investigation of stress-strain relationship of confined concrete in hollow

8/13/2019 Investigation of Stress-Strain Relationship of Confined Concrete in Hollow

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330 Copyright 2002 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959.

REFERENCE: Mo, Y. L., Hung, H. Y., and Zhong, J., Investi-gation of Stress-Strain Relationship of Confined Concrete in

Hollow Bridge Columns Using Neural Networks, Journal ofTesting and Evaluation, JTEVA, Vol. 30, No. 4, July 2002, pp.330339.

ABSTRACT: Typically, material modeling has involved the de-velopment of mathematical models of material behavior derivedfrom human observation of experimental data. An alternative pro-cedure, discussed in this paper, is to use a computation and knowl-edge representation paradigm, called a network, to model materialbehavior. The main benefits in using a neural network is that the net-work is built directly from experimental data using the self-orga-nizing capabilities of the neural network, i.e., the network is pre-sented with the experimental data and learns the relationshipsbetween stresses and strains. Such a modeling strategy has impor-tant implications for modeling the behavior of complex materials. Inthis paper, the stress-strain relationship of confined concrete in hol-

low bridge columns is modeled with a back-propagation neural net-work. The results of using networks to study the behavior of con-fined concrete look very promising.

KEYWORDS: neural networks, bridge columns, confined con-crete, mathematical modeling

Nomenclature

Ach area of a member measured outside-to-outside of lateral

steel

Ag gross area of section

Ash total area of lateral steel within spacing s and perpendic-ular to dimension hc

dbl diameter of longitudinal rebarsc concrete compressive strength

yh yield stress of lateral steel

yl yield stress of longitudinal rebarsu ultimate strength of longitudinal rebars

hc dimension of column core measured center-to-center ofconfining reinforcement

P axial force

s spacing of lateral steell ratio of longitudinal rebars

Introduction

The high speed rail project to improve Taiwans transportation

systems is a part of the efforts of the Taiwan Government to further

the islands economic development. The planned route is 345 km

long, and the viaducts and bridges are approximately 207 km long.

Hence, there are many bridge columns in the project. To maximize

structural efficiency in terms of the strength/mass and stiffness/

mass ratios and to reduce the mass contribution of the column to

seismic response, it is desirable to use a hollow section for the

columns. The hollow section used in the bridge columns in Taiwan

is as shown in Fig. 1a. However, the configuration of lateral steel

in the hollow columns studied in the past, as shown in Fig. 1 b1h[17] is different from that used in Taiwan. Note that ties and

closed stirrups are used in Fig. 1a and Fig. 1b, respectively.

On the other hand, during the past three decades, various studies

on the confinement effects of lateral reinforcement in columns

have been conducted. Kent and Park [8] proposed a stress-strain

model consisting of a second-order parabola ascending branch and

a straight line descending branch. In this model, the effects of con-

finement were reflected by adjusting the slope of the descending

branch. Park et al. [9] revised the prototype model to introduce the

increase in concrete strength caused by confinement. It was as-

sumed that the confinement effect is proportional to the volumetric

ratio and yield strength of confining reinforcement. Sheikh and

Uzumeri [10,11] proposed a stress-strain model that reflects the

confinement effect by adjusting the peak stress and a confinementeffectiveness coefficient. The confinement effectiveness coeffi-

cient depends on the configuration of the confining reinforcement.

The deterioration rate of the falling branch is similar to that in the

model of Park et al. [9]. Mander et al. [12] proposed a fractional ex-

pression to represent both the ascending and falling branches of the

stress-strain curve. To evaluate the peak stress, a confinement ef-

fectiveness coefficient for circular, square, and wall-type sections

was introduced based on a theory similar to that of Sheikh and Uza-

meri [10,11]. Furthermore, a constitutive model involving a speci-

fied ultimate strength surface for multiaxial compressive stresses

was applied in the model, which enabled development of a theoret-

ical model without dependence on a statistical analysis of test re-

sults. This model was found to provide a good prediction of test re-sults [13]. Saatcioglu and Razvi [14,15] proposed a parabolic

Y. L. Mo,1H. Y. Hung,2 and Jianxia Zhong3

Investigation of Stress-Strain Relationshipof Confined Concrete in Hollow Bridge ColumnsUsing Neural Networks

Manuscript received 12/22/98; accepted for publication 2/19/02.1 Professor, Department of Civil and Environmental Engineering, University

of Houston, Houston, TX.2 Structural Engineer, T. Y. Lin International, Taipei, Taiwan.3

Ph. D. Student, Department of Civil and Environmental Engineering, Uni-versity of Houston, Houston, TX.


