Simplified computational
method
for non-linear
seismic
analysis
of bridges
F.J. Montans E. Alarcon
SUMMARY - After the experience gained during the past years it seems clear that nonlinear analysis of bridges are very
impor-tant to compute ductility demands and to localize potential hing-es. This is specially true for irregular bridges in which it is not clear weather or not it is possible to use a linear computation followed by a correction using a behaviour factor. To simplify
the numerical effort several approximate methods have been proposed. Among them, the so-called Dynamic Plastic Hinge
Method in which an evolutionary shape function is used to re-duce the structure to a single degree of freedom system seems to mantein a good balance between accuracy and simplicity. This paper presents results obtained in a parametric study conducted
under the auspicies of PREC-8 european research program.
Simplified
ods; Seismic response.
1. Introduction and objectives
In Dynamics of Structures the selection of the
com-putational Method impinges directly on the accuracy and cost of a Dynamic Analysis. This is specially true in Seismic Engineering where it is ussually accepted that a certain amount of damage can happen; for
in-stance damage is accepted in bridges if after the shock the structure can sustain the emergency traffic. That implies the use of nonlinear analysis.
The computational methods generally in use are: Equivalent static procedures, modal superposition or Spectral analysis, step by step integration in the time domain.
The advantage of the first group of procedures is their simplicity and this is why they were adopted in Seismic Codes. The main disadvantage is related to the approximate character of their results and as a
conse-quence, they are randomly distributed around true val-ues.
The developments in computer hardware and soft-ware, as well as those related to the rationalization of spectra have contributed to the increase of modal com-putations that, being generally based on a linear ap-proach combined with rules related to mode truncation and combination as well as to a global non-linear be-haviour through ductility factors, can only be used in
structures relatively regular where non linear effects are well distributed. They can not describe the evolution of the structure along with the seismic action what in
some occassions can be crucial, for instance when there are zones with different ductilities or, more often, when it is neccessary to make estimations of the ductility demand.
All those difficulties are overpassed thanks to the step by step methods which due to the effort needed are only used for special structures.
For bridges, in addition to the occasion in which an individual bridge has to be carefully studied, there ex-ists the possible need of a repetitive study on a set of bridges to analize the need of retrofitting and to iden-tify potential areas in which a relatively fine study is needed. In those situations one is confronted with the need of reducing the duration and cost of the analysis what proscribes the use of a very complicated model.
This is why this paper is dedicated to study the
pos-sible advantages of a simplified method that could combine the advantages of the step by step methods with fast and simple computations easy to follow by practising engineers and useful for parametric studies in order to analize the effects of the different retrofitting measures.
The motivating ideas was published as an Annex to one of the drafts of EUROCOCE 8 Part II (ref 1) al-though the origins can be found in the famous Bigg's book (ref. 2). The computations shown here follow the philosophy of that approach although some
-tions or alternatives are introduced in order to improve
their performances.
In the end of the paper several examples on regular
and irregular bridges are used to show the possibilities
and limitations of the method.
2. The dynamic plastic hinge method for framed
structures
2.1. GENERAL SCHEME
It is intended to obtain the structural response using
a single degree of freedom system related to the main
vibration mode or to a static deformation collecting the
main contribution of the eigenmodes in the direction of
the seismic action.
The shape function is evolutionary in the sense that
plastic hinges can be formed during the earthquake and
therefore a new shape has to be taken to reproduce the
displacements.
The method is based on the following points
1) Using a evolutionary vibration period to represent
the dynamic response of the structure;
2) Assuming that the hinges can be produced only
in the end member sections and that the behavior is
perfectly elastoplastic;
3) Modifying the structural model as it is being
de-graded, changing the shape function, the stiffness and
obtaining the response as a combination of a sequence
of Equivalent Substituting Systems.
The moments in which the structure changes its
con-figuration are identified through its acceleration and
can be collected in a poligonal line that is called
«Mo-dal Load-Deflection Line (MLDL)».
As can be deduced from the previous lines the
anal-ysis is organized into three blocks (figure 1).
a) Computation of the MLDL identifying the
substi-tute systems through which the structure can pass until
its last configuration;
b) Transient step by step analysis of the equivalent
evolutionary one-degree-of-freedom system;
c) Postprocessing. Computation of stresses and
dis-placements at different points of the structure.
