35 197 1284
Fig. 1. The Solarino building.
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Numerical solutions for nonlinear HDRBs 49
Fig. 2. (a) High damping rubber bearing (HDRB) and (b) low friction sliding bearing (LFSB)
of the Solarino building.
Fig. 3. Single-degree-of-freedom (SDOF) system of the Solarino building.
The compatibility, equilibrium and constitutive equations of THM are presented
in Table 2, where fTdenotes the force in THM. The constitutive equation of the
LCFM is given by the following expression:
(2.1) fF= (ff+kf|u|)sign( ˙u),
where fFdenotes the force in the LCFM, udenotes the displacement, ˙uthe velocity,
ffdenotes the force at zero displacement and kfthe slope, see Fig. 5. The rela-
tionships between the mechanical parameters shown in Fig. 4(a) and (e) of the THM
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50 A.A. Markou, G.D. Manolis
Fig. 4. Trilinear hysteretic model (THM) for the description of the shear behaviour of high
damping rubber bearings: (a) mechanical model, (b) force-displacement graph of element 1,
(c) force-displacement graph of element 2, (d) force-displacement graph of element 3 and (e)
force-displacement graph of THM.
Table 2. Compatibility, equilibrium and constitutive equations of THM
Compatibility u=ue+uh
Equilibrium fT=fe1=fe2+fe3
Constitutive law
fe1=keue
fe2( ˙uh6= 0) = fssgn( ˙uh)
fe2( ˙uh= 0) = fe1−fe3
fe3(|uh| ≤ uc) = kh1uh
fe3(|uh|> uc) = (kh1uc+kh2(|uh| − uc))sgn(uh)
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Numerical solutions for nonlinear HDRBs 51
Table 3. Relationships between mechanical and mathematical parameters of the THM, see
Figs. 4(a),(e)
ke=k0;kh1=k1
k0
k0−k1
;kh2=k2
k0
k0−k2
;fs=k0uy;uc= (u3−uy)k0−k1
k0
are given in Table 3. The constitutive equation of the LVD is given by the following
expression:
(2.2) fV=c˙u ,
where fVdenotes the force in the LVD and cdenotes the damping coefficient.
Fig. 5. Linear Coulomb friction model (LCFM) for the description of the shear behaviour
of low friction sliding bearings: (a) mechanical model and (b) force-displacement graph of
LCFM.
3. NEW MA RK ’S METHOD WITH NEW TON-RAPHON ITERATION
Newmark’s method with Newton-Raphson iteration belongs to the group of implicit
time-stepping procedures for nonlinear systems that are based on assumed variation
of the acceleration, see [7]. The original formulation assumes either a constant aver-
age or a linear variation of the acceleration within a time increment ∆t. The general
form of the method is based on a Taylor series expansion of the velocities and dis-
placements, respectively, at time step i+ 1 in terms of the accelerations as follows:
(3.1) ˙ui+1 = ˙ui+ [(1 −γ)∆t] ¨ui+ (γ∆t) ¨ui+1 ,
(3.2) ui+1 =ui+ (∆t) ˙ui+(0.5−β)(∆t)2¨ui+β(∆t)2¨ui+1 .
For the constant average acceleration method, the parameters introduced by New-
mark are set as γ=1
2and β=1
4, while for the linear acceleration method these
are set as γ=1
2and β=1
6. The above equations combined with the equilibrium
equation at the end of the time step iwill permit us to calculate the revised values
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52 A.A. Markou, G.D. Manolis
ui+1,˙ui+1,¨ui+1 . In particular, the equilibrium equation of the SDOF system of the
Solarino building can be written as follows:
(3.3) m¨u+fV+fT+fF=p ,
where pdenotes the external force. Equation 3.3 can be written in incremental form
(3.4) m¨ui+1 +c˙ui+1 + (fT)i+1 + (fF)i+1 =pi+1 .
Solving Eq. 3.1 for ¨ui+1 , we get a relationship with ui+1
(3.5) ¨ui+1 =1
β(∆t)2(ui+1 −ui)−1
β∆t˙ui−1
2β−1¨ui.
Now substituting Eq. 3.5 to Eq. 3.1, we get
(3.6) ˙ui+1 =γ
β∆t(ui+1 −ui) + 1−γ
β˙ui+ ∆t1−γ
2β¨ui.