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MO ET AL. ON HOLLOW BRIDGE COLUMNS 331

ascending branch, followed by a linear falling branch. The falling

branch was a function of the strain corresponding to 85% of the

peak stress. Many test data were evaluated to establish the param-

eters of this analytical model. Muguruma at al. [16] proposed a

model of the stress-strain curve constructed by two second-orderparabolas. The confinement effect was evaluated in terms of a con-

finement effectiveness coefficient. An evaluation method for the

peak stress and the ultimate strain was proposed based on a statis-

tical study of test results [17]. Fuji et al. [18] proposed a model con-

sisting of a second-order parabola and a third-order curve for the

ascending branch. A confinement effectiveness coefficient, which

was based on the model by Park et al. [9], was proposed. The peak

stress and the deterioration rate were expressed as a linear function

of the confinement effectiveness coefficient, based on a regression

analysis of test results. Hoshikuma et al. [19] proposed a model

consisting of three parts, i.e., an ascending branch, falling branch,

and sustaining branch. The stress of concrete is represented by a

higher-order function that satisfies the four boundary conditions.Nine selected models are summarized in Table 1. More recently,

Sheikh and Khoury [20] proposed a performance-based approach

for the design of confining steel in tied columns.

Application of artificial neural networks in civil engineering was

begun in the late 1980s [21], but already covers a range of topics as

diverse as process optimization [22], determining the loads on the

axles of fast-moving trucks [23], construction simulation [24], es-

timating construction costs [25], selection of a vertical formworksystem [26], damage detection of steel frames [27], seismic hazard

prediction [28], structural analysis of plates [29], prediction of

framed shear wall behavior [30], and modeling of concrete behav-

ior [31]. Basic principles of neural networks and their applications

have been described [32,33] in detail.

In this paper, neutral networks were created and trained to study

the stress-strain relationship of confined concrete in hollow bridge

columns using Professional II Plus, a development tool kit for neu-

tral networks [34]. Experimentally-generated sample cases for both

training and testing of the networks were obtained from 24 panel

tests. The application of Professional II Plus to the confined con-

crete is then described. It is found that the stress-strain relationship

of confined concrete in hollow bridge columns can be predicted

quite well by neutral networks.

Fundamentals

Back-propagation is a very powerful technique for constructing

nonlinear transfer functions between several continuously valued

inputs and one or more continuously valued output. Therefore,

back-propagation networks are employed in this paper. A detailed

description of back-propagation networks can be found in Rumel-

hart et al. [35] and nonlinear functional mapping capabilities in La-

pedes and Farber [36]. In this paper, various forms of the transfer

functions are used, such as sine, tanh, linear, and sigmoid, as shown

in Fig. 2.

The effect of learning rules on network learning will be critically

examined in the application to confined concrete. Four different

learning rules will be considered. They are: a) Delta-Rule: to re-

duce the error between the actual output of a processing element

and its desired output by modifying incoming connection weights;

b) Norm-Cum-Delta-Rule: to accumulate the weight changes over

several training presentations and make the application all at once,

where the learning coefficient is divided by the square root of the

epoch size (setting the epoch value to the size of the training set

will allow the graph to show the performance of the entire training

set); c) Delta-Bar-Delta (DBD): this rule is an attempt to address

the speed of convergence issue, via the heuristic route. By using

past values of the gradient, heuristics can be applied to infer the

curvature of the local error surface. With this type of information,

intelligent steps can be taken in the weight space using a number ofstraightforward rules; d) Extended DBD: in addition to DBD, Ex-

tended DBD contains several other modifications of momentum

adjustment, based on heuristics, in an attempt to increase the rate of

learning. Other learning rules will be mentioned, but not dwelt

upon.

Experimental Program

Specimens

Figure 3a indicates the dimensions of a hollow bridge column.

When a single wall is cut from the hollow column, it is shown in

Fig. 3b. Note that no concrete cover is considered in Fig. 3. Three

types of configurations of lateral steel, shown in Fig. 4, were tested,namely, Types A, B, and C. Type B is often employed in bridge en-

Fig. 1Configuration of lateral steel in hollow columns.


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332 JOURNAL OF TESTING AND EVALUATION

gineering in Taiwan. As shown in Table 2, 24 specimens have been

designed for compression tests. N or H in the first character of the

designation stands for normal or higher strength concrete, respec-

tively. A, B, or C in the second character of the designation speci-

fies the configuration of lateral steel with Types A, B, or C, re-

spectively, and 3 or 4 in the third character of the designation

represents the spacing of lateral steel of 3 cm or 4 cm, respectively.

