bridge foundation stiffness...

Bridge Foundation Stiffness Identification

Masoud Sanayei1 and Nathan Davis2

1Professor, Dept. of Civil and Environmental Engineering, Tufts Univ., Medford, MA 02155. E-mail: [email protected]

2Doctoral Candidate, Dept. of Civil and Environmental Engineering, Tufts Univ., Medford, MA 02155. E-mail: [email protected]

Abstract

There are great economic incentives motivating the reuse of existing foundations when deteriorated and damaged superstructures are replaced. The increasing reuse of existing bridge foundations warrants investigation into verifying foundation performance and condition of in service bridge foundations. A method is proposed which characterizes the load-response behavior of existing bridge foundations during daily vehicular loading without impeding traffic. It uses a limited number of substructure response measurements from an in service bridge taken during daily traffic heavy truck loadings. The results from this method can be used to verify the vertical, horizontal and rotational load-deformation behavior during daily loadings. Potential applications for the results obtained through this approach are: the detection of structural damage, identification of scour, identification of unknown foundations, and validation the load-deformation behavior predicted by finite element analysis. While loading during the tests would be limited to the magnitude which is caused by daily truck passage, the behavior observed from this loading can be used to update the parameters governing soil behavior which will lead to an improved assessment of the foundation behavior during failure loading or extreme events. A preliminary study is presented as a feasibly study for future use of the proposed method for bridge foundation stiffness identification using substructure measurements.

Keywords: Bridges; Foundation stiffness identification; Foundation reuse; Capacity rating; Full-scale bridge testing.

Geotechnical and Structural Engineering Congress 2016 1897

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Geotechnical and Structural Engineering Congress 2016

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Introduction

The objective of the research described in this paper is to develop a simple method for determining the load-response behavior of a bridge foundation under daily traffic loading. The present research is limited to bridges supported on piers comprised of a single or multiple rectangular or circular columns; however the method is applicable to any type of foundation including caissons, piles, or shallow footings.

Methods have long existed (Matlock and Reese, 1960) to approximate the load deformation response of laterally loaded piles. Bowles (1999) also gives methods for computing the lateral and horizontal deformations expected for deep and shallow foundations. These curves are generally designed to provide an envelope of behavior from initial loading to failure. Such methods can provide an approximation of the total deflection expected at failure or at service levels; it can be difficult to extract the boundary condition which exists service levels from these methods. This boundary condition is expected to behave as a linear spring for small, repeated loading, as shown in Figure 1.

Previous research has attempted to identify this boundary condition created at the ground surface using parameter estimation, (Santini, 1999), (Olson 2005), employing optimization to identify the foundation boundary condition which causes the entire model to have the best fit with measured data. These techniques are susceptible to modeling errors, especially when modeling bearing pads or pin/roller connections because they are in-between the measurements at the superstructure and the parameters being sought at the foundation.

To help reduce these potential sources of error, this research focuses on devising a test setup which reduces the need for structural modeling as much as possible. This method proposes creating a free body diagram of the bridge pier shown in 2D in Figure 2 which contains only the internal forces within the pier and the soil-structure reaction at ground surface to these forces. By measuring the 3 internal forces and 3 internal moments (for 3D measurements) at a point within a pier column, a free body

Figure 1: Transforming shallow foundation into springs

,m mY YM θ

,m mZ ZF Q,m m

X XF Q

SSSK

CK

Figure 2: Forces, moments, displacements and rotations for a 2D bridge foundation


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diagram can be drawn which eliminates the need to model the entire bridge above the free body diagram; including the top part of the pier, pier caps, bearing pads, girders, bridge deck and abutments. For the 2D case, only 2 translational and 1 rotational degrees of freedom are shown at the top of the free body diagram. Rotations and displacements of the bridge structure can then be measured within this free body diagram which can allow for direct calculation of a stiffness matrix governing the “Soil-Substructure Superelement” ([KSSS]), shown in Equation 1 (Maser, 1998). The individual entries of [KSSS] represent a simplification of the load-deformation behavior of the soil and foundation system into a series of spring constants at the ground surface as shown in Figure 1. The spring constants in the KSSS matrix are intended for use as boundary conditions during structural modeling, prediction of deflection during surface loading, condition assessment of existing foundations, and identifying foundation type when it is unknown. (Maser, 2001) investigated using the ratios of individual entries into this matrix to determine foundation type.

