Seismic Damage Prediction of Reinforced Concrete Building
Structures based on the Energy Balance Method
Jorge GRANADOS SOTO*,

Supervisor: Takashi HASEGAWA†

MEE06010
ABSTRACTThe Energy Balance Method is presented and applied to the analysis of reinforced concrete structures; theseismic demand is represented by the damage input energy which is directly related to the Pseudovelocity spectrum found in many of the current seismic codes, several factors that influence thedistribution of the energy in the structure, such as the damage concentration factor and the relationshipbetween ductility and cumulative plastic deformation, are studied in more depth, finally illustrative
examples of moment resisting frames and frames with hysteretic dampers analyzed by the Energy
Balance Method are presented; these examples show good agreement of the results when compared toother analysis methods that are widely adopted in many seismic codes such as the Capacity Spectrum
Method and the Nonlinear Time History Analysis.
Keywords: Energy Balance, Seismic Design, Reinforced Concrete Structures, Hysteretic Dampers.
INTRODUCTION
The earthquake resistant design methodology in most countries is regulated by the National Seismic Codeor Building law, most of these codes use force based design procedures, these procedures establish that inorder to calculate the force that an earthquake induces in the structure, a base shear needs to be calculated;
this shear is obtained by multiplying the weight of the structure by a seismic coefficient.
One fundamental issue that is not taken into account in current design practice is the earthquake
energy input to the structure, earthquake input energy accounts for both the ground motion characteristics(frequency content, intensity, and duration of strong motion) and structural properties (ductility, damping,period, and hysteretic behavior), whereas PGA, EPA, etc. account for some but not all thesecharacteristics.
The Concept of Energy has been studied by many researchers but the basic theory was proposedby G. Housner (1956) and later was developed and extended by H. Akiyama (1985), most of the work has
been done for steel structures, therefore more investigation into RC structures is needed.
This concept is based on the premise that both the demand of energy due to a ground motion andthe energy supply of a structure can be established, therefore a satisfactory design implies that the energy
supply must be larger than the energy demand.
This research is aimed towards the development of a methodology for the design and predictionof damage of reinforced concrete structures subjected to earthquakes according to the energy balance
theory, in order to achieve this objective several specific topics have to be studied such as, theapplicability of the energy spectrum to reinforced concrete structures using nonlinear models, theinfluence of the beam to column strength ratio in the damage concentration, the relationship between themaximum ductility μ and the average cumulative plastic deformation , and finally the response
prediction based on the energy balance method using analytical models.
*†

Structural Engineer, Arquitecture and Engineering Division, Caja Costarricense del Seguro Social
Senior Researcher, Building Research Institute1

THEORETICAL BACKGROUND
The equation of motion for a structure subjected to a horizontal ground motion at the base were M is thethe horizontal groundmass of the system, C the damping, F(u) the restoring force on the spring andacceleration, can be expressed in an energy form according to equation (1).
(1)
The term on the right hand side of equation (1), represents the total input energy E, on the left
side, the first term represent the kinetic energy Ek, the second is the energy consumed by damping Ed andthe third is the strain energy Es, the strain energy is composed of cumulative plastic strain energy Ep andelastic strain energy, the elastic strain energy and the kinetic energy represent the elastic vibration energyof the system Ee, Housner (1956) assumed that in elastic-plastic systems, the damping energy does notcontribute to the damage of the structure, therefore the following equation holds:(2)

It has been demonstrated that the energy input into a structure is a very stable parameter that is
exclusively dependant of the natural period and the weight of the system. The total energy E and theenergy attributable to damage ED can be converted into equivalent velocities as follows:(3)
Damage Distribution
The energy attributable to damage ED has to be absorbed by the structural elements without suffering
excessive damage in order to have a safe design, this damage energy has to be distributed to every storyof the structure, small changes in strength and/or stiffness between resisting elements in different stories
will cause an energy concentration in the weak portion of the structure, this concentration will lead toundesirable behavior of the structure and possible collapse, in this study the damage distribution proposed
by Akiyama (1985) will be used,
Energy Balance Method for the Analysis and Design of Structures
The energy balance method basically consists of 5 steps,
1. Determine the total damage input energy.
2. Determine the elastic energy that the structure can absorb.
3. The difference between the energy from step 1 and 2 is the plastic energy.
4. Distribute this plastic energy to every story in a rational way. (Distribution law)
5. Convert the plastic energy in each story into displacement and check with the code limits.
Some important parameters of this method are studied in more depth through the report; also thisprocedure is implemented to analyze various types of model structures.

