Modal criticality index of an existing reinforced cement...

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 4, No 3, 2014

© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0

Research article ISSN 0976 – 4399

Received on August 2013, Published on March 2014 389

Modal criticality index of an existing reinforced cement concrete T-beam bridge

Roseenid Teresa Amaladosson1, Umarani Gunasekaran2, Premavathi Narayanan3

1-Ph.D. Research Scholar, Anna University, Chennai and Associate Professor, St. Joseph’s

College of Engineering, Chennai.

2-Associate Professor, Anna University, Chennai.

3-Assistant Professor, Bannari Amman Institute of Technology, Sathyamangalam.

[email protected]

doi: 10.6088/ijcser.201304010038

ABSTRACT

An existing reinforced cement concrete T-beam bridge is evaluated using inelastic analysis

procedures namely capacity spectrum method (CSM) and modal pushover procedure (MPA).

In this paper, MPA is performed in both the transverse and the longitudinal directions of the

bridge structure independently. The capacity curves that represent the response of the bridge

in transverse direction for the particular modes of the vibration are generated using MPA.

Modal Criticality Index (MCI), a value which is the ratio of the Spectral acceleration (Sa) for

demand and the Spectral acceleration (Sa) for capacity is determined. Modal criticality index

value was calculated, considering the initial stiffness and the secant stiffness of the bridge

structure, for identifying the critical mode. From the calculated modal criticality index value,

the critical mode of the structure was inferred to be the higher mode irrespective of the

stiffness (initial/secant) adopted.

Keywords: Modal Pushover procedure, capacity Spectrum method, T-beam bridge, higher

mode effects, Modal Criticality Index, acceleration-displacement response spectrum.

1. Introduction

Roads are the lifelines of modern transport, and bridges are an integral part thereof. They are

susceptible to failure if their structural deficiencies are unidentified. A large number of

bridges constructed around the world were designed during the period, when bridge codes

had no seismic design provisions, or when these provisions were insufficient according to the

current standards. The 1971 M6.6 San Fernando Earthquake, 1989 M7.1 Loma Prieta

Earthquake, 1994 M6.7 Northridge Earthquake, 1995 M7.2 Hanshin-Awaji Kobe Earthquake,

and the most recent 2011 M9.0 Tohoku Earthquake (Japan), are some of the earthquakes

which have caused severe damage to a considerable number of bridges that had little or no

design consideration to seismic resistance, leading to huge loss of lives and property. The

2001 M7.6 Bhuj Earthquake that shook the Indian Province of Gujarat was the most deadly in

India's recorded history.

This disaster has created awareness among the engineers to determine the structural

vulnerability of the bridges which were built before 2001 to develop the required retrofit

measures. So it is the need of the hour to identify the critical mode of the structure to find the

response of the bridge when subjected to an earthquake. Thus, the objective of this paper is to

identify the critical mode of an existing T-Beam cum Slab Bridge using nonlinear static


Roseenid Teresa Amaladosson

International Journal of Civil and Structural Engineering

Volume 4 Issue 3 2014

390

pushover analysis in which the structural performance levels are well understood over a

wider range than only at the first yield or near collapse.

In the present study, an existing Reinforced Cement Concrete (R.C.C) T-beam and slab Road

Bridge located in Chennai City, Tamilnadu, India, was evaluated, using nonlinear static

method, for assessing the seismic behaviour under imposed earthquake ground motions of

high magnitude. Initially, a live load test on the study bridge was conducted to measure the

flexural responses of the longitudinal and the cross girders. The strain histories obtained from

the experimental investigation indicated the nonlinear response of the longitudinal girder and

linear response of the cross girder. A three-dimensional nonlinear finite-element model of the

reinforced cement concrete bridge, the Koyambedu Bridge, was modeled and analysed, using

the nonlinear software package, SAP2000. The nonlinear pushover analysis was performed

on the bridge structure to determine the inelastic response. Modal pushover analyses were

carried out in both the transverse and the longitudinal directions. As the higher modes had

significant effect on the seismic behaviour, the modal pushover analysis with the capacity

spectrum method was employed, to identify the critical mode which causes the failure of the

bridge structure. The criticality of the bridge structure was evaluated by calculating the

Modal Criticality Index (MCI) value.