(2 )VV VH V

DSSS HV HH H

V H

K K K

K K K K

K K K

θ

θ

θ θ θθ

=

(1)

Foundation Stiffness Identification

The objective of this research is to analyze the behavior of a bridge foundation during daily vehicular loading. Daily vehicular loading refers to the vehicular loads induced during typical traffic conditions. The sources of these loads can range from small passenger cars to large trucks. The loads induced during vehicular loading can be relatively small in comparison with dead loads, and are generally expected to be away from the failure load. The small range over which daily vehicular loading occurs means that the responses to these loads cannot be highly non-linear in nature.

The total loading on a foundation during vehicle passage consists of a superposition of both dead and live loads. The dead load generally consists of a large, stationary compressive vertical force caused by the weight of the superstructure. Live load can have a number of sources, including vehicular load, wind load, rain/snow loads, etc. Since the dead load is stationary it cannot be explicitly measured, and this condition is used as the reference state of the bridge pier. Additional live load from the above mentioned causes will cause a change in internal stresses and position which can be measured using strain gauges, LVDTs, and tiltmeters. It has been observed in this research that while trucks passing at normal traffic speeds do cause some dynamic responses from the bridge, the majority of the observed response is pseudo static in nature, and bridge dynamics that occur during passage can easily be filtered out and disregarded. This allows daily traffic to be used as the excitation source to which the response is measured, eliminating the need to close the bridge to traffic for load


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testing or provide alternative sources of loading. Since the forces being applied to the foundation element are directly measured, it is not necessary to weigh these passing vehicles for this method; however a larger passing vehicle will produce forces and displacements further away from the noise floor, providing better results. Other forces such as wind and temperature induced loads primarily act on the superstructure, and will contribute to the forces measured at the top of the FBD. These forces are considered to be small in normal circumstances.

A passing vehicle will produce a variety of live load forces which will cause measurable strains in an instrumented pier column. The weight of the vehicle will produce additional compressive load in the column when it is near the column, and potentially a small tensile load if the vehicle on a non-adjacent span (although the net load considering the dead load will still be compressive). Additionally, as the girders in the bridge are subjected to bending forces, the deck will go into compression while the bottom fiber of the girder will go into tension and elongate. For simple girder supported bridge decks, the neutral axis of the composite deck/girder system is at a vertical offset with the top of the pier cap support (e.g., pin, rocker, or bearing pad). The elongation of the lower fibers of the girders will push the pier away from the vehicle and in turn create a horizontal force on the pier. A horizontal force applied to the top of a pier column will induce a linearly varying bending moment along the length of the column as the moment arm increases. Since bridge piers are connected to girder using a variety of connection details, the amount of force and moment transferred into a column will vary based on bridge type. For certain connection details, such as neoprene bearing pads, the amount of force transferred can be dependent on temperature, making it somewhat difficult to predict the exact loads caused at the foundation level from vehicular load. If the girder support imparts a moment into the top of the pier, it will be constant through the length of the column. When a vehicle passes over the bridge offset transversely to the centerline, shear forces and moments are developed transverse to the bridge deck. The amount of transverse force and moment which is developed is highly dependent on the geometry of the bridge, connection details, and the exact path the truck takes. In general the above discussion applies to both steel and reinforced concrete girders supported on pier caps.

Powder Mill Bridge

The Powder Mill Bridge (PMB), in Barre MA was selected as a test case for the proposed method is shown in Figure 3. The PMB is a 3-span continuous steel girder bridge in composite action with the reinforced concrete deck. It is a straight, non-skewed bridge supported by 2 abutments, each founded on 5 caissons and 2 piers, each comprised of 3 vertical 0.915m diameter reinforced concrete columns, with each founded on its own 1m diameter caisson.


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Finite Element models have been created, refined and calibrated to match the dynamic responses observed during testing (Sanayei, 2012), (Bell, 2013), (Sipple, 2014). These models served as the basis of the numerical simulations for predictions of response to vehicular loading on the bridge. In actual testing for determination of the foundation stiffness using this method, finite element models are not required.

