structural identification using the applied element method advantages and case study application.pdf

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Chapter 29

Structural Identification Using the Applied Element Method:Advantages and Case Study Application

Matthew J. Whelan, Timothy P. Kernicky, and David C. Weggel

Nomenclature

f a Column vector of analytical natural frequency estimates

f e Column vector of experimental natural frequency estimates

a Matrix of analytical mode shape estimates

e Matrix of experimental mode shape estimates

W f Weighting matrix for eigenvalue residuals

W Weighting matrix for mode shape residuals

MAC Modal Assurance Criterion

Abstract Structural identification has continued to develop into a versatile tool for developing high fidelity analytical

models of large civil structures that accurately reflect the measured in-service response. The results of successful structural

identification have been applied to validate the performance of innovative systems and improve assessments of response

analysis for operational and extreme loads. Furthermore, the developing field of vibration-based damage detection has

sought to employ structural identification for long-term performance monitoring and condition assessment of aged structures.

Overwhelmingly, the finite element method has served as the analytical framework for such models. However, alternative

physics engines, such as the Applied Element Method, offer distinct advantages over the finite element method both with

respect to the computational considerations in the identification process and with respect to the use of the calibrated model forassessment of structural response to extreme loads. A general framework for structural identification with applied elements

is discussed, and advantages are contrasted with traditional finite element approaches. A case study application, a prestressed

concrete double-tee joist roof tested in a full-scale building, is presented to demonstrate the approach and emphasize these

advantages.

Keywords Structural identification • Applied Element Method • Extreme load analysis • Field testing of structures

• Double-tee joist roof

29.1 Introduction

Structural identification (St-Id) has emerged as a priority area of research and development in the field of civil engineeringto facilitate a transition to performance-based engineering and design. The framework promotes techniques within

which experimental measurements from in-service structures are obtained and then utilized to field calibrate analytical

models. Application results in enhanced fidelity of the prediction models and, consequently, improved phenomenological

understanding of built systems. However, as enumerated by a recent publication by the ASCE SEI Committee on Structural

Identification of Constructed Systems, the framework can also be leveraged: (1) in design through performance assessment

M.J. Whelan () • T.P. Kernicky • D.C. Weggel

Department of Civil and Environmental Engineering, University of North Carolina at Charlotte,

9201 University City Boulevard, Charlotte, NC 28223-0001, USA

e-mail: [email protected]

F.N. Catbas (ed.), Dynamics of Civil Structures, Volume 4: Proceedings of the 32nd IMAC, A Conference and Expositionon Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series,

DOI 10.1007/978-3-319-04546-7__29, © The Society for Experimental Mechanics, Inc. 2014

255

mailto:[email protected]

mailto:[email protected]



256 M.J. Whelan et al.

and verification of new or not well understood systems; (2) as a tool for performance-based quality assurance during

construction; (3) within periodic load rating and condition assessment of aged systems for asset management; and (4) to

facilitate long-term structural health monitoring [1]. The structural identification framework presents such a significant

potential to address a wide range of persistent challenges in life-cycle engineering and hazard mitigation that leading figures

in the field have called for the mandatory inclusion of St-Id in civil engineering curricula in addition to urging federal

agencies to consider this field a priority area for civil engineering [2].

Fundamentally, structural identification recognizes that there are too many uncertainties in structural systems to

completely and accurately predict the behavior without experimentally calibrating the analytical model. However, the currentpractice of structural identification does not completely address the full spectrum of uncertainty calibration as its application

by the current state-of-the-art yields only a calibrated linear model. While this significantly improves simulations taken to

the nonlinear regime by ensuring that the linear elastic region of response is reasonably predicted, the ultimate fidelity of

simulations for extreme loads and hazards as well as nonlinear limit states is dependent on the accuracy and completeness of

the simulation framework used in the final analysis and decision-making.The finite element method has been long established

as the predominant framework for structural identification, so much so that the ability to apply structural identification to

alternative analytical models and the associated advantages are often overlooked in the selection and construction of physics-

based models for calibration. However, application of structural identification can be expanded to a variety of analytical

frameworks that may offer distinct advantages when the calibrated model is used for decision-making, such as improved

simulation fidelity to extreme loads. In practice, emphasis should be placed in Step 5: Selection, Calibration, and Validation

of Physics-Based Models of the St-Id framework proposed in [1] to not only consider the size, complexity, and completeness

of the model, but also the framework within which it is developed so as to be most useful and reliable in the ultimatedecision-making stage.

