This document discusses using nonlinear combinations of intensity measures (IMs) to more accurately predict engineering demand parameters (EDPs) for response analysis of reinforced concrete (RC) buildings. It presents an evolutionary polynomial regression (EPR) technique to model complex nonlinear relationships between IMs and EDPs without assumptions about the form of the relationship. The EPR technique is applied to dynamic analyses of an RC framed building to predict maximum inter-story drift ratio and maximum floor acceleration under earthquake ground motions, demonstrating more accurate predictions compared to a single IM, especially for base-isolated buildings. The document concludes more accurate EDP predictions can be obtained through nonlinear IM combinations and advocates improving data-driven modeling before proposing new IMs.

1. Nonlinear combination of intensity measures for
response prediction of RC buildings
Alessandra Fiore1, Fabrizio Mollaioli2, Giuseppe Quaranta2,
Giuseppe C. Marano3,1
1Department of Science of Civil Engineering and Architecture, Technical
University of Bari, Italy
2Department of Structural and Geotechnical Engineering, Sapienza University of
Rome, Italy
3College of Civil Engineering, Fuzhou University, China

2. Introduction
A key issue in probabilistic performance-based earthquake
engineering is the evaluation of the p[EDP|IM] probabilities.
The stronger is the correlation between the selected IM and the
designated EDP, the larger is the accuracy of the structural
assessment.

3. Finding reliable EDP(IM) models
Predictive models EDP(IM) are evaluated through a data-driven
approach.
In doing so, a predefined model suitable for linear regression-based
calibration is commonly adopted:
EDP = aIMb
.
Nonlinear approaches involving different IM, however, can provide
more accurate results.

4. Artificial Neural Network (ANN)
An ANN can model complex, nonlinear patterns without
assumptions about the relationship between input and output.
Typical issues:
• the structure of an ANN
must be identified a priori
(e.g., model inputs, number
of hidden layers, etc),
• over-fitting.

5. Genetic Programming (GP)
The most frequently used GP method is the symbolic regression,
which creates expressions using an evolutionary process.
xx 2
*
_
*
*
x y
x2_
2xy
xx
+
:
2
(x2
+y)/2
y*
xx 2
*
_
*
xx
+
y*
x2_
2(x2
+y)
:
2*
x y
xy/2
Typical issues:
• it tends to produce functions
that grow in length over the
evolutionary process,
• not very powerful in finding
constants.

6. Evolutionary Polynomial Regression (EPR)
Within the EPR technique, the column of N × 1 predictions ˆY can
be expressed using a general structure such as:
ˆY = a0+
m
j=1
aj · X
ES(j,1)
1 · . . . · X
ES(j,k)
k · f X
ES(j,k+1)
1 · . . . · X
ES(j,2k)
k
where
• f (·) is a generic function and a0, aj are adjustable parameters,
• X = X1 . . . Xi . . . Xk is the matrix of inputs (each
N × 1 column represents the ith model variable),
• ES is a matrix of exponents whose elements can assume any
real value within EX, e.g. EX = . . . −0.5 0 0.5 . . . .

7. Features and implementation of the EPR
The EPR is a hybrid, nonlinear, global, stepwise regression method
for data-driven modeling. Given the model structure:
ˆY = a0+
m
j=1
aj · X
ES(j,1)
1 · . . . · X
ES(j,k)
k · f X
ES(j,k+1)
1 · . . . · X
ES(j,2k)
k
• the constants a0, aj are computed using linear least-squares
(LS) method,
• the matrix ES (i.e., the best structure of the model) is
determined via genetic algorithm (GA).

8. Accuracy vs. Complexity
Among a set of otherwise equivalent models, one should choose
the simplest one to explain a set of data.
Principle of parsimony:
• prevent over-fitting,
• easy-to-use and interpretable
models.
The trade-off is achieved by
means of a multi-objective
GA-based framework
(EPR-MOGA).

9. Generation of reference data
Nonlinear dynamic analyses by means of OpenSees 2.2.2.
3.53.53.53.53.53.5
6.0 6.0 6.0
RC framed building and ground
motions:
• first three periods equal to
1.17, 0.4 and 0.24 s,
• ordinary and pulse-like
earthquakes.
F (�=0)
DDy
K
KdFd
Friction pendulum isolators
(Dy = 0.00 mm):
• Fd = 0.03W (W is the
seismic weight),
• Kd such that the period is
3.0, 3.5, 4.0 and 4.5 s.

10. Selected EDP and candidate IM
Several IM have been selected. They are classified as follows:
• non-structure-specific IM:
• acceleration-related IM,
• velocity-related IM,
• displacement-related IM,
• structure-specific IM:
• spectral IM,
• integral IM.
The selected EDP are the following:
• maximum inter-storey drift ratio,
• maximum floor acceleration.

11. Maximum inter-storey drift ratio
Fixed−base Base−isolated
0
10
20
30
40
50
60
70
80
90
100
COD(%) Ordinary ground motions
Linear
Non−linear
Fixed−base Base−isolated
0
10
20
30
40
50
60
70
80
90
100
COD(%)
Pulse−like ground motions
Linear
Non−linear

12. Maximum floor acceleration
Fixed−base Base−isolated
0
10
20
30
40
50
60
70
80
90
100
COD(%) Ordinary ground motions
Linear
Non−linear
Fixed−base Base−isolated
0
10
20
30
40
50
60
70
80
90
100
COD(%)
Pulse−like ground motions
Linear
Non−linear

13. Conclusions
Main results:
• more accurate predictions of the EDP can be obtained
through the nonlinear combinations of several IM (especially
for base-isolated buildings),
• let’s improve data-driven modeling using advanced
computational techniques before proposing new IM!
To be continued:
• efficiency and sufficiency evaluation,
• validation using a different set of data.