The superstructure was instrumented during construction with 200 sensors which are not used in this research. Only a limited number of new sensors were installed on one bridge pier at the PMB and used for preliminary testing as described below.

Instrumentation

As shown in Figure 2, for two-dimensional foundation stiffness identification, six quantities are required at top of the free body diagram: the vertical force and displacement, the horizontal force and displacement, the bending moment and rotation. Figure 4 gives a schematic for a 2D test setup using 4 strain gauges, 2 LVDTs, and a tiltmeter for a total of 7 transducers per pier. These simultaneous measurements will be used to determine internal forces and deformations at 3 DOFs. Performing the test in 3D requires 8 strain gauges, 3 LVDTs, and 2 tiltmeters for a total of 13 transducers per pier. These can be used simultaneously to measure forces and deformations at 5 DOFs, not including shaft’s torsion. Twisting about the vertical axis is normally not activated.

While out of plane and torsional degrees of freedom exist for a 3 dimensional column, for many foundation types the out of plane forces will not interact with the in-plane forces in a significant way, allowing them to be decoupled. Since the PMB is a simple caisson foundation, no coupling is expected between the longitudinal (in traffic direction) and the transverse (perpendicular to traffic direction) horizontal directions, so the 2-D instrumentation scheme was employed. It is important to note

Figure 4:2D sensor setup for bridge column

Figure 3: Powder Mill Bridge in Barre, MA


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that this does not imply that the forces being transmitted to the foundation are uncoupled, only that the foundation below the top of the FBD has independent, uncoupled longitudinal and transverse behavior. For this reason, even though this method is being researched on the straight, non-skewed PMB, it will be applicable to many skewed and curved bridges. By performing the test twice, both the longitudinal and transverse stiffness matrices can be measured in separate tests. Since the vertical DOF is present in each direction, the tests are expected to produce a consistent vertical stiffness value. Additionally, since the foundation piers are circular, the cross section of the column and foundation are nearly identical when comparing the 2D longitudinal view with the 2D transverse view. For this reason the behavior is expected to be largely similar between the two views of the pier and foundation.

Measuring the pier torsional moment or rotation about the vertical axis would be very difficult, but it is not expected to be much torsional excitement during vehicular passage and this DOF is not expected to interact with the longitudinal or transverse displacement, rotation, or stresses. As a result, the torsional DOF is deemed insignificant and neglected in this research.

Determining forces at top of FBD

Strain gauges are capable of measuring strain, ε. By installing a pair of strain gauges vertically on the pier as shown in Figure 5, and using the estimated modulus of elasticity of the concrete, the vertical stress σz can be found. It is known from statics that the vertical stresses in a linearly elastic column can be given by:

Y XZ

Y X

M x M yP

A I Iσ = ± ± (2)

Where P is the applied vertical force, A is the transformed area, MY and MX are the moments about the X and Y axis, x and y are the distances from the neutral axis, and IY and IX are the transformed moments of inertia about the X and Y axes. By placing the strain gauges in plane and vertically, the equations for σ1 and σ2 simplify to:

1Y

Y

M rP

A Iσ = + (3)

2Y

Y

M rP

A Iσ = − (4)

Figure 5: Cross section of column with 2D SG setup


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Therefore, by finding σ1 and σ2, and knowing the transformed A, r, and IY, it is possible to find the vertical force and moment at each strain gauge pair with the following equations:

1 2

2P A

σ σ+= (5)

1 2

2Y YM Ir

σ σ−= (6)

By knowing the distance between each strain gauge pair it is possible to estimate the shear force which is constant throughout the column:

( )t bt b m Y Y

X X X

M MV V V

L

− −= = = (7)

Since the moment varies linearly, the moment at the FBD is:

( )

2

t bm Y YY

M MM

+= (8)

Measuring Displacements with LVDTs

The vertical and horizontal displacement can be obtained using LVDT’s, mounted vertically and horizontally, as shown in Figure 7. Since the displacements can be obtained directly at the top of the FBD, the displacements obtained are directly used in the estimation of [KSSS].