29.2 Applied Element Method

One of the emerging alternative analytical approaches to the finite element method offering distinct advantages for the

analysis of structures subject to extreme loads is the Applied Element Method (AEM), developed by Meguro and Tagel-Din

[3]. The method, like the Finite Element Method (FEM), discretizes continua into a finite array of elements but an innovative

rethinking of element connectivity permits for natural handling of crack initiation and element separation. Unlike in FEM

where degrees of freedom are shared between elements at the nodes, the AEM approach assigns independent degrees of

freedom to each element at the element’s centroid. Element connectivity is facilitated through a matrix of normal and shear

interface springs that can have arbitrary nonlinearity to define the constitutive laws of the material or materials in contact

(Fig. 29.1). Since stress and strain for individual springs can readily be computed using the end displacements of the springs

and assigned material properties, the approach can identify localized fracture according to material constitutive models and

computed principal stresses [4]. Cracking is then naturally handled by removal of individual matrix springs and redistribution

of spring forces in the next time step of the simulation. Furthermore, fracture can continue until elements completely separate

and then automatically transition to discrete elements with motion dictated by Newtonian mechanics, collision forces, and

element contact forces. In the context of blast simulation and analysis, this is an exceptionally powerful combination since

it permits for seamlessly addressing both whether induced stresses will yield failure as well as completely describing the

failure mechanics through fragmentation and collapse. Comparison between AEM and LS-DYNA® finite element analysis

for reinforced concrete walls and columns subjected to blast loading has revealed strong agreement in the elastic and early

inelastic response with significantly less modeling and computational time required for the AEM simulations. Additionally,

AEM simulations have exhibited higher fidelity to physical experiments after initiation of fracture and damage by permitting

physical separation of elements into ejecta thereby releasing trapped momentum [5]. An AEM code has been implementedinto a commercial software package, Extreme Loading® for Structures, that has a Department of Homeland Security (DHS)

qualified anti-terrorism technology designation and has been demonstrated as an effective tool for the study of blast effects

on critical bridge components [6], forensics analysis of collapse [7], and protective design against progressive collapse [8].

Although the formulation of element connectivity and kinematics differs from FEM, the Applied Element Method still

develops structural stiffness, mass, and proportional damping matrices for simulation of dynamic response [9]. Consequently,

strategies for structural identification using experimentally measured natural frequencies and mode shapes can be applied to

AEM models without any reformulation of the theory. However, beyond producing a calibrated model suitable for higher

fidelity blast simulation, structural identification within the AEM framework provides several computational advantages

over FEM that should be emphasized. These computational advantages arise from the rethinking of shared nodal degrees of

freedom and element connectivity and are enumerated below:



29 Structural Identification Using the Applied Element Method 257

Fig. 29.1 Matrix springs and

element connectivity in the

Applied Element Method

Fig. 29.2 Computational

advantages of AEM over FEM as

a framework for structural

identification resulting from

reduction of number of matrix

equations due to: (a) less degrees

of freedom required per element;

(b) relaxed modeling

requirements for element

connectivity

1. Foremost, significantly reduced solution time is achieved by the reduced number of total matrix equations required to

discretize a structure into equivalent virtual elements (Fig. 29.2a). A three-dimensional applied element has six associated

degrees of freedom: three translational and three rotational about the centroid of the element. In contrast, the analogous

8-node brick element commonly employed in three-dimensional finite element analyses features three translational

degrees of freedom per node for a total of 24. It should be pointed out that while every unconstrained applied element

in the model introduces six degrees of freedom (i.e. a model with N applied elements has 6N equations), the shared

degrees of freedom introduced by nodal connectivity in finite elements means that the number of equations in the model

with N 8-node brick elements is significantly less than 24N . Therefore, the computational advantage achieved is problem

specific. Although a direct AEM to FEM model comparison was not developed for the case study examined in this paper,

prior comparison of equivalent discretizations of the masonry infill from the same experimental test program highlighted

a 44% reduction in the number of matrix equations and 55% reduction in the average eigenproblem solution time for the

AEM framework over the FEM model [10].