To obtain the equivalent systems (a) it is neccessary
to use three types of computations.
a,L Modal shape and frequency of vibration. This
has been done using two alternatives: by a classical
eigenvalue analysis choosing the mode with higher
participation factor and by Rayleigh method using the
static deformation of the structure subjected to a force
proportional to the mass distribution and acting in the
direction of the earthquake as a shape function.
a.2. Computation of the static forces produced in
every degreee of freedom by a unit acceleration so that
a multiple of the eigenvalue is obtained.
a.3. Computation of the forces and displacements at
different points of the structure. With the knowledge of
the plastic moments, it is easy to determine the
locali-zation of the next plastic hinge.
Once the new hinge is produced the system will
change its behavior increasing the displacements but
manteining the level of efforts at the yielded section.
The new configuration incorporating the new hinge is
the Substitute System with which the analysis will be
continued.
Going back to point a.l. the behaviour of the new
system is computed until all pieces of the MLDL curve
have been obtained. After that it is only neccessary to
use an step by step integration method to obtain the
response of the equivalent single-degree-of-freedom
system. In that respect it is interesting to localize the
stiffness changes in the exact form by using a
consist-ent evconsist-ent localization technique (CELT), as described
MODEL OF THE STRUCTURE
EQUIVALENT SINGLE DEGREE OF FREEDOM SYSTEM
EQ. STATIC
FORCE PER UNIT MASS
MODAL-LOAD-DEFLECTION-LINE
EACH LINE REPRESENTS THE BEHAVIOUR OF EACH EQUIVALENT SUBSTITUTE SYSTEM
MODIFICATION OF THE STRUCTURE, BY INCLUDING A
NEW HINGE
711 \
YES
DETERM. OF THE DISPL. AND STRESSES IN THE
NEXT PLASTIC HINGE
MODAL DISPL NORMALIZED BY
THE PARTIC. FACTOR
%
TRANSIENT ANALYSIS OF A SDOF
STEP INT.
NO
END OF ANALYSIS
NO
YES
S THERE A CHANGE OF SUBSTITUTE
SYSTEM ?
\K
YES
COMPUTATION OF THE SUBSTEP _
AND CHANGE OF PARAMETERS
Y
POSTPROCESS
for instance in ref. 5, to detect either the unloading or the formation of a new plastic hinge (overloading).
Finally the post processing (c) does not present any special difficulty because the response in terms of a generalized one is already known as it is the equivalent
substitute system, that allows the use of typical tech-niques of matrix analysis for the computation of mem-ber forces and displacements, as well as the rotation of every plastic hinge.
In particular the knowledge of the plastic hinge rota-tions allows the knowledge of local ductility demands
and the computation of several damage measures.
2.2. STRUCTURAL MODELLING
To model the structure one can considerer the fol-lowing statements
a) The stiffness matrix for a static analysis is «ex-act» and independent of the discretization;
b) The lumped mass matrix depends on the discreti-zation and improves with finer meshes;
c) The computational time and the memory required increase slightly with the mesh refinement because the transient analysis is independent of the number of ele-ments, although the establishment of the equivalent subtitute systems is mesh-dependent. It is worthwhile to point out that the use of a Rayleigh method in place of a modal analysis is more convenient in terms of com-putational time and memory requirements.
(where £ is the damping ratio o> the eingenfrequency:
O) 2
0T K0
0T m0
and T the participation factor)
0 TmJ
r
0T 1710
(5)
is required to approximately represent the structural response.
In Rayleigh method, the starting point is relationship 3 where 0 is a reasonable estimate of the response that in our case will be obtained by loading the structure proportionally to the mass distribution in the direction of the earthquake.
To avoid the influence of different configurations it is advisable to write equation (4) as
ft\
r
/ L \
+ 2£o>
v A J jj
+ (0 X • * (6)
where — represents displacements continuous from a
substitute system to other one. In addition the grouping.
f>\
j ,
* *
+ x
r
t\
+ 2£o>/
r
0) 2\
V " J
r
(7)
V 1 J
2.3. COMPUTATION OF THE EQUIVALENT SUSBTITUTE SYSTEMS
To establish the curve describing the behaviour of the equivalent 1 d.o.f. it is convenient to start with the equilibrium equation of the n d.o.f. structure
allows the interpretation of the right hand side as a pseudoaceleration as plotted in figure 2. The abscissa
is the generalized coordinate and the vertical axis
represents the pseudoaceleration or equivalent static force per unit mass.