Equation 3.4 can also be written as
(3.7) (fS)i+1 =pi+1 ,
where
(3.8) (fS)i+1 =m¨ui+1 +c˙ui+1 + (fT)i+1 + (fF)i+1 .
By using the Taylor series expansion to expand the force (fS)i+1 about the displace-
ment ui+1, which is unknown, and by dropping the higher-order terms, we get
(3.9) (fS)(j+1)
i+1 '(fS)(j)
i+1 +(∂fS)(j)
i+1
∂ui+1
∆u(j)=pi+1 .
By differentiating Eq. 3.8 with respect to the unknown displacement u(j)
i+1, we get
(3.10) ∂(fS)i+1
∂ui+1
=m∂¨ui+1
∂ui+1
+c∂˙ui+1
∂ui+1
+∂(fT)i+1
∂ui+1
+∂(fF)i+1
∂ui+1
.
The derivative of ¨ucan be calculated from Eq. 3.5 and the derivative of ˙ucan be
calculated from Eq. 3.6 we recover the relations
(3.11) ∂¨ui+1
∂ui+1
=1
β(∆t)2;∂˙ui+1
∂ui+1
=γ
β∆t.
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Numerical solutions for nonlinear HDRBs 53
Next, from the above two equations, we get the tangent stiffness of the SDOF as
(3.12) (kS)(j)
i+1 =1
β(∆t)2m+γ
β∆tc+ (kT)(j)
i+1 + (kF)(j)
i+1 ,
where
(3.13) (kT)(j)
i+1 =∂(fT)i+1
∂ui+1 (j)
; (kF)(j)
i+1 =∂(fF)i+1
∂ui+1 (j)
.
Note that kTcan be set equal to k0, which is the largest stiffness in the THM, see
Fig. 4, while kFcan be set equal to kf, see Fig. 5. Equation 3.9 can be written as
(3.14) (kS)(j)
i+1∆u(j)=pi+1 −(fS)(j)
i+1 ≡R(j)
i+1 .
Substituting Eq. 3.5, 3.6 in Eq. 3.8 and then in Eq. 3.13, we get
(3.15) R(j)
i+1 =pi+1 −(fT)(j)
i+1 −(fF)(j)
i+1 −1
β(∆t)2m+γ
β∆tcu(j)
i+1 −ui
+1
β∆tm+γ
β−1c˙ui+ 1
2β−1m+ ∆tγ
2β−1c¨ui.
The whole procedure of programming the process of the Newmark method is
presented in Table 4 by following [7].
In order to define the force in THM, upper and lower limit bounds need to be
established, depending whether the velocity is positive or negative, see Fig. 6. When
the velocity is positive the force cannot be larger than fup
lim, while when the velocity
is negative the force cannot be smaller than flo
lim. In Appendix the pseudo-code for
the definition of fup
lim and flo
lim is presented.
Fig. 6. Upper and lower limit forces for THM.
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54 A.A. Markou, G.D. Manolis
Table 4. Newmark method: for the SDOF system of Solarino building, following [7]
Special cases
(1) Constant average acceleration method γ=1
2, β =1
4
(2) Linear acceleration method (γ= 1/2, β = 1/6)
1.0 Initial conditions
1.1 Check whether |pi|> ff⇒initiation of motion, set i= 0.
1.2 State determination (fT)0,(kT)0,(fF)0and (kF)0.
1.3 Solve m¨u0=p0−cu0−(kT)0u0−(kF)0u0.
1.4 Select ∆t.
1.5 a1=1
β(∆t)2m+γ
β∆tc;
a2=1
β∆tm+γ
β−1c;
a3=1
2β−1m+ ∆tγ
2β−1c.
2.0 Calculations for each time instant
2.1 Initialize j= 1,
u(j)
i+1 =ui,(fT)(j)
i+1 = (fT)(j)
i,(fF)(j)
i+1 = (fF)(j)
i,
(kT)(j)
i+1 = (kT)(j)
i,(kF)(j)
i+1 = (kF)(j)
i.
2.2 ˆpi+1 =pi+1 +a1ui+a2˙ui+a3¨ui.
3.0 For each iteration j= 1,2,3, . . .
3.1 R(j)
i+1 = ˆpi+1 −a1u(j)
i+1 −(fT)(j)
i+1 −(fF)(j)
i+1.