Because each case has two specimens, the last character of 1 or 2

means specimen 1 or 2, respectively. It should be noted that the

spacings of lateral steel of both 3 cm and 4 cm satisfy the require-ment of the ACI code [37]. However, when the following equation

proposed by Priestley et al. [38] is used,

s 3 6 yul 1 dbl (1)where

s spacing of lateral steelu ultimate strength of longitudinal rebars

yl yield stress of longitudinal rebarsdbl diameter of longitudinal rebars

it is found that the spacing should not be greater than 3.3 cm. Withregard to the lateral steel, the test specimens satisfy the require-Fig. 2Four kinds of transfer functions.

TABLE 1Summary of previous stress-strain models for confined concrete.

Stress-Strain Model for Confined Concrete ApplicableCross-Sectional

Researcher Ascending Branch Descending Branch Residual Stress Shape

Kent and Park [8] c c

0.

2

0

02

0.0

02

2

c c[1 Z( 0.002) Square

Confined Kent and Park [8] c c 0.2002 0.0022

c c[1 Z( 0.002)] 20% of c SquareModified Kent and Park [8] c Kc 0.0202K 0.002K

2

c Kc [1 Zm ( 0.002K)] 20% of Kc SquareMuguruma et al. [17] c Ec

c

E

c2

cc 2 c

c

c

u

u

c

c

c

c ( cc) cc Circle

c (

c

c

c

c

c

c

)2 ( cc)

2 cc

Square

Sheikh and Uzumeri [11] c cc 2s

1 s1

2

c cc [1 Z( s2)] 30% of cc SquareMander et al. [12, 13] c

r

c

1

cx

r

xr c

r

c

1

cx

r

xr Circle

SquareWall-type

Fuji et al. [18] c Ec cc

c2

Ecc 2 c cc ( cc) 20% of cc Circle

c (

c

c

c

c

c

c

)3 ( cc)

3 cc

Square

Saatcioglu and Razvi [14] c cc 2cc

cc

2

1/(12K)

c cc

0

8

.

5

1

5

c

c

c

c ( cc) 20% of cc Circle

SquareWall-type

Hoshikuma et al. [19] c Ec 1 1n ccn1

c cc Edes ( cc) 50% of cc CircleSquare


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ments of both the ACI code [37]

Ash 0.3shc

y

c

h AAcgh 1 (2)

and

Ash 0.09shc

y

ch

(3)

where

Ash total area of lateral steel within spacing s and perpendicu-

lar to dimension hcs spacing of lateral steel

hc dimension of column core measured center-to-center of

the confining the reinforcement

Ach area of a member measured out-to-out of lateral steel

Fig. 3Dimensions of specimens (unit: mm).

Fig. 4Configuration of lateral steel (unit: mm).


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Development of Neural Networks

Neural Works Professional II/PLUS (Nworks) provides a com-

plete neural network development environment, and it has the fol-

lowing advantages: (1) Multiparadigms: Dozens of well-known,

built-in network types can be quickly generated; (2) Custom net-

works: by writing customized script files, the network types can be

created; (3) Instrumentation: powerful diagnostic capabilities al-

low anyone to monitor weights, errors, activation, and more with

prebuilt and custom on-screen graphic displays; (4) Network IO:

data for networks can be supplied from the keyboard or an ASCII

file. The dialog box of the network builder is shown in Fig. 5. Us-

ing Nworks, the algorithm to build a network is explained as fol-

lows:

1. Create networkto set the network type, layer number, num-

ber of processing elements, network connection, and network

parameters. The network parameters include (1) control strat-

egyto control the flow of processing through the network;

(2) learning and recall scheduleto hold many of the ad-

justable parameters that will be used by the network; and (3)MinMax tablein which the lows and highs of each data field

for all the input data files are stored.

2. Set instrumentsto send diagnostic information from a net-

work to the screen, a file, or a user-written program.

3. Set network IOto be used during learning and testing.

4. Initialize networkto initialize the weights in all layers to

random values between the low and high values specified in

each layer.

5. Network learningto use the learning rule for the weight up-

dates. For back-propagation, the Delta-Rule, Norm-Cum-

Delta rule, Ext DBD rule, and Delta-Bar-Delta rule will be

studied.

6. Network testingthe network is trained using the test cases,

and then the test cases serve as way of measuring network

performance. During this test phase, the test cases are pre-

sented to the network and the network provides results. If the

correct response is known, the network performance can be

measured.

Application to Confined Concrete

Before a neural network is employed, the experimental stress-

strain relationship of each of Specimens NA3, NB3, and HC4 is

compared with that predicted by all nine models presented previ-

ously (Table 1), as shown in Figs. 6 and 7. It can be seen from Figs.