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ENERGY INPUT INTO REINFORCED CONCRETE STRUCTURES
As was shown previously, the total energy can be transformed into an equivalent velocity, byusing a numerical procedure to solve the equation of motion; we can calculate the total input energy of aground motion for several SDOF systems with different natural periods and a certain damping factor andplot an equivalent velocity spectrum,according to Akiyama (1985), the equivalent velocity VE thatneeds to be used to check the ultimate strength of structures against earthquakes can be obtained by theenvelope of the velocity spectrum of an elastic system with a 10% damping factor, this premise wasestablished after the comparison of the response of several Elastoplastic SDOF systems, also Housner(1956) assumed that Vmax was the upper bound of the damage energy, this assumption was validated by
Akiyama for Elastoplastic systems, therefore we shall determine is this is true for hysteresis models that
represent the behavior and damping of RC structures.

El Centro NS
Figure 1, Total Velocity VE Spectra and damage velocity for Takeda Model, Damping 3%In figure 1 we can observe that the values of total input energy represented by the equivalentvelocity of the model with a damping factor of 3% are approximated by the elastic VE spectrum for 10%
damping factor, also when plastification occurs, the natural period of the structure is amplified, and it was
found that the damage energy values of structures with the amplified period can be predicted by thepseudo velocity spectrum which is available in every seismic code. (It can be obtained from theacceleration spectra)
INFLUENCE OF BEAM-COLUMN STRENGHT RATIO ON THE DAMAGE
CONCENTRATION IN REINFORCED CONCRETE STRUCTURES
The influence of the beam to column strength ratio , on the damage concentration is investigated, this isvery important because the beam to column strength ratio determines the type of collapse mechanism ofthe structure, in order to determine the damage concentration factor n.
The ratio of the plastic energy of each story to the total plastic energy in a given story i can becomputed is denoted as a, then the strength of the story i is deliberately reduced by multiplying thestrength by a factor pd, in this case we will use pd=0.8 in the third story of our 5 story model, the ratio of
the plastic energy of each story to the total plastic energy in the story i can be computed is denoted as b,
from these two cases, the damage concentration factor n can be obtained.
This procedure was carried out for 8 different values of column to beam strength ratios Rc varying
from 0.8 which is a weak column system, to 1.5 which is a weak beam system, for 3 types of nonlinear
behavior and for 3 ground motion records scaled to a maximum velocity of 50 kine. (See figure 2)
We can obtain the damage concentration factor n, from these analyses we can plot the damage
concentration factor as a function of the beam to column strength ratio Rc as shown in figure 3.

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Figure 2, Distribution of Plastic Energy, Degrading Trilinear Model, El Centro NS, Left Column pd=1.0, RightColumn pd=0.8 in the 3rd story.

Figure 3, Damage concentration factor n

RELATIONSHIP BETWEEN DUCTILITY AND AVERAGE CUMULATIVE PLASTIC
DEFORMATION FOR REINFORCED CONCRETE STRUCTURES
To determine the relationship between the equivalent frequency of plasticity and the ductility of astructure an analysis was carried out for the 3 ground motions and the 3 types of hysteresis models thathave been used in the previously, it’s important to mention that the ground motions used were scaled tohave an equivalent damage velocity VD equal to the pseudo velocity obtained using the Costa Rica
Seismic Code design spectrum as shown in figure 4, this was done with a special function of the program
Shearms developed by Professor Ogawa from Kumamoto University.
In the earthquake response analysis, the plastic energy of each story is determined and theaverage cumulative plastic deformation ratio is obtained, also the ductility μ of each story can be easily
obtained dividing the maximum displacement by the yield displacement; we must obtain the value of μ-1
because we are only interested in the inelastic part of the deformation, for each story, a relationshipbetween μ-1 and can be obtain, the relationship between these two parameters represents the equivalentfrequency of plasticity of the story i.
From the earthquake response analysis performed to several building models we can concludeof 2.0 can be used for normal level earthquakes, andthat a value of equivalent frequency of plasticity
2.3 for strong level earthquakes in reinforced concrete structures, in the case of Elastoplastic response that
happens mainly in steel buildings, the average values are lower than the degrading hysteresis models,4