2. Description and modeling of the study bridge

The bridge is built across Coovam river in Koyambedu, Chennai and connects Guindy and

Thirumangalam. It is a multi-span (eight equal spans of 16.21m) simply supported R.C.C.

slab cum T-Beam bridge of 129.7 m total length. Each span of the superstructure consists of

four longitudinal T-Beam girders and five cross girders. It is supported on multi-column

bents over neoprene bearing pads. Each multi-column bent has four columns which are

transversely connected by the bent cap. The bridge piers and abutments are supported on well

foundations. The cross sectional details of the bridge components are given in Table 1. The

longitudinal view and the sectional elevations of the bridge are shown in Figure. 1, Figure. 2

and Figure. 3.

Table 1: Cross sectional details of the bridge components

Sl. No. Description Size (mm)

1.

Longitudinal Girder

Top Flange 2500 x 220

Bottom Flange 500 x 300

Web 250 x 1400

2. Cross Girder 200 x 1400

3.

Bent Cap

Cross Section 1400 x 600

Length 8800

4.

Bent Column

Diameter 800

Height 5476

5. Bearing Pad 500 x 320 x 33.5

A three dimensional finite-element model of the bridge was created using SAP 2000. A spine

model was employed to model the superstructure. The deck edges in each simply supported

span were considered rigid. Due to the large in-plane rigidity, the superstructure was assumed

as a rigid body for lateral loadings. The bridge consists of seven multi-column bents and





391

every bent was modeled as a plane frame. The framing action and coupling between columns

in the multi-column bent provides seismic resistance in terms of strength and stiffness.

Figureure 1: Longitudinal view of the bridge

400.00 3400 .00 6800 .00

650

.00

60

0.0

0

8800.00

48

67.0

0

600

.00

33.5

0

T-B eam cum slab

B ent C olum n

B ent C ap

Foot path

C apping slab

W ell Foundation

W earing coat

E lastom eric bearing

16

05.0

0

800.00

Figure 2: Cross sectional elevation of the bridge

Seat type

abutmentHand rails

Transverse

girders

Longitudinal

girders Bent

Seat type

abutment

Cast-in-situ bored piles

Well foundation Well foundation

Figure 3: Longitudinal Elevation of the bridge

All Dimensions are in





392

All these effects were incorporated in a planar frame model along with the bent axis. The

bent cap and the columns were modeled as beam-column elements. Effective moment of

inertia was taken as 0.7 times gross moment of inertia (Ig) for Reinforced Concrete (RC)

columns which were modeled using Section Designer (Sub programme in SAP 2000). The

interface between each column and centroid of bent cap was considered rigid. The default

hinge properties (PMM -Biaxial Moment hinge, SAP 2000) were assigned to each end of the

columns. The base of the column was assumed as fixed. The model was created using SAP

2000 and is shown in Figure 4.

The horizontal sliding behaviour of the interface between the bearing and the girder or cap

beam was modeled, using the linear spring element or link element, as shown in Figure. 5. A

link element is composed of lateral, vertical and rotational stiffness component. Translational

or effective stiffness is used to consider the nonlinear behaviour of elastomeric bearing. Shear

modulus values for elastomers in bridge bearings range between 0.8 MPa and 1.20 MPa

depending on their hardness. The effective shear modulus of pads was taken as 0.9 MPa. The

bridge is longitudinally free up to the maximum elastomer flexibility. The expansion joints

between the deck slabs, the abutment and deck slab were modeled as Gap elements (Figure.

6). Gap element is a compression only element such that it will contribute resistance when the

relative distance between the adjacent structures is smaller than the initial gap of 25.40 mm.

When the gap closes, pounding occurs and the gap element offers infinite stiffness. The

support provided by the abutment was assumed as fixed against vertical translation and the

stiffness properties of the translational spring in the longitudinal and transverse directions

were given as per Caltrans design aid. The active and passive soil earth pressures were not

considered in the abutment modeling.

Figure 4: (Continued)

Detail A





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Superstructure (Spine element)

Expansion joint (Gap element)

Elastomeric bearing (Link element)

Edge of the deck (Rigid end)

Bent cap beam (Inelastic beam-column element)

Bent column (Inelastic beam-column element)

Rigid link

Figure 4: Modeling of the bridge using SAP2000 (Detail A)

Figure 5: Elastomeric Bearing Pad Model

Figure 6: Expansion Joint Model

3. Modal Pushover Analysis (MPA) and Capacity Spectrum Analysis (CSM)

Modal Pushover analysis is an extension of the “standard’ pushover analysis (SPA).