6 and 7 that the predictions from all nine models are not in good

agreement with the test result. Therefore, application of a neuralnetwork is used below.

If a network is to be trained, sufficient training examples are re-

quired. The 24 examples used in Nworks are acquired directly from

the panel experiments and used for training and testing simultane-

ously. Using Nworks, the relationship between stress and strain can

be explained as follows:

1. Example parameterThere are four input parameters: con-

crete compressive strength, spacing of lateral steel, configu-

ration of lateral steel, and strain values; the output parameters

are stress values.

2. Example collectionsAs shown in Table 2, the results of 24

tests are indicated. The 24 tests produced 5226 sets of data.Among these data, 4393 sets of data are used for network

c concrete compressive strengthyh yield stress of lateral steel

and the equation proposed by Priestley et al. [38]:

Ash 0.12shc ych

0.5

1.c2A5gP

0.13(l 0.01) (4)

where

P axial forcel ratio of longitudinal rebars

Note that the diameters of the longitudinal and lateral bars are 6

mm and 4 mm, respectively.

Materials

The concrete was supplied by a local ready-mix concrete plant

with two kinds of compressive strengths, namely, 22 MPa and 41

MPa. The concrete strength values, also shown in Table 2, were ob-tained by using 150 by 300 mm cylinders at the test age, which was

about 30 days after casting. The bars with the diameters of 6 mm

and 4 mm had yield strengths of 459.5 MPa and 400.0 MPa, re-

spectively.

Test Setup

The specimens were tested under a universal compression ma-

chine that has a capacity of 500 tons. A thick steel plate is added to

the top and the bottom of the specimen. At the bottom of the thick

steel plate, a load cell is connected to read the values of axial force.

In each face of the specimen there are two LVDTs, and their loca-

tions are shown in Fig. 3b. The axial force is monotonically in-

creased until the maximum force of the specimen. After the peakforce is reached, the tests were controlled by displacement.

TABLE 2Experimental results.

Maximum UltimateSpecimen fc Stress Maximum Stress Ultimate

No. (MPa) (MPa) Strain (MPa) Strain

NA3-1 22 34.00 0.0050 27.20 0.0085

NA3-2 22 34.81 0.0065 27.85 0.0105NA4-1 22 32.73 0.0041 26.18 0.0078NA4-2 22 33.79 0.0037 27.03 0.0060NB3-1 22 35.33 0.0066 28.27 0.0101NB3-2 22 31.79 0.0057 25.43 0.0114NB4-1 22 31.96 0.0048 25.57 0.0086NB4-2 22 26.20 0.0045 20.96 0.0078NC3-1 22 36.22 0.0061 28.97 0.0106NC3-2 22 29.34 0.0078 23.47 0.0123NC4-1 22 35.48 0.0057 28.38 0.0096NC4-2 22 34.37 0.0061 27.50 0.0087HA3-1 41 50.38 0.0035 42.98 0.0048HA3-2 41 49.39 0.0023 36.31 0.0035HA4-1 41 52.94 0.0026 42.35 0.0045HA4-2 41 52.67 0.0023 42.14 0.0033HB3-1 41 55.62 0.0028 44.49 0.0048HB3-2 41 55.12 0.0029 44.10 0.0045

HB4-1 41 54.87 0.0036 43.90 0.0063HB4-2 41 47.11 0.0019 37.69 0.0031HC3-1 41 55.86 0.0036 44.69 0.0037HC3-2 41 52.27 0.0050 49.95 0.0051HC4-1 41 56.51 0.0038 45.21 0.0059HC4-2 41 48.37 0.0019 38.69 0.0028


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Fig. 5Dialog Box of InstaNet/Black propagation.

Fig. 6Experimental results compared to predictions from all nine models for specimens NA3 and NB3.


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Fig. 7Experimental Results Compared to Predictions from All Nine Models for Specimen HC4

training and the remaining 833 sets of data are used for check-

ing the accuracy of the trained network.

3. Example processingA MinMax Table specifies the ranges

within which the real world data lie. Specification of the tar-

get network ranges is also needed to define the mapping. A

single range is allowed for the input layer and a separate sin-

gle range is allowed for the output buffer. The MinMax table

defines a linear mapping that maps the minimum value of the

real world range to the minimum value of the network rangeand the maximum value of the real world range to the maxi-

mum value of the network range. Then the mappings from the

real world to the network are performed.