mx+ cx+kx mJx (1)
where m, c and k are respectively the mass, damping and stiffness matrices. The vectors and x are the relative displacement, velocity and acceleration vectors of the structural d.o.f., J is the influence vector result-ing from the projection of the d.o.f. on the earthquake
direction and xs(t) the earthquake accelerogram. As it
is well know the modal projection
x =
*!>£
(2)(where I|J is the matrix which columns are the eigen-vectors and £ are the modal coordinates) produces in the case of classical damping, an uncoupled system. If,
in addition a certain mode 0 is predominant, it is posi-ble to write
x - 4 0 (3)
and only the following relationship
H
4 h
SA = (-p-) +2CW (± X,
SA> = SAi +A SAi
SAJ = S A I + A S A J
A
-o A
(f)
>
M
Rd
c
y Reinforced Concrete
c
uc
Moment - Curvature of cross - section
where N i - I V i-l M i- 1 D i-1 are the forces and
dis-plecements when the substituting system (i-l) was left and ni, vi, mi, di the efforts and displecements for a unit acceleration in the substituting system (i)
b) As the pseudoaccelerations at the starting point ef every substituting system are
f y \ / ; \
ASA = &x„ +
V
r
+ 2£a)
/
v ^
(9)
it is seen that it depends on the value taken for £. At one point of MLDL curve before the formation of the plastic hinge i it is possible to write
/ i: A
£A<-> Axs + A
r
ir i- \
+ 2^o)' A
*
V A /
vr.
+ SA, (10)
Fig. 3 - Elastoplastic behavior of the sections.
Every straight line will be valid until the response includes a new plastic hinge.
From here on the member forces can be computed using the equivalent static forces in static analysis and fixing an upper-threshold «a posteriori» by a
conven-ient scaling.
2.3.c. CHANGE OF THE STRUCTURAL MODEL
Once the plastic moment of a section has been reached the structural response changes due to the
for-mation of a hinge for the incremental loads. In the hereby presented approach the behavior of the sections is assumed to be perfectly elasto-plastic. Then, it is neccesary to modify the stiffness matrix of the member in which the hinge is formed using techniques well known in matrix structural analysis.
The MLDL is obtained for every situation as shown in figure 2. If is assumed that the curve is symmetric with respect to the origin, what implies, among other things, symmetric reinforcement of concrete sections.
Despite of other ones, it is possible to make the fol-lowing considerations
a) The straight line representing the behaviour of the substituting system (n) starts at the end point of system ( n - 1), and system (1) starts from the origin. Superpo-sition is applied in an incremental fashion, so that if
o / \ j9 0/1.2 ••• ^-**n
are the pseudoaccelerations for which the plastic hinges 1,2, ... n appear the member forces and displacements for a load
SAi=l SA* < SAi
will be
Axial force N* Shear force V*
*
nl (SA
V (SA
'i- 1
*
SAt x) + Nl
• 5 A / . 1 ) + V - 1
Bending Moment M* m( (SA *
Displecement D* dl (SA * SA
: 1 —
SA; j) + M i) + r»-1
i - 1 (8)
where tj, o>', P are the damping ratio, natural frequency
and participation factor for system i; and Axs,
r y\
A
r
/r 1 \
and A
v / \~
r
J 1are the increments of soil acceleration and acceleration and velocities since the formation of the last plastic hinge.
Then, it is seen that different damping rations can be chosen for different configurations as an alternative to the usual assumption of a constant damping ratio throughout the whole time interval.
2.3.d. COMPUTATION OF THE MAIN EIGENVALUE AND
EI-GENVECTOR
As can be deduced from the last paragraphs, one key factor of the procedure is the selection of a shape
func-tion 0 (eq. 3) - Two ways have been chosen to do so: An eigenvalue analysis by the subspace iteration mode selecting the one with higher mobilized mass,
what is repeated for every substituting system.
A Rayleigh method using the static deflections pro-duced by a load proportional to the mass matrix acting
along the earthquake direction.
It is founded that the last method is more convenient from a computational viewpoint and, as the parametric
study has shown, the results are comparable and in occasions better than using the main mode.
The steps needed to determine the load-displacement curve (MLDL line) can be summarized in the following points.
a) An eingevalue analysis of the configuration (which includes the existing plastic hinges) or a static analysis is conducted to determine the selected shape function 0 and its associated value
a> 2 0
Tk0
b) The equivalent static forces per unit acceleration are computed using the shape vector 0;
c) A static analysis under the above mentioned forc-es is conducted;
d) The response acceleration of the structure capable of producing the next plastic hinge is computed. The new points of the MLDL line will be
A
n
I+
ASA,O) 2 »
i
(12)
SA n SA; i- 1 + ASA,- (13)
e) Using the previous acceleration new member forces and displacements are computed;
f) The new substituting system is modeled;
g) If the new system is not a mechanism the process is repeated. In our case the deck is assumed to remain in the elastic range so the procedure is terminated when all the possible plastic hinges have developed in the piers.