According to the MPA procedure, standard pushover analysis is performed for each mode

independently, wherein invariant seismic load patterns are defined according to the elastic

modal forces. Bridges are the structures where higher modes usually play a more vital role

and hence developing a modal procedure for such structures is even more challenging than in

case of buildings. The purpose of the modal pushover procedure is to generate capacity

Detail A





394

curves that represent the response of the building for particular modes of vibration. The

Capacity spectrum analysis (CSM), originally developed by Freeman et al., is a graphical

procedure for estimating structure load–deformation characteristics and for predicting

earthquake damage and structure survivability. The capacity curves are determined by

statically loading the structure with lateral forces to calculate base shear values and roof

displacements that define the global force-displacement characteristics of the structure. The

procedure to develop these curves has been referred to as the pushover analysis. In this paper

Capacity Spectrum method is employed using the modal pushover procedure (MPA) and the

Acceleration-Displacement Response Spectrum format to evaluate the response of the bridge

to different earthquake conditions.

4. Modal Criticality Index (MCI) using Modal Pushover Analysis

Though the study bridge (Koyembedu Bridge) is symmetric, both the longitudinal and

transverse responses are significant because the lateral load may lead to stability problem

along the transverse direction and unseating of deck is most common along the longitudinal

direction. Therefore, a three dimensional finite element model of the bridge was developed.

The modal analysis of the bridge was carried out to find the dynamic characteristics of the

bridge such as modal participation, mode shapes etc. Table 2 shows the modal period and

mass participation of the bridge in both the longitudinal and transverse directions. The mode

shapes in transverse direction are shown in Figure. 7 and Figure. 8 respectively. The mode in

the longitudinal direction is shown in Figure. 9. In the Longitudinal direction the whole of

90% mass participation is captured by mode#2, which is the critical mode in the longitudinal

direction. Modal Pushover analysis was not in need in the Longitudinal direction. In this

study, the MCI value was calculated and critical mode was identified in the transverse

direction.

Table 2: Modal Periods and Mass participation of the bridge

Sl.No. Mode

Number

Period

(second)

Mass excited in

longitudinal direction (%)

Mass excited in

transverse direction (%)

1. 1 0.412 0 84.31

2. 2 0.276 93.57 0

3. 8 0.104 0 1.38

Figure 7: Mode shape in the transverse direction (mode#1)





395

Figure 8: Mode shape in the transverse direction (mode#8)

The modal pushover analysis was conducted in both the transverse and the longitudinal

directions.

Figure 9: Mode shape in the Longitudinal direction (mode#2)

The capacity curve (pushover curve) is the graphical plot of the total lateral force or base

shear (Vb) on a structure against the lateral deflection (δ) of the control node of the bridge

structure. The pushover curve for mode#1 is shown in Figure. 10. The Figureure indicates

that the first yielding occurred at a base shear of 7961.26kN with the control node

displacement of 19.7mm. Beyond the first yield, the control node displacement increases with

the increase in base shear. The softening of the pushover curves associated with the

progressive formation of plastic hinges was noticed in the multi-column bents of the bridge

structure, with increasing lateral forces. The first mode caused a global plastic mechanism

and increasing force intensity, leading to the rotation of the bridge structure about its base

(bottom local plastic mechanism). The control node continued to move in the direction of the

application of lateral force. The pushover curve displayed normal behaviour without any

reversal. The formation of mechanism reduced the stiffness and caused an incremental





396

displacement. The structure experienced a maximum displacement of 115mm with the base

shear of 11229.20kN with the indication of the loss of lateral stiffness.

Figure 10: Base shear vs. Displacement curve (mode#1- Transverse Direction)

The capacity curve for mode 2 is shown in Figure. 11. From the pushover curve it was found

that the overall strength of the system appeared to be higher (i.e. yielding occurred at a higher

level of base shear. In the longitudinal pushover analysis, when the push load was applied in

the longitudinal direction, the expansion joints which are provided between the adjacent sides

of a deck joint, permitted relative translations and rotations at both sides of the bridge decks.

The first yield occurred at a base shear of 34644.03kN, and a control node displacement of

22.3mm was observed. The structure experienced a maximum displacement of 35.2mm with

the base shear of 52610.17kN.

Figure 11: Base shear vs. Displacement curve (mode#2- Longitudinal Direction)





397

The capacity curve for mode#8 is shown in Figure. 12. In the pushover analysis performed in

the transverse direction for mode#8, the first yield occurred at a base shear of 4468.55kN

with the control node displacement of 22.8mm. The bridge structure had displaced to a

maximum of 115.9mm, with maximum base shear value of 6517.11kN. The bridge structure

had displaced far into the inelastic range with significant degradation in the lateral capacity.