While obtaining the displacement measurements is more straightforward than obtaining forces, there are additional limitations imposed by the test setup and instrumentation. When using LVDTs, it is necessary to create a reference frame from which movement is being measured. As seen in Figure 7, it is possible to simply mount the horizontal and vertical LVDTs onto a tripod in the vicinity of the column being instrumented. This setup is simple and quick, however the LVDTs are still within the zone of influence of the column, so the reference frame may in fact move slightly with respect to the column. Attempts to move the LVDTs further from the column generally led to less stiff test frames which resulted in noisy measurements. It is generally expected that the movement of the ground will be very small.

Figure 6: Forces applied to section of column containing SGs

Figure 7: Tiltmeter (left), two LVDTs (bottom right), and strain gauge (top right) mounted to PMB column


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Tiltmeters and filtering of time domain signals

Tilt measurements are taken with tiltmeters which measure the electrical resistance of a bubble in fluid. While this is a reference free measurement, these tiltmeters are unable to measure the higher frequencies (~3Hz and up), which the LVDTs and strain gauges will capture. It has been observed from measurements at the PMB that only very small rotations occur at the measurement location. The high-gain tiltmeter used for these measurements still only required a range of a few mV, while a range of ±10V was used. It is expected much better measurements can be made by using a smaller voltage range on the DAQ.

Since all of these measurements will be used together to estimate the foundation stiffness matrix, [KSSS], it is important that they all contain the same data. During normal vehicle passage, vibrations can be seen at multiple low natural frequencies of the bridge in the calculated forcing functions and displacement measurements, but not the tiltmeters due to their natural filter (Figure 8).

By filtering all of the signals down to approximately 2.5Hz-3Hz, information about the vibrations at the bridge natural frequencies are removed, along with any sensor noise at frequencies above 3 Hz (Figure 9). Since a typical vehicle passage can take from 4 to 10 seconds, the vast majority of the energy is in the forcing, displacement, and rotation time histories are between 0.1 to 0.25Hz. Even when the energy above 3Hz is filtered out, the vast majority of the energy in the signal is still retained.

Estimating KSSS from Measured Data

Equations 2 through 8 allow the determination of the forces and moments applied to the foundation at the top of the FBD shown in Figure 2 at any instant in time. For a

Figure 8: (A) Vertical force (B) Vertical displacement (C) Longitudinal tilt time histories

Figure 9: Filtered (A) Vertical force (B) vertical displacement (C) Longitudinal tilt time histories


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single measurement, n forces and n displacements are obtained, where n is the number of DOFs, equal to 3 for the 2-D case. The force and displacements can be grouped into two vectors, {Fm} and {Qm}, which are both of size (n x 1). In can be shown through statics that the force and displacement vectors are related by the stiffness matrix of the entire system contained with the FBD, such that:

{ } [ ]{ }FBDF K Q= (9)

Where [KFBD] is a 2n x 2n matrix representing the stiffness matrix of the entire system contained in the free body diagram. {F} and {Q} are 2n x 1 vectors which contain {Fm} and {Qm}, as well as force and displacement vectors at the ground surface, {Fu} and {Qu}, such that:

{ }m

u

FF

F

=

(10)

{ }m

u

QQ

Q

=

(11)

Where the lower-case superscripts, m and u represent the measured location (top of FBD) and the unmeasured location, which is the ground surface. By this convention, it is important to note that {Fu} (n x 1) = 0, and {Qu} (n x 1) ≠ 0.

The stiffness matrix of the free body diagram, KFBD, is a formed from a combination of the stiffness matrix above ground column segment, [KC], and the soil-substructure-superelement stiffness matrix, [KSSS], as shown in Equation 11. Since our interest is only in the solution to [KSSS], it is necessary to partition [KFBD] in Equation 12 to allow the column stiffness to be condensed out, and a solution for [KSSS] can be obtained which does not require the measurement of the displacements at ground surface, [Qu]

mm mumm muC CFBD FBD

FBD mu uu uuum uuC C SSSFBD FBD

K KK KK

K K KK K

= = +

(12)

where,

mm mum mC Cum uu uuu uC C SSS

K KF Q

K K KF Q

= +

(13)

In order to solve for an n-DOF system, the number of measurement points, p, must be greater than or equal to the number of DOFs, n. These sets of measurements can be appended into 2 matrices, [FM] and [QM], which are both size (n x p).