2.4. TRANSIENT ANALYSIS OF THE EQUIVALENT S.D.O.F
SYSTEM
As soon as the equivalent substitute systems have been determined it is possible to obtain the MLDL curve governing the response of the s.d.o.f. subjected to the action of a specific accelerogram. One of the ad-vantages of the method is the linear behaviour between events, so that a simple Newmark-jS method can be used along with the Consistent Event Location Tech-nique (CELT) described in ref. 5.
2.5. POSTPROCESSING. COMPUTATION OF DISPLECEMENTS, ROTATIONS AND MEMBER FORCES
S u b s t i t u t e s y s t e m s r e c o r d
3
1
r^LiMI
Equivalent single d.o.f
displacement record
S h a p e functions of
forces and d i s p l a c e m e n t s of the different
e q u i v a l e n t s u b s t i t u t e s y s t e m s
Fig. 4 - Postpreccesing.
p
o
s
T
p R
o
c
E
s
s
History of forces
rotations & displ.
in the diff. nodes
of the structure
2.6. IMPLEMENTATION OF THE METHOD IN A COMPUTER
PROGRAM
To develop a study in order to validate the method, a program called AROSA written in Fortran VMS has been developed. The global process is shown in Figure
5 and has four modules: Input, Computation of the MLDL line, transient analysis and postprocessing, all of them following the sheme developed in the previous points.
The transient analysis provides the response in
mo-dal coordinates, i.e.: the displacement histories
£,
r
i thepseudoaccelerations SA; and the equivalent system j valid at each instant are known. As shown in figure 4 it is and easy matter to obtain the time histories of dis-placements, member forces, etc: Once the configuration
/ (t) is known, following the previous notation it is possible to write
/
x(t) = I 0 A
r=0 ~i(«)
V>
vr,v
r,-„
(0N(t)= I n ASA(f)
t
M(t)= S m ASA(0 (14)
t
V(t)= I v ASA(t)
3. Examples
To analize the validity of the method several earth-quakes were simulated and the two alternatives above mentioned were checked one against each other and also against the results obtained using a classical Finite Element approach using the ANSYS commercial pro-gram. Several checks were also done using DRAIN 2D code.
The bridge models were those proposed by the Prenormative Research on Eurocode 8 (PREC-8)
rein-I MODEL DATA READrein-ING]
MODAL ANALYSIS
MODAL STATIC STATIC
ANALYSIS TRANSIENT
SUBSTITUTE SYSTEMS DETERMINA TION
MODAL RAYLEIGH
SUBSPACE METHOD RAYLEIGHMETHOD
T
MAIN EIGENVECTOR AND EIGENVALUE
T
STATIC DEFL AND MAIN EIGENVALUE
EQUIVALENT STATIC FORCES
STRUCTURAL FORCES AND DISPLACEMENTS
DETERMINATION OF THE PART OF THE STRUCTURE IN WHICH THE NEXTPLASTIC HINGE 1$ FORMED AND ITS
CORRESPONDING SOIL ACCELERATION
MODIFICATION OF THE STRUCTURE BY INCLUDING A NEW HINGE IN THAT LOCATION
NO
A MECHANISM FORMED?
" '
TRANSIENT ANALYSIS
SOIL ACCELERA TION DA TA READING
ANALYSI E STEP
NO
NO
YES
IS THERE
ANY CHANGE OF SUBSTITUTE SYSTEM BECAUSE OF A NEW PLASVC HINGE
OR UNLOAD
YES
counuJtOH OF r«£ uti
SUJSTEP innmxi
COMPUTA TION OF THl REMAINING SUBSTEP
CHANGE OF THE SYSTEM PROPERTIES
P6&TPRM&&
RECORDS AND PLOTS WITH STRUCTURAL FORCES,
ROTATIONS AND DISPLACEMENTS (MAX. AND VERSUS TIME)
Fig. 5 - Scheme of the implementation of the Dynamic Plastic Hinge Method (D.P.H.M.).
forcement in the bridge response, different degrees of reinforcement have been used in the models called 213 A, 213 B and 213 C. Bridge 213 A follows the usual criteria (0.50-0.92-0.50%); Bridge 213 B is used to
analize the increase of reinforcement in the central pier (0.50-1.69-0.50%) while mainteining the lateral ones and bridge 213 C is used to see the effect of lateral piers very reinforced and the central one with lower reinforcement (1.15-0.50-1.15%).
In the developed study, two groups of earthquakes were used; four historical ones
TER 1: ElCentro 1940
TER 4: San Fernando 1971
TER 5: Imperial Valley 1979 (James Rd.)