Figure 12: Base shear vs. Displacement curve (mode#8- Transverse Direction)

Modal criticality index (MCI) [15, 18] is a value, which is used to identify the critical

vibration mode which causes the failure of the structure. Mathematically, it is the ratio of the

spectral acceleration (Sa) value for demand and the spectral acceleration (Sa) value for

capacity. In the present study, the response of the bridge structure to the El Centro earthquake

ground motion was evaluated by the capacity spectrum method. In the fundamental mode

(mode#1) there was transverse displacement with the modal mass participation ratio of

84.32%. In the longitudinal direction 93.57% of mass participation was captured by mode#2,

which is the critical mode that may cause the failure of the bridge in the longitudinal direction.

To consider the vibration modes capturing at least 90% of the total mass of the bridge

structure in the transverse direction, the higher mode, i.e., mode#8 was considered, as the

modal mass participating ratios of the other intermediate modes 3,4,5,6 and 7 were not

significant. In mode#8 the total mass excited was 1.4%.

Though the bridge structure was found to be a fundamental mode dominant structure, a

higher mode, mode#8, was also considered in this study to evaluate the criticality of the

structure. The modal pushover analysis was performed for mode#1 and mode#8, and the

capacity curves were plotted. The capacity curves were converted to the Acceleration

Displacement Response Spectrum format (ADRS). The demand spectrum of the El Centro

Earthquake was overlaid with the capacity spectrum of the bridge structure. The intersection

of the capacity curves and the demand curves, when plotted in the same graph in Sa vs. Sd

format, approximates the response and the performance of the structure for that particular

earthquake.





398

Displacement based methods that incorporate the initial stiffness have some approximate

relation between the elastic and inelastic responses. The concept of initial stiffness for a

structure, responding to the inelastic range to a displacement δD and strength level Vb is

shown in Figure. 13.

Lateral displacement, δ

Secant stiffnessLat

eral

Forc

e (k

N)

Vb

Initial stiffness

δD

Figure 13: Capacity curve with initial and secant stiffness

A commonly adopted relation between the elastic and the inelastic responses is the equal

displacement approximation (Displacement Coefficient method – Applied Technology

Council (ATC) [1]), in which the capacity spectrum and elastic response spectrum (5%

damped spectrum) are overlaid. This approximation argues that the displacement of the

elastic system of initial stiffness, will be equal to that of the inelastic system.

The secant or effective stiffness (Keff), is the ratio of the strength Vb to the maximum

displacement δD. It is presented in Figure. 13. To facilitate the design using the linear secant

stiffness, an equivalent viscous damping coefficient was used to account for the energy

dissipated during the actual non-linear structural response. The capacity spectrum intersects

various damping curves, but the point at which the capacity spectrum possesses the same

equivalent damping value as that of the demand curve is identified as the performance point.

This intersection point indicates the structural response that is expected to develop during the

design earthquake. In the present study, the MCI value of the bridge structure was calculated

using the CSM (secant stiffness based procedure) and the MPA, to identify the critical mode.

The MCI value obtained using secant stiffness, was compared with the value calculated by

employing the procedure, which was used by Paret et al [15, 18]. The pushover curves of

mode#1 and the higher mode mode#8 were developed (using SAP2000).

For finding the MCI using the initial stiffness, the 5% damped response spectrum of the El

Centro Earthquake (elastic spectrum) was employed. As the capacity curves of the

fundamental mode (mode#1) and the higher mode (mode#8) intersected the demand curve

well beyond the linear elastic region (Figure. 14), the initial stiffness (equal displacement

approximation) was used to obtain the corresponding spectral demand (Sd). The bridge

experienced significant yielding when subjected to the earthquake ground motion. The

yielding of the structure was concentrated at the bottom of all the columns in all the bents.

From Figure. 14, the MCI value for mode#1 was found to be 7.56, whereas the MCI value for

the higher mode mode#8 was found to be 7.67. Though the MCI value of the higher mode is





399

much closer to the value corresponding to the fundamental mode, the critical vibration mode

which would cause the bridge structure to fail would be the higher mode.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3

Sp

ectr

al accele

rati

on

, S

a(g

)

Spectral displacement, Sd, m

Capacity spectrum for mode#1


5% damped response spectrum (ElCentro)Demand for mode#1 capacity spectrum

1st significant yield for mode#1

Demand for mode#8 capacity spectrum


Initial stiffness for mode#1

Initial stiffness for mode#8

Figure 14: MCI using the Initial Stiffness

While using the capacity spectrum method (secant stiffness procedure), the response

spectrum curve reduced from the inherent viscous damping should be used. For the study

bridge, the effective damping of the structure in the fundamental mode (mode#1) was found

to be 25.8%. In the higher mode (mode#8) the effective damping value was found to be