therefore the designer should take this into account when analyzing a structure, its important to observe, a larger value of displacement isthat the smaller the value of the equivalent frequency of plasticity
obtained therefore a conservative design is achieved.

Figure 4, Pseudo-Velocity Spectra

Figure 5, Correspondence between maximum ductility μ-1 and average cumulative plastic deformation .

PRACTICAL APPLICATION OF THE ENERGY BALANCE METHOD
Two model buildings are used in this chapter to evaluate the Energy Balance Method; both buildings havethe same plan configuration (in the X direction 3 bays of 7 m and in the Y direction 3 bays of 6 m) butone is 8 stories and the other one is 4 stories (3.5m high), both were designed according to the Costa Rica
Seismic Code.
A nonlinear static pushover analysis of the two building models was done using the program
Stera 3D developed by Dr. Taiki Saito, for each model and for each story a force displacement curve wasobtained, and the values of the most important parameters, like cracking, yielding and ultimate force anddisplacement were obtained by a procedure of equivalent linearization, these properties were used in theenergy balance procedure mentioned previously in order to predict the interstory drift of the structure.
The computer program Stera 3D was used to obtain the earthquake response of the structure, theanalysis was carried out for 3 ground motion records scaled in order to produce an equivalent velocity
equal to the pseudo velocity obtained using the Costa Rica Seismic Code design spectrum this is donewith a special function of the program Shearms.
After performing the analysis of the building models with all the procedures mentioned before,
the final results for interstory drift were obtained and compared.
The analysis procedure for a structure with hysteretic dampers is basically the same as for a regularframe structure; this method was described previously, although some parts must be modified in order totake into account the damper system.
As mentioned before the values from the Time History Analysis will be treated as the exact value
for comparison purposes.

5Figure 6, Maximum Drift Values
According to the results obtained we can observe that the Energy Balance method is in the safe
side of the approximation of the interstory drift of the “exact” THA procedure whereas the Capacity
Spectrum Method seems too conservative.
This result validate the use of the Energy Balance Method as an alternative way to predict theseismic damage in reinforced concrete structures, and also has the benefit of being able to be
implemented very easily into any design code since the input data can be obtain directly from the pseudo
velocity design spectra.
CONCLUSIONSThe Energy Balance Method has been explained and applied to model structures in order to predict thedamage in reinforced concrete structures produced by an earthquake, some conclusions are presented next• The input energy represented by the equivalent velocity spectra and damage energy represented
by the pseudo velocity spectrum; can be applied to reinforced concrete structures.
• The damage concentration and the collapse mechanism in structures is highly dependant on the
beam to column strength relationship .
= 1.4 or higher, there is practically no damage concentration in the structure• For values ofwhen one story changes suddenly its strength.
• From a series of nonlinear time history analysis it was found that for a normal type of earthquake
a value of equivalent frequency of plasticity= 2.0 is appropriate while for a strong earthquake
a value of 2.3 must be used.
• The Energy Balance Method (EBM) was applied to model structures with and without hysteretic
dampers, and it was found to be in good agreement and on the safe side with the results ofNonlinear Time History Analysis (NTHA).

ACKNOWLEDGMENT
Professor Ogawa of Kumamoto University for letting me use his programs FISH and SHEARMS, also Dr.
Saito who answered all of my questions and also provided me with many computer programs to use in mystudy.
REFERENCE

Akiyama, H. (1985), Earthquake-Resistant Limit State Design for Buildings, University of Tokyo
Press, Tokyo, Japan.
CFIA. (2002) “Código Sísmico de Costa Rica 2002” Editorial Tecnológica de Costa Rica.
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