TER 6: Imperial Valley 1979 (Bonds Corner)
and seven artificial ones (ACC 1 to ACC 7) that were generated using the program SIMQKE and were com-patible with the EC-8-spectra used for the modal
anal-ysis.
The natural frequencies and mobilized masses of the first three modes can be seen in Table I.
It is interesting to see that the predominant part of the mobilized mass corresponds generally to one mode which shape can be approximated by a static loading
Bridge
232 213 A
213 B 213 C
MODE 1 Freq.
(Hz) 1.164 1.885 1.897 1.919
Mobiliz. mass 87.8% 12.0% 6.5% 30.0%
MODE 2 Freq.
(Hz) 3.953 2.263 2.394 2.276
Mobiliz. mass 10.8% 84.9% 89.9% 67.6%
MODE 3 Freq.
(Hz) 10.38 10.79 10.94 10.72
Mobiliz. mass 1.4%
2.8% 3.3% 2.4%
Table 1 - Eigenvalues and mobilized masses of the analyzed bridges
although only for that symmetric bridge the first mode is predominant.
To compare the answers obtained by different meth-ods the following magnitudes have been selected:
- maximum displacements & rotations;
- ductility demands at plastified sections; - dissipated energy at plastic hinges;
- member forces.
In the following figures the comparison between the results obtained using AROSA program (Dynamic plas-tic hinge method) and FEM ANSYS V5.0 is shown.
Figure 7.a shows the comparison between FEM and Rayleigh approach for the rotations at the plastic hinge in the central pier of bridge 232. It is seen that AROSA produces a very good approach of maxima, minima and general time-histories of thet measures in all piers.
In the other bridges the results are not so clear. Fig-ure 7.b. shows the comparison for bridge 213 A and earthquake ACCl. Although larger differences than for the symmetric hinges are detected, it is possible to
es-tablish the following facts related to the simplified method:
- Peak values are registered at exact time instants and the values fit in an acceptable way to the correct ones.
- In the intervals in which the differences are larger, the higher modes are excited by the accelerogram so
the proportional contribution of the main mode-shape is different. Nevertheless the amplification of those modes seems to contribute only slightly to the maximum
re-sponses.
BRIDGES ACCEL. RECORDS
ANALYZED RESULTS
DISPLACEMENTS AT EACH PIER TOP
P232
S
M
O ROTATIONS OF THE P U S
-TIC HINGES
O DUCTILITY DEMANDS
A213 B213 C213
Damping = 0 %
ACCl . . . ACC7 ENERGY DISIPATED BY ° THE PLASTIC HINGES
3
CO
MAXIMUM POWER DIS. ° BY THE PL. HINGES
A213 with ACCl
x 5.5 x7,5
x9.5
TER 1: ElCentro 1940
TER4: San Fernando 1971
TER5: Imperial Valley 1979 (J.R.) TER6: Imperial Valley 1979 (B.C.)
MAXIMUM & RECORDS
B8IDCE 232 - 21m PIER- ARTP. ACC. 1
x i o 3
2.00
1.00 .
Hi
o
s o.oo.
-i.oo .
-2.00 .
-3.00
C-AROSA-4. C-FE.M. I
•i~' i i ' t
0.00 0.50
T 1 1 I 1 I " 1 T™ ! ' • I ' T' '1 f 1 ' T- • I I | - r " t ' I I I
1.00 1.50 TIKE (s)
2.00 2.50 3-00 X 10
BRIDGE 213A - 7m PIER- ARTF. ACC. 1
x io2
n
1.00
UJ
a.
UJ 0.50 .
o
0.00
C-AROSA-4.
C-F-I.M. ;
-0.50 .
- 1 . 0 0
t » »
0.00
1 I r I I I I V I V' T I I
0.50 1.00 1-50
TIME (s)
r
2.00
"T"
2.50 3.00 X 10
Fig. 7 - Comparison Finite Element Method (FEM) and DPHM. Rota-tion at the base of the central pier of bridge 232 (above) on 213 A (be-low) under the artificial accelerogram ACC1.
- It can be said that when the plastification is
im-portant the differences are reduced.
Figure 8 shows the comparison of the maximum
dis-placements at the top of the piers while figure 9
col-lects the rotation in the plastic hinges. In abscisas we
represent the earthquake and in vertical axis the
maxi-mum values obtained. On those plots (printed with
continuous line for convenience) it is possible to see
that the differences among computational methods are
less than those due to the use of different
accelero-grams.
It is also interesting to see that bridge 213 C, that in
principle could be problematic because the used mode
mobilizes less mass than the others, presents a very
good accuracy in the rotation of the plastic hinge of the
central pier, and with conservative values.