25.9%. In order to account for the hysteretic damping and the nonlinear effects, the demand

spectra of the El Centro Earthquake was reduced to 25.9% effective damping. Thus, from

Figureure 15, the MCI value for mode#1 was found to be 1.183, whereas the MCI value for

the higher mode, mode#8 was found to be 1.71. Thus, the MCI value of the higher mode is

the critical vibration mode, which would cause the bridge structure to fail.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.05 0.1 0.15

Sp

ectr

al a

ccel

rati

on,

Sa(

g)

Spectral displacement, Sd, m



25.9% damped demand spectrum

Demand for mode#8 capacityspectrum1st significant yield for mode#1


Demand for mode#1 capacityspectrumSecant stiffness for mode#1

Secant stiffness for mode#8

Figure 15: MCI using the Secant Stiffness





400

Table 3 shows the Modal Criticality Index (MCI) value calculated with the Initial stiffness

and Secant stiffness.

Table 3 Modal criticality index value

Sl. No Stiffness Mode#1 Mode#8

1. Initial Stiffness 7.56 7.67

2. Secant Stiffness 1.183 1.71

5. Conclusion

The criticality of the structure was found by calculating Modal Criticality Index value.

Modal Criticality Index (MCI), which is the ratio of the spectral acceleration value of the

demand and the spectral acceleration of the yield, was determined using both the initial

stiffness and the secant stiffness of the structure. MCI value was found to be greater in the

higher mode (mode#8) with a value of 7.67 while using initial stiffness, and a value of 1.71,

while using secant stiffness. Though the MCI value differs for initial and secant stiffnesses,

the critical mode which would lead to the failure of the structure is the higher mode (mode#8).

6. References

1. Applied Technology council. Seismic evaluation and retrofit concrete buildings,

ATC-40, Report No. SSC96-01, Seismic safety commission, Redwood City,

California, 1996.

2. Caltrans. Seismic design of highway bridge foundations, Training course manual,

California Department of Transportation, Sacramento, California, 1996.

3. Can Akogul and Oguz C Celik. (2008), Effect of elastomeric bearing modeling

parameters on the seismic design of RC highway bridges with precast concrete girders,

The 14th World Conference on Earthquake engineering, Beijing, China.

4. Chopra ΑΚ Goel RK. A,(2002), modal pushover analysis procedure for estimating

seismic demands for buildings, Earthquake Engineering & Structural Dynamics, 31(3),

pp 561-582.

5. Fajfar P. (1999), Capacity Spectrum method based on inelastic demand spectra,

earthquake engineering and structural dynamics,28, pp 979-993.

6. Freeman SA, Nicoletti J, and Tyrell JV,( 1975), Evaluation of existing buildings for

seismic risk- A case study of Puget Sound Naval shipyard, Bremerton, Washington,

Proceedings of first U.S National Conference on Earthquake Eng., EERI, Berkeley,

pp 113-122.

7. Freeman SA. (2004), Review of the development of the capacity spectrum method,

ISET Journal of Earthquake Technology,438, 41(1), pp 1-13.

8. IRC: 83 (Part III). Standard specifications and code of practice for road bridges,

Section: IX Bearings. The Indian Roads Congress, 2002.





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9. Kappos AJ, Paraskeva TS, Sextos AG. (2005), Modal pushover analysis as a means

for the seismic assessment of bridge structures, Proceedings of the 4th European

Workshop on the seismic Behaviour of Irregular and Complex structures, Greece.

10. Keri L Ryan, Brian Richins (2011), Design, analysis, and seismic performance of a

hypothetical seismically isolated bridge on legacy highway, Report No. UT-11.01,

prepared for utah department of transportation research division,

11. Mahaney JA, Paret TF, Kehoe BE, Freeman SA. The capacity spectrum method for

evaluating structural response using the Loma Prieta Earthquake, National earthquake

conference, Memphis, 1993.

12. Mohamed ElGawady, William F Cofer, Reza Shafiei-Tahrany. (2000), Seismic

assessment of WSDOT bridge with prestressed hollow core piles, Technical research

report, Washington state transportation Centre (TRAC), Pullman, Washington,

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Technology,

14. Nasim Shatarat, (2003), Adel Assaf. Seismic behavior and capacity/demand analyses

of a simply-supported multi-span precast bridge, International journal of engineering

and applied sciences, pp 220-226.

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procedures to identify failure mechanisms from higher mode effects, Paper No. 966,

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17. Priestly MJ, Seible F, Calvi GM (1996), Seismic design and retrofit of bridges, John

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A method to identify the effects of higher modes in a pushover analysis, Proc. of the

U.S. National Conference on Earthquake Engineering,

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