2. Nodal connectivity required in modeling continuous media with finite elements will often introduce excessive degrees of

freedom in instances where the solution does not require the level of mesh refinement that is required simply to generate

the desired geometry. This is often the case at the interface between materials of significant difference in stiffness (such as

soil-structure interaction problems) as well as in modeling the transition from regions of stress concentrations to regions of

low stress gradient. Since element connectivity in AEM is developed by matrix springs along the interface of the elements,

the connectivity requirements are relaxed to only requiring contact along the face of elements. This permits elements of

different size to be in contact without further mesh refinement as would be required in FEM (Fig. 29.2b). Consequently,AEM models can typically be constructed with far fewer elements than required by FEM to simply maintain connectivity,

which results in further reduction in the number of matrix equations and corresponding solution time.

Structural identification is predominantly performed using either iterative, sensitivity-based optimization strategies or

large search solvers, such as genetic algorithm. These techniques require a very large number of forward solutions of

the matrix eigenproblem for dynamic systems, so computational efficiencies associated with reduced model size produce

significant reductions in the time required to acquire a calibrated model. Regardless of whether the simulation capabilities

required in the ultimate decision-making stage of the structural identification routine require the advanced fracture and

element separation features afforded by AEM, the computational efficiencies provided by the modeling approach, as

enumerated above, should be considered a significant advantage when performing model updating of large and/or complex

structures.




29.3 Details of the Case Study and Model

In the current study, structural identification is applied to a prestressed concrete double-tee joist roof. The modal parameters

for this roof were estimated through multiple-input-multiple-output (MIMO) vibration testing and the system identification

details can be found in the companion paper [11]. The vibration testing was performed both prior to and after subjecting the

building to an internal blast load from an explosive charge placed approximately 1.8 m (6 ft) below the bottom surface of

the double-tees. Modal parameter estimates developed from vibration testing of the structure prior to the blast are presentedin Fig. 29.3 for comparison with the calibrated model developed through the structural identification described later in this

paper. Double-tee joists are designed with prestressing strand patterns for flexural resistance in positive curvature and are

not intended to be subjected to loads capable of producing significant negative moment in the joists. However, the blast

pressures produced by a charge positioned under the roof could produce significant negative moment in the joists and result

in cracking of the concrete in the flanges of the joists. Furthermore, the strength and ductility of connections in prestressed

concrete structures is often a critical design aspect under blast loading [ 12], so it is likely that the connectivity of joists may

have been compromised during the application of the blast load. The companion paper presented at this conference provides

modal parameter estimates obtained after the blast event that exhibit modest reductions in undamped natural frequency as

well as noticeable changes in the higher-order mode shapes relative to those obtained on the building prior to the blast.

To support the plausibility of speculations on the performance of the roof and residual condition developed from the

vibration testing, analytical modeling of the nonlinear response of the roof to the measured blast load is desired. However,

since this analysis is necessarily dynamic, a structural identification should be performed to field calibrate the model to

the measured modal properties. This is particularly necessary to establish reasonable boundary conditions, which have asignificant influence on blast resistance [13, 14]. As described in the companion paper, a three-dimensional AEM model of

the roof was developed within the Extreme Loading® for Structures software environment (Fig. 29.4). The model includes

Mode 1: f n=3.62 Hz x =12% Mode 2: f n=6.06 Hz x =4.8% Mode 3: f n=9.3 Hz x =3.2% Mode 4: f n=9.9 Hz x =1.5%

Mode 5: f n=10.44 Hz x =1.7% Mode 6: f n=11.17 Hz x =2.5% Mode 7: f n=12.05 Hz x =2.9% Mode 8: f n=13.27 Hz x =4%

Mode 9: f n=14.69 Hz x =4.3% Mode 10: f n=16.19 Hz x =3.1% Mode 11: f n=18.02 Hz x =3.9% Mode 12: f n=20.37 Hz x =4%

Mode 13: f n=21.19 Hz x =1.9% Mode 14: f n=22.63 Hz x =1.8%

Fig. 29.3 Experimental modal parameter estimates obtained from roof prior to blast l oading




Fig. 29.4 AEM model of the

prestressed concrete double-tee

joist roof developed within the

Extreme Loading® for Structures

software package (callouts

correspond to features

highlighted in Fig. 29.5)