The comparison of the ductility demands is
reflect-ed in figure 10 were we show their values are shown
both in displacements and in curvatures for the bridge
piers. Once again it is seen that the differences among
the different computational methods are less than those
due to the use of different accelerograms compatible
with the same response spectrum.
Ductility gives an idea of the maximum excursion in
the plastic range, but it does not reflect the number of
times of exceedance so it can be interesting to analize
the energy and power dissipated by the plastic hinge.
Figure 11 shows the comparison in terms of
dissipat-ed energy and although the differences between the
aproximated and the Finite Element method are larger
than in the previous measures the same observations
related to the use of different accelerograms apply.
Generaly the Dynamic Plastic Hinge Method
(DPHM) detects which earthquake is worst in energy
or power.
COMPARISON FEM.VS.AROSA
BRIDGE 213A - Maximum displacement
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TER1 EARTHQUAKE
TER4 TER5 TER6
FEM- 14m — *FEM-7m " ' " F€M-21m AflOSA-14m AROSA-7m - — AROSA-21m
COMPARISON FEM.VS.AROSA
BRIDGE B213 - Maximum displacement
14
« A . • 4 • 4 • * # »
10- —i <-• 4 +
+-fr
4
« *
•
4
*
•
*
« t 4
«
f
» «
4 4
•
•
4 * 4
1
4
* *
I
4 4
« « 4
4 4 4
4
i
f* *
/ :
t
«
<= ----**<-/-—1 -• • • k i L # * * * * *
t t
ACC1 ACC2 ACC3 ACC4 ACCS ACC6 ACC7 MEAN TER1 TER4 TER5 TER6 EARTHQUAKE
FEM-14m FEM7m AROSA-14m AROSA-7m
FEM-2lm
AROSA-2lm
COMPARISON FEM.vs.AROSA
BRIDGE 232 - Maximum displacement
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN EARTHQUAKE
TER1 TER4 TER5 TER6
COMPARISON FEM.vs.AROSA
BRIDGE A213 - Maximum pl.hinge rotation COMPARISON FEM.VS.AROSA BRIDGE B213 - Max. pi. hinge rotation
4 4 4 4 a * > * i i t * a i i * a a * • 4 a
•f * # a * * * « * * J » > a * '
4 a • • a a i a i +
t I
a t a i •
I
a a a i a • • i T 9 1 • i a a a a i i » a * * * a * a * 4 * 4 ft t * a a a a • t *» * * • * * * • i
t J t t i J • -a>. • * t a *
l ^ V » » » J * -I ...y
•«+•
a
• a
•
• • • > • • * » * i
t a
i s T * {
4 * - 4 * J 7 * # * * ^ f c » ^ » * f * » » • 4 a t * * • • l
a t a * a a • a l •i l i a
<• * L ,
a a a * a i • i k t a t a • • a I a 4 a * a « . . . • » • • *
• t-*Wr i
..^a
i.
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TER1 TER4 TER5 TER6 EARTHQUAKE
ACC1 ACC2 ACC3 ACC4 ACC5 ACC8 ACC7 MEAN TER1 TER4 TER5 TER6 EARTHQUAKE
FEM-14m — • FEM-7m
AROSA-7m ---AROSA-14m
FEM-21 m AR0SA-2lm
FEM-14m ™ F E M 7 m AflOSA-14m — AR06A-7m
FEM-21m AROSA-21m
COMPARISON FEM.VS.AROSA
BRIDGE C213 - Max. pi. hinge rotation
COMPARISON FEM.VS.AROSA
BRIDGE 232 - Max. pi. hinge rotation
25
^ 2 0
- 15- 10-• • a * a w I i i i t a a • * a a • i • • • a • a a t a t t V * i i a * a a a a a a a a • a 4 i • • a i • i a i i i a a a • * a t 4 a • » * • • t a a a a a * • * » a I t • > a i a • i • a i » a i a i a a • a I i t •F I I I I 4 f i I a
•4 * * « • • • • * a
I a
*t -fr.*» i^ **
a • t a » a i a • » i a l t r * a a t * a a a a
i f t j * f t j * * > * * *
I a a a a • a i t '*M•• • i t a i a a a t r • i i a r a • t > a a * > • • « * # • «
I a f • 4 I a a " a a t a a a a a • * \ — \
• 4 • 4» • » * • • » 4
• ' a
* / >
v s>
/ / ' :
/ - a a
t 9 # i
t a * J • • • * 1 • •
I i • • % * 4 « % h * * * 1 * & * • '
a a a
ACC1 ACC2 ACC3 ACC4 ACC6 ACC6 ACC7 MEAN TER1 TER4 TER5 TER6
EARTHQUAKE ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TERl TER4 TER5 TER6 EARTHQUAKE
FEM-I4m — - F£M7m AROSA-14m - — AROSA-7m
FEM-21m
AROSA-21 m FEM-14m « - F E M - 2 1 m AROSA-14m - - - AR0SA-21m
Fig. 9 - Comparison FEM-DPHM. Maximum rotation at the plastic hinges in all the analyzed cases.