3.2 Check convergence;
if the acceptance criteria are not satisfied, implement steps 3.3 to 3.7;
otherwise go directly to step 4.0.
3.3 (kS)(j)
i+1 =a1+ (kT)(j)
i+1 + (kF)(j)
i+1.
3.4 Solve (kS)(j)
i+1∆u(j)=R(j)
i+1 ⇒∆u(j).
3.5 u(j+1)
i+1 =u(j)
i+1 + ∆u(j).
3.6 State determination (fT)(j+1)
i+1 ,(kT)(j+1)
i+1 ,(fF)(j+1)
i+1 and quad (kF)(j+1)
i+1 .
3.7 Replace jby j+ 1 and repeat steps 3.1 to 3.6; denote final ui+1.
4.0 Calculation for velocity and acceleration
4.1 ˙ui+1 =γ
β∆t(ui+1 −ui) + 1−γ
β˙ui+ ∆t1−γ
2β¨ui.
4.2 ¨ui+1 =1
β(∆t)2(ui+1 −ui)−1
β∆t˙ui−1
2β−1¨ui.
5.0 Repetition for next time step.
Replace iby i+ 1 and
implement steps 2.0 to 4.0 for the next time step.
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Numerical solutions for nonlinear HDRBs 55
4. COMPARISON BETWEEN SEMI-ANALYTICAL AND NUMERICAL SOLUTION
The equation of motion, see Eq. 3.3, can be solved semi-analytically as it is shown
in [6]. In this section, a comparison between the semi-analytical and the numerical
solution, provided by the Newmark method, will be presented. The excitation pin
Eq. 3.3 will be
(4.1) p=−0.025mg sin(2πfrt),
where mis the mass of the structure (see Table 1), gis the gravitational acceleration
and fr= 0.41 Hz. It should also be noted that the time step used in the numerical
simulation is Dt = 0.005 sec. The results shown in Fig. 7 show that the results are
identical and the numerical solution is extremely accurate. The difference between
the maximum displacements in the steady-state stage of the excitation is 0.2 mm
(237 mm for the semi-analytical, 237.2 mm for the numerical). In Table 5 the time
required for the solution by the semi-analytical and the numerical solution is pre-
sented. Finally, the results show that the numerical solution requires almost double
implementation time compared to the semi-analytical one.
Table 5. Implementation time required for the solution with the semi-analytical solution and
the numerical one
Semi-analytical Numerical
[sec] [sec]
1.85 3.44
5. CONCLUSIONS
For the first time, the numerical solution for the THM model is presented by using
the Newmark method. The numerical solution is compared with the semi-analytical
solution, provided by [6], for the SDOF system of the Solarino building. The results
show that the agreement between numerical and semi-analytical solution is practi-
cally identical. The disadvantage for the numerical solution is that it requires almost
double time to solve the problem. Most importantly, the numerical solution can be
used for extension of the trilinear mechanical models to bidirectional horizontal mo-
tion, to time-varying vertical loads, to multi-degree-of-freedom-systems as well to
generalized models connected in parallel, see [4], where only numerical solutions are
possible.
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56 A.A. Markou, G.D. Manolis
-300 -200 -100 0 100 200 300
u (mm)
-2000
-1000
0
1000
2000
fT (kN)
Numerical Analytical
(a)
0 20 40 60 80
t (s)
-300
-200
-100
0
100
200
300
u (mm)
Numerical Analytical
(b)
-300 -200 -100 0 100 200 300
u (mm)
-50
0
50
fF (kN)
Numerical Analytical
(c)
0 20 40 60 80
t (s)
-1000
-500
0
500
1000
v (mm/s)
Numerical Analytical
(d)
-300 -200 -100 0 100 200 300
u (mm)
-150
-100
-50
0
50
100
150
fV (kN)
Numerical Analytical
(e)
0 20 40 60 80
t (s)
-0.2
-0.1
0
0.1
0.2
a (g)
Numerical Analytical
(f)
Fig. 7. (a) Force-displacement graph of THM, (b) displacement-time graph of SDOF system,
(c) force-displacement graph of LCFM, (d) velocity-time graph of the SDOF system, (e)
Force-displacement graph of LVD, (f) acceleration-time graph of the SDOF system under
harmonic excitation.