4. Network creationA single hidden layer is sufficient. The

number of processing elements in the hidden layer is three,

which is approximately equal to the number of input and out-

put parameters divided by 2. The transfer function used is the

tanh function. The momentum term to act as a low-pass filter

is selected to be 0.4, and the learning coefficient is 0.5.

5. Network learningThe error-training cycle curves of net-

work learning will be studied later in this paper. It is found

that network learning converges very fast, and the error is

close to 0.5.

6. Network testingIn Fig. 8, the relationship between stressand strain of Specimen NA3 represented by the thin line was

determined by the results of the experiments; the same rela-

tionship represented by the curve with crosses was deter-

mined by network tests. Similar results are indicated in Fig. 9

for Specimen HC4. It can be seen from Figs. 8 and 9 that the

networks produce results with acceptable accuracy.

7. Effect of network types on network learningThe error-

training cycle curves based on the six learning rules men-

tioned previously are shown in Fig. 10. It can be seen from

Fig. 10 that the error converges for Delta-Rule and Max-

Prop-Rule when the number of training cycles approaches

8000. It can also be seen from these figures that the Delta-

Rule or the Max-Prop-Rule provides acceptable results andthe results from the Delta-Bar-Delta rule are not acceptable.

All three transfer functions are used, namely, the sine, tanh, and

sigmoid functions. The error-training cycle curves based on these

three transfer functions are studied. It is found that using the tanh

transfer function provides the best learning results.

The number of processing elements in the hidden layer is se-

lected to be 2, 4, and 20, separately. The error-training cycle curves

based on these three cases are also investigated. It is found that a

greater number of processing elements in the hidden layer does not

provide better network learning.

Conclusions

According to the available test results, the stress-strain relation-

ship of confined concrete in hollow bridge columns can be pre-

dicted with acceptable accuracy when neural networks are used.

The method presented in this paper can be applied to study the

stress-strain relationship of confined concrete affected by other

factors. However, the application of the developed network is lim-

ited to the range of the tested panels. Beyond this range, test data

are required. It should be noted that once the networks have been

tuned, they have to be retuned for different types of problems. For-

tunately, the tuning that is needed to get good networks is simple

because Nwork is very user-friendly.Whether the learning and prediction capabilities are acceptable

depends on appropriate network creation and suitable examples

for training and testing. It is found from this study that the effect

of the transfer function and learning rules on the network learn-

ing is significant. In contrast, the effect of the number of pro-

cessing elements in the hidden layer on the network learning is in-

significant.

Acknowledgment

The research in this report was funded by the National Science

Council, Taiwan through research grant NSC 87-2621-P-006-011.

The authors wish to thank S. H. Yao and S. J. Wang for their assis-tance in construction and testing of the specimen.


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Fig. 9Experimental stress-strain relationship compared to neural network prediction for Specimen HC4.

Fig. 8Experimental-stress strain relationship compared to neural network prediction for Specimen NA3.


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References

[1] Mander, J. B., Priestley, M. J. N., and Park, R., Behavior ofDuctile Hollow Reinforced Concrete Column, Bulletin of

the New Zealand National Society for Earthquake Engineer-

ing, Vol. 16, No. 4, December 1983, pp. 273290.[2] Taylor, A. W. and Breen, J. E., Design Recommendations

for Thin-Walled Box Piers and Pylons, Concrete Interna-tional, American Concrete Institute, December 1994, pp.3641.

[3] Suda, K., Murayama, Y., Amano, R., and Ichinomiya, T.,Effect of Torsion on Ductility of Reinforced Concrete Piers

with Hollow Section, Proceedings of the 51st Annual Con-ference of JSCE, No. 5, September 1995, pp. 840841.

[4] Suda, K., Shimbo, H., Masukawa, J., and Murayama, Y.,

New Method for Restraint of Longitudinal Bar Buckling inRectangular Columns with Hollow Section, Proceedings ofTechnical Conference of the Great Hanshin-Awaji Earth-

quake, JSCE, Tokyo, 1996, pp. 579582.

[5] Matsudal, T., Yukawa, Y., Yasumatsu, T., Tsukiyama, Y.,

Ishihara, S., Suda, K., et al., Seismic Model Tests of Rein-

forced Concrete Hollow Piers, Proceedings of 12th US-

Japan Bridge Engineering Workshop, 1996, pp. 407421.

[6] Suda, K., Shimbo, H., Masukawa, J., and Murayama, Y., Re-

inforcing Method to Improve Ductility of RC Column with

Hollow Section, Proceedings of JCI, Vol. 18, No. 2, 1996,pp. 725730.

Fig. 10Error-training cycle curves based on different learning rules.

(b) Learning rules: MaxProp, QuickProp, and Ext-DBD.


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