COMPARISON FEM.VS.AROSA
BRIDGE A213 - Displ. ductility demand COMPARISON FEM.VS.AROSA BRIDGE B213 - Displ. ductility demand
14T2 - 10-a a a • V I t r r i a • a a • a a a • • » a »H a a a T t t 4 > V « t a a * I a a a a a a • * ' a a ( a a a a t a a a a a • a a 4 i a a i I a a i t * i » a a • i • * "
i • t 4 4 I i » t a a > 4 4 I a a a » • a t a r • • a a 9\
• * » • • * ' I * * ' a a a a a ; / ! •\-ft
• * * K . ^ * • a a /
-a V ^ -a t -a / * *
a a *• t a > a a a I a a a 1 *>• a a a a
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TER1 TER4 TER5 TER6
EARTHQUAKE ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TER1 TEB4 TER5 TCR6
EARTHQUAKE
FEM-14m — • FEM-7m FEM-21m
AROSA-14m---AROSA-7m AROSA-2lm FEM-14m — FEM-7m
AROSA.14m — AR06A-7m
FEM-2101
AROSA-21 m
COMPARISON FEM.VS.AROSA
BRIDGE C213 - Displ. ductility demand COMPARISON FEM.vs.AROSA BRIDGE P232 - Displ. ductility demand
14-1 a a I a I a i a 4 4 a a i t a * a a a a a a t 4 a 4 a a 4 4 1 4 • I I I I I
b ~ > 4 * * f * B * » « d f c « l - * 4 * * B
I I
4
1£^
**-^—**-8- - v
f 64
A-m > > • 4 * + » « t
2-•••1
t t t t t t t t
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN EARTHQUAKE
TER1 TER4 TER5 TER6
ACC1 ACC2 ACC3 ACC4
FEM-14m - - - FEM-7m
AROSA-14m — AROSA-7m
FEM-21 m AROSA-21 m
ACC5 ACC8 ACC7 MEAN TER1 EARTHQUAKE
TER4 TER5 TER6
FEM-14m FEM-21m AROSA-14IT1 - - - AR06A.2im
COMPARISON FEM.VS.AROSA
BRIDGE A213 - Dissipated energy
7-to B&4 • id —t • t 1 * * • 4< a I I * a 4 t • 4 *
h*# * t '
• 4 4 * ' a • I t t t i i : * a t * : * » t * • 4 * • 4 4 4 • • t t 4 4 i * 4 # 4 I f I 4 I
* < I I ' # * I
t
a
I I
t n
/ • • t t I t 1 1 1 I 4 I
• I I • • / • t l l t » ( » * * * * # • > » • , • • • • • * * » af» * • » a * * * * f " a * * * » • t • * • * • • * • * * •
• * i < 4 *
4 I I I • • *T * 4 t i 4 • • ' I *
4 • I • » • J - * • I I 1 • I * i 4 ^ 4 - • • ' / * *
% * • / I * • V * ^ • •*. • • • / * •
• \ • # * \ • ' *# J *
• v a j k f f • • # # • • 4 » » # * * » • a * a a a < t > * a a * a » * a a • • » * a a * > a » * * * * * * h * * * # - • * • • * > *
* • v * * * - r » * * • r J *
#* • ^ v V i » * • v* I i
• ^ f c » » • * • * •
* v ' ' L **1 ! / ! ' •
ACC1 ACC2 ACC3 ACC* ACC5 ACC6 ACC7 MEAN EARTHQUAKE
TER1 TER4 TER5 TER6
COMPARISON FEM.VS.AROSA
BRIDGE B213 - Dissipated energy
7 -i±L64 5-* ! • * * » * 4 : t r a * « i • 4 4 * • I B i
' • > • • * * • 4 ' * * # » # • • [ P * 4 a • t • i
• » a 4 * t * * a a *
• 1
k * * » * * # • * <
a • t i i i •r
/ : s
' ; P * * • • • * « • •
t I I I I • I 1 • f t t 4 a 1 • 4 4 « 4 r • • ! ' 1 a • 4 I • 4 4 1 1' ( I a a • • a a t * * * * • * - , »1 a * * * » B « a « B ^ B # * • * • * • « , * a * * « i
a
4 7:
• / « X I I i »
• j * * • J a
* * i r " * - * j y * » * * * • * * • • • • • » • * - f c * Y * * 4 » * # » * * * » * W # « * » * • • * * • • * • * • *
" ' 4 ^ 4 / 4 * a
1 * * \ * M * 1 *
• % • • * * * » * * * • * • * * * a « W « - - » * • »-4> » \ 4 * * p * V * • » * • • # * » * • • • • > « • * * • » * • * •
™ • fc < 4 * T * 1
• tj • *
a a t a > a a '§>** a a t * a a » * > * • .