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Least square fitting for [KSSS]

When the number of measurement points, p is equal to n, the number of DOFs, the [FM] and [QM] matrices are both square and invertible. When this is the case, there is only a single solution to the system of equations which can be obtained by inverting [QM]. While this solution is “exact” considering the data provided, in the presence of measurement errors, additional measurements allow for a “best fit” estimate to be created which reduces errors. In general, the process of finding a best fit stiffness matrix involves using a set of measurements so that p is larger than n. Such a system of equations is referred to as overdetermined and several methods are available for solving this problem.

Predictions of KSSS

Numerical predictions of KSSS were made using dynamic stiffness values given by Poulos (1980) and Novak (1974). Dynamic stiffness values for embedded shallow foundations can be found in Beredugo and Novak (1972). Dobry and Gazetas (1986) extend this work to include arbitrarily shaped foundations, while Dobry and Gazetas (1988) give dynamic stiffness and damping values for floating pile groups. These formulations were used to find an approximate dynamic stiffness for the PMB bridge foundation, given in (14) with units for each term:

2.5 9 0 0 / / * /

0 6 7 2.041 8 / / * /

0 2.041 8 9.1 8 / / * /SSS

e N m N m N m m

K e e N m N m N m m

e e N rad N rad N m rad

= − −

(14)

Monte Carlo Simulations

In order to assess the sensitivity of this method to measurement error, simulations of the proposed method were performed using a SAP2000 model of the PMB. Simulated measurements were obtained from the model and contaminated with a normally distributed random “error” having a standard deviation of 5% of the measurement taken; corresponding to 32% of the data having more than 5% error while 95% of the data has less than 10% error. The [KSSS] matrix was estimated from 5000 Monte Carlo observations, which were comprised of the simulated data with 5000 different sets of the described contaminations added. The results from the Monte Carlo analysis allow the method to be verified, the sensitivity of the math to measurement errors to be quantified, and any bias created by the method to be observed.

The simulated measurements were obtained from a multi-step static analysis of a simulated 72K truck driving over the PMB. A coupled spring, using the constants in Table 1, was used as the boundary condition for the bottom of the each column, essentially representing the [KSSS] spring constant expected to be found under


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operating conditions. Nodes were placed at the location where sensors will be placed during actual testing and SAP2000 is capable of outputting the simulated displacement, rotations and stresses of the pier column. The aforementioned contaminations were applied to the measured stresses and deformations and then [KSSS] was estimated using the forces and obtained from contaminated stresses and from the contaminated displacement and slope outputs. Table 1 shows the results from a Monte Carlo analysis of the PMB for the 2-dimensional test setup in the longitudinal direction.

Table 1. Monte Carlo analysis of the PMB for 2D test setup in the longitudinal direction

Stiffness coefficient

Mean of Stiffness

Bias (% diff)

Standard Deviation

Spread of Stiffness (%)

KVV 2.50 e9 (N/m) 0% 1.60 e7(N/m) 0.64%

KHH 6.02 e7 (N/m) 0.33% 1.21 e6(N/m) 2.02%

Kθθ 9.24 e8 (N*m) 1.55% 8.67 e7(N*m) 9.53%

KHθ -2.06 e8 (N) -0.82% 1.02 e7(N) 2.05%

Table 1 shows that this method was very resilient to errors when estimating the vertical stiffness, with virtually no bias and a standard deviation of 0.64% of the actual stiffness observed during the Monte Carlo simulation. This implies approximately 95% of the Monte Carlo observations gave a result for the vertical stiffness within ±1.28% of the actual value. The horizontal and cross rotational stiffness coefficients had a standard deviation of about 2%, implying this method estimated those coefficients within ±4% with a 95% confidence interval. The rotational stiffness was the most poorly estimated quantity with this method, with a 9.53% standard deviation.

Preliminary Testing

Testing has been performed and continues at the PMB employing this technique. Initial testing has been reliably estimating a consistent vertical stiffness, with horizontal stiffness, rotational stiffness, and cross rotational stiffness estimations having a greater spread.