ACKNOWLEDGMENT
The authors wish to acknowledge financial support from Horizon 2020 MSCA-RISE-
2015 project No. 691213 entitled “Experimental Computational Hybrid Assessment
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Numerical solutions for nonlinear HDRBs 57
of Natural Gas pipelines Exposed to seismic RISK” (EXCHANGE-RISK), Assoc.
Prof. A. Sextos, Principal Investigator.
REFERENCES
[1] HO UB OLT, J. C. A Recurrence Matrix Solution for the Dynamic Response of Elastic
Aircraft. J. Aeronaut. Sci.,17 (1950), 540-550.
[2] NE WM ARK, N. M. A Method of Computation for Structural Dynamics. ASCE J. En-
grg Mech. Div., 85 (1959), No. 3, 67-94.
[3] BE LYTS CH KO, T., T. J. R. HUG HE S. (editors), Computational Methods for Transient
Analysis, North Holland, Amsterdam, 1983.
[4] MA RKO U, A . A., G. D . MA NO LI S. Mechanical Formulations for Bilinear and Tri-
linear Hysteretic Models used in Base Isolators. B. Earthq. Eng., 14 (2016), No. 12,
3591-3611.
[5] OL IVETO , G., M. GRANATA, G. BUDA, P. SCIACCA. Preliminary Results from Full-
scale Free Vibration Tests on a Four Story Reinforced Concrete Building after Seismic
Rehabilitation by Base Isolation, In: JSSI 10th Anniversary Symposium on Perfor-
mance of Response Controlled Buildings, Japan, Yokohama, 2004.
[6] MA RKO U, A. A., G. OL IV ETO , A. ATHAN ASIOU. Response Simulation of Hybrid
Base Isolation Systems under Earthquake Excitation. Soil Dyn. Earthq Eng.,84 (2016),
120-133.
[7] CH OPRA, A. K. Dynamics of Structures: Theory and Applications to Earthquake
Engineering, NJ, Prentice-Hall:Englewood Cliffs, 2012.
APPENDIX
In order to define the upper and lower limits of the force in the THM the conditions described
in the following pseudo-code (in Matlab/Octave) need to be applied. Note that ujt denotes
the current displacement amplitude.
% C a l c u l a t e f l i m u p −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
i f −uy h +2 ∗uy <= u j t && u j t <= uy h
i f uy h >=2∗uy
f l i m u p = Q + k1 ∗u j t ;
e l s e i f uyh<2∗uy
f l i m u p = Fy + k 1 ∗( u j t −uy ) ;
end
e l s e i f u j t >uyh
i f uy h >=2∗uy
f l i m u p = Q + k1 ∗uy h + k2 ∗( u j t −uyh ) ;
e l s e i f uyh<2∗uy
f l i m u p = Fy + k 1 ∗( uyh −u y ) + k2 ∗( u j t −uy h ) ;
end
e l s e i f u j t <−u yh + 2 ∗uy
i f uy h >=2∗uy
f l i m u p = Q + k 1 ∗(−uy h + 2∗u y ) + k2 ∗( u j t −(−uy h + 2∗uy ) ) ;
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58 A.A. Markou, G.D. Manolis
e l s e i f uyh<2∗uy
f l i m u p = Fy −k 1 ∗( uyh −uy ) + k2 ∗( u j t −(−uyh + 2 ∗uy ) ) ;
end
end
% C a l c u l a t e f l i m l o −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
i f −uyh <= u j t && u j t <= uy h −2∗uy
i f uy h >=2∗uy
flimlo = −Q + k 1 ∗u j t ;
e l s e i f uyh<2∗uy
flimlo = −F y + k 1 ∗( u j t + uy ) ;
end
e l s e i f u yh−2∗uy <ujt
i f uy h >=2∗uy
flimlo = −Q + k 1 ∗( uyh −2∗uy ) + k2 ∗( u j t −( uy h −2∗u y ) ) ;
e l s e i f u yh<2∗uy
flimlo = −F y + k 1 ∗( uyh −u y ) + k2 ∗( u j t −( uy h −2∗u y ) ) ;
end
e l s e i f u j t <−uyh
i f uy h >=2∗uy
flimlo = −Q−k1 ∗uy h + k2 ∗( u j t + u yh ) ;
e l s e i f u yh<2∗uy
flimlo = −Fy −k1 ∗( u yh −uy ) + k 2 ∗( u j t + u yh ) ;
end
end
%−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
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