t » 4 t I a a I
1 1 "
I * I ' a > i » : ' » : / / : « t t / ' '•
; *$ i
: / '
• • * • * • g » + • • • f * '
7 ^
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 K/EAN TER1 TER4 TER5 TER6
EARTHQUAKE
FEM-14m FEM-7m AROSA-14m - - - AR0SA-7m
COMPARISON FEM.VS.AROSA
BRIDGE C213 - Dissipated energy
»
6-i
I**
a m
a a
* * • * ^ p * • * ^ 4
I a a
i
i
tea » * •
'•*u • 4 > * * • * • * * * | t » * * 4 ' 4 ' 4 i * * f » >
- 4 - ! * * ' i ; l fV
i f a a a a
i i » a a a
1 4 a a » I
4 a • a * i
i I * * a V * * •
V . O . B . ^ V ^ . V - ^ / » a * ^ * . a • >\
: .»;» : t v •* ^ V !
7
; > *
a a a I a a a a a a a i l i i i
• • * * * * * t * * * * * i t t i • * * * * * ^ * * • • • i i i * a t a > a • t a 4 4 4 4 i. I I a i V i • a l a a * < a a a a a a
* * * * * * # 4 .
: : j *
a a ^ »
C I a
if ! • # • # • « • ! * * * * * v h * 4"*1
: f * i
t p > « 4 • • • • * » * > * • * ^r *f c*
4 f * I
4
**}• » * * • • B *44 ^
ACCl ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TER1 TER* TER5 TER6 EARTHQUAKE
FEM-14m - - * FEM-7m AROSA.14m - - - AR0SA-7m
SUMA SUM*
COMPARISON FEM.vsAROSA
BRIDGE P232 • Dissipated energy
4.5
4 • — > • * * *
-3^- — 3- 25- 2.—W-1.5- ••-V-1- —i UJ
0.5-> • * •
* * * • i a t p 4 a 4 » t a a i a a a 4 V * • * • w
ACC1 ACC2 ACC3 ACC4 ACC5 ACC6 ACC7 MEAN TERl TER4 TER5 TER6 EARTHQUAKE
FEM-Hm — FEM-7m AROSA-14m — - AROSA-7m
FEM-21m AROSA-21m
Fig. 11 - Comparison FEM-DPHM. Energy dissipated by plastic hinges in all the analyzed cases.
If the ductility demand is important the quantifica-tion of the dissipated energy by the DPHM is
conserv-ative.
Finally, it seems that there are no significant
differ-ences with the dissipated power predicted by both methods.
5. Acknowledgements
4. Conclusions
From the observed results and, at least, for the ana-lyzed tipology it is possible to conclude that
- The D.P.H.M. produces displacements, rotations and energy histories very close to that of more accurate methods specially if one considers the variability in-duced by the excitations;
- In relation to the sections with greater ductility demands or dissipated energy the DPHM is a
conserv-ative method;
- The DPHM produces realistic degradation mech-anisms that are very useful for a fast analysis in the case of seismic retroffiting. In those applications the
method can be recommended as an analytical tool to discriminate which bridges need a more in-depth study and in which it is possible to accept the indications on the damaged areas.
The authors should like to make explicit their indebt-eness with prof. Ramon del Cuvillo who pointed out the interest of the subject and has been continuously
supporting its developments, as well as to the engineers of the Spanish Ministry of Public Works, Mr Hinojosa and Mr. Elvira who promoted those studies.
The authors are also grateful to prof. Calvi from Pavla University that accepted to include the research
inside PREC 8 and was continuously helping its devel-opment.
Last but not least it has to be noticed that some parts of the works have been possible thanks to the help of the Direccion General de Investigacion Cietlfica y Tec-nica of the Spanish Ministry os Science & Education under grant Pb 93-0201.
6. References
[ 1 ] - : «Eurocode 8. Structures in Seismic Regions-De-sign. Part2-Bridges». Commission of the European
Communities. 1990 Draft.
[ 2 ] - : Biggs. J.: introduction to Structural Dynamics» Mc-Graw Hill 1964.