Model Selection

Finite element modeling has become an increasingly common tool in the geotechnical field. While very high quality models can be made in commercial packages such as PLAXIS or ABAQUS, the variable nature of soil means these models can only be as good as the assumptions made about soil parameters, subsurface geometry, and structural parameters. While for the PMB, the geometry and structural material behavior is known with some confidence, there is less confidence when modeling soil


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behavior. One of the most simplistic approaches to modeling this foundation is to use a Mohr-Coulomb failure envelope to represent the soil behavior. There are 5 parameters of importance in a Mohr-Coulomb model: the modulus of elasticity, the cohesion intercept, the friction angle, the dilation angle, and Poisson’s ratio (Lade, 2005). By varying these inputs into a model created in PLAXIS or ABAQUS, the dependence of the bridge under vehicular load to these input parameters can be ascertained.

Future Work

Ongoing testing at the PMB and refinement of the sensor setup procedures are being performed. The immediate goal is to determine the repeatability of the measurements being performed and comparing them with the empirical formulations mentioned earlier, as well as with finite element models created of the PMB foundation. Further ahead, it is important to understand how sensitive [KSSS] is to the input values in the model, and whether or not the estimated [KSSS] can be used to improve confidence in the model, differentiate between foundation types, and identify different forms of damage.

References

Bell, E., P. Lefebvre, M. Sanayei, B. Brenner, J. Sipple, and J. Peddle. (2013). "Objective Load Rating of a Steel-Girder Bridge using Structural Modeling and Health Monitoring." Journal of Structural Engineering 139 (10): 1771-1779.

Beredugo, Y. O. and M. Novak. (1972). "Coupled Horizontal and Rocking Vibration of Embedded Footings." Canadian Geotechnical Journal 9 (4): 477-497.

Bowles, Joseph E. 1996. Foundation Analysis and Design. 5th ed. New York: McGraw-Hill.

Dobry, Ricardo, Gazetas, George, (1986). “Dynamic Response of Arbitrary Shaped Foundations”, Journal of Geotechnical Engineering, Vol. 112, No. 2, 109-135.

Dobry, R. & Gazetas, G. (1988). “Simple method for dynamic stiffness and damping of floating pile groups.” Géotechnique 38, No. 4, 557-574

Lade, P. (2005). "Overview of Constitutive Models for Soils." GSP 128, 1-34: American Society of Civil Engineers.

Maser, K. R., Sanayei, M., Lichtenstein, A., and Chase, S. B., (1998). "Determination of Bridge Foundation Type from Structural Response Measurements," SPIE, Proceedings of the Nondestructive Evaluation Techniques for Aging Infrastructure & Manufacturing, March 31-April 2, 1998, San Antonio, TX, SPIE Vol. 3400, pp 55-67. DOI: 10.1117/12.300125.


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Maser, K., Sanayei, M., Klimkewicz, L., Abedi, H. and Edgers, L., (2001). "Unknown Foundation Identification Using Stiffness Data From Field Measurements," Proceedings of Transportation Research Board, 80th Annual Meeting, TRB ID Number: 01-3388, January 7-11, 2001.

Matlock, H. and L. C. Reese. (1960). "Generalized Solutions for Laterally Loaded Piles." ASCE -- Proceedings -- Journal of the Soil Mechanics and Foundations Division 86: 63-91.

Novak, M. (1974). "Dynamic Stiffness and Damping of Piles." Canadian Geotechnical Journal 11 (4): 574-598.

Olson, L. D. (2005). Dynamic Bridge Substructure Evaluation and Monitoring. United States.

Poulos, H.G., Davis, E.H., (1980) Pile Foundation Analysis and Design, John Wiley & Sons, Inc., New York, NY.

Sanayei, M., Phelps, J.E., Sipple, J.D., Bell, E.S., and Brenner, B.R. (2012). "Instrumentation, nondestructive testing, and FEM updating for bridge evaluation using strain measurements." Journal of Bridge Engineering. Vol. 17, No. 1, January/February 2012, pp. 130-138.

Sipple, J. and Sanayei, M., (2014). "Full Scale Bridge Finite Element Model Calibration Using Measured Frequency Response Functions," Journal of Bridge Engineering, ASCE, J. Bridge Engineering, 04014103-11.


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