Fig. 29.5 Summary of the uncertain modeling parameters included in the structural identification process: (a) stiffness of bearing support of select

stems at infill wall and superimposed mass of roofing material; ( b) stiffness of end bearing supports; (c) intermediate weld connection stiffness

and stiffness of filler material between adjacent joist flanges; (d) end weld connection stiffness

the 12 double-tee joists, which are welded together along the flanges at the quarter-points of the span as well as welded

to the exterior walls at the quarter-points. The only exception was a single exterior quarter-point weld in the bay adjacent

to the blast chamber, which was not present due to an opening in the wall that extended up to the roof. These welds were

modeled with rectangular cuboid elements either spanning across the joists, in the case of the intermediate connection welds,

or spanning over to constrained elements, in the case of the exterior connection welds. These modeling details can be seen in

Fig. 29.5. The bearing condition at both ends of the joists were similarly modeled using elastic elements sandwiched between

the base of the stem of the joists and constrained elements. Likewise, a similar approach was used to model the bearing of only two of the stems in the entire roof on the interior masonry infill wall, as identified in the field with a feeler gauge. Filler

material between the joists was modeled with as elastic elements in continuous contact with the flanges of adjacent joists. It is

important to note that boundary conditions in AEM are applied at the centroid of the element instead of at nodes. Therefore,

in order to introduce boundary springs within the applied element model, the elastic elements described are created to model

partial fixity between constrained elements and the structure. Accelerometers were explicitly modeled as elements bonded to

the bottom stems of the double-tee joists to allow for the model to produce outputs directly at the centroid of the sensors. In

the model, the unit weight and elastic modulus of the concrete in the double-tees were assigned fixed values of 1,922 kg/m3

(120 pcf) and 18.6 GPa (2,695ksi), respectively, as determined from measurements on a triplet of core samples obtained from

a double-tee flange. For this case study, prestressing strands were modeled, but nonlinear geometric effects on the stiffness

matrix introduced by the development of camber under the prestressing were not considered during the model updating for

reasons explained in the subsequent section.




29.4 Application of Structural Identification

Parameterization of the model included six uncertain parameters uniformly modified across the model, which are called out

in Fig. 29.4 and detailed in Fig. 29.5. These uncertain parameters include: (1) the elastic modulus of the bearing surface over

the interior masonry infill wall for the two stems bearing on the wall, (2) the superimposed mass of the roofing material,

(3) the elastic modulus of the bearing surface to adjust the partial restraint of the boundary, (4) stiffness of the intermediate

connection welds between joists, (5) elastic modulus of the filler material between adjacent joists, and (6) stiffness of theexterior connection welds. For the uncertain parameters related to elastic modulus, the shear modulus was also adjusted

proportionally according to an assumed Poisson ratio for each material. This selection of uncertain parameters was developed

using engineering judgment to create sufficient flexibility of the model to be calibrated to the field measurements, but also to

maintain a sufficiently small number of uncertain parameters to maintain a reasonable size search space.

A parallel implementation of genetic algorithm was applied for structural identification using a 25-core commodity

computing cluster running a distributed computing program developed in-house within the Matlab software environment.

This implementation assembles mass and stiffness matrices for candidate solutions using the linear superposition of weighted

parameter mass and stiffness contributions to a lower-bound model. This linear superposition approach allows for the

rapid assembly of candidate solutions outside of the Extreme Loading® for Structures software environment, thereby

eliminating the need for any Application Programming Interface (API) to interface with the model to facilitate the structural

identification. For this particular problem, the linear superposition approach does require that nonlinear geometric effects

associated with the camber produced by the prestressing force be neglected. However, these effects are generally insignificant

on the modal parameters of prestressed beams [15] and it has been common practice in structural identification to computeeigenproperties with respect to the undeformed condition.

The objective function used in the optimization scheme was defined as:

J D W f

f a f e

f e

CW

1

jfagH fegj

2

fagH fagfegH feg

(29.1)

where the first term computes the residual of the predicted and measured undamped naturalfrequencies as a percentage, while

the second compares the mode shape correlation using the Modal Assurance Criterion (MAC). The eigenvalue weighting

matrix and eigenvector weighting matrix were established proportional to the inverse of the measured undamped natural

frequency using the relation:

W f i;j

W i;j D 1

0:5

1

f eiıi;j (29.2)

where i and j are the indices associated with the elements in the weighting matrix ( i is also associated with the i -th

mode). The reduced weighting applied to the MAC residual emphasizes greater certainty in the undamped natural frequency

estimates than the mode shapes, which are more difficult to consistently measure. A population size of 1,000 individuals was

used in the genetic algorithm, which was bounded and used a heuristic algorithm for generation of the crossover population.

Other assignments used in the implementation of the genetic algorithm were an elite count of 25 and crossover fraction

of 80%. The optimization was run for 100 generations for a total of 100,000 forward solutions of the eigenproblem. The

minimization produced the model correlations presented in Fig. 29.6 and summarized in Table 29.1. It should be noted that

the mode shapes presented in this figure are plotted using only the model predictions from the locations of the sensors in

the experimental testing so as to produce a fair comparison with Fig. 29.3. The tradeoff in this presentation style is that joist

response may appear to be less smooth than predicted by the full model due to the effects of spatial reconstruction with

the limited data. In general, the calibrated model exhibits strong correlations with the experimentally measured natural

frequencies and mode shapes, particularly in light of the generally poor correlations exhibited with assumed materialproperties and idealized boundary conditions as described in the companion paper [11]. Natural frequency and modal vector

correlations are comparable with similar finite element-based applications of structural identification, although it should

be emphasized in this study that a large number of modal parameters were used in the structural identification and it is

challenging to achieve strong correlations with such a large set. The most significant discrepancy exists with the natural

frequency prediction for the second mode, however it can be seen in the stabilization diagram and frequency response

function presented in the companion paper that this mode was poorly excited and therefore the experimental estimate may

be inaccurate.




Mode 1: f n=3.62 Hz Mode 2: f n=5.1 Hz Mode 3: f n=9.7 Hz Mode 4: f n=10.15 Hz

Mode 5: f n=10.91 Hz Mode 6: f n=11.94 Hz Mode 7: f n=13.15 Hz Mode 8: f n=14.39 Hz

Mode 9: f n=15.66 Hz Mode 10: f n=16.9 Hz Mode 11: f n=18.05 Hz

Mode 13: f n=22.05 Hz Mode 14: f n=22.51 Hz

Mode 12: f n=19.04 Hz

Fig. 29.6 Modal parameter estimates developed from calibrated AEM model corresponding to the mode shapes experimentally measured

Table 29.1 Results from St-ID

by genetic algorithmMode f e (Hz) f a (Hz) (%) MAC Mode f e (Hz) f a (Hz) (%) MAC

1 3.62 3.62 0.0 0.969 8 13.27 14.39 C8.5 0.771

2 6.06 5.10 15.8 0.919 9 14.69 15.66 C6.6 0.685

3 9.30 9.70 C4.3 0.929 10 16.19 16.90 C4.4 0.625

4 9.90 10.16 C2.6 0.871 11 18.02 18.05 C0.2 0.602

5 10.44 10.91 C4.5 0.837 12 20.37 19.04 6.5 0.633

6 11.17 11.94 C6.9 0.809 13 21.19 22.05 C4.1 0.896

7 12.05 13.15 C9.1 0.745 14 22.63 22.51 0.5 0.535

29.5 Conclusion

Structural identification is developing an increasing role in the development of performance-based civil engineering by

affording the opportunity to field calibrate analytical models of large and complex civil structures to the actual in-service

response. Predominantly, the finite element method has served as the analytical framework for these models, however

practitioners should consider the ultimate fidelity expected during the use of the calibrated model in decision-making

when selecting an appropriate analytical framework. The Applied Element Method provides a proven, practitioner-friendly

framework for reliably analyzing the performance of structures and structural components to extreme loads, such as blast,

progressive collapse, and impact. Advantages of this method include the ability to handle crack generation, propagation, and

separation which are critical to the faithfulness of the simulation beyond elastic behavior and up to nonlinear limit states.

Within the implementation of structural identification, this method also provides computational advantages due to reduced

number of matrix equations and relaxed modeling requirements for element connectivity that can significantly reduce the

solution time of large optimizations. This paper has presented the application of structural identification to an AEM model




of a full-scale prestressed concrete double-tee joist roof using 14 undamped natural frequency and mode shape estimates.

The model correlation achieved through the model updating is comparable with similar finite element-based studies and

would likely improve the faithfulness of blast simulations run on the analytical model of the roof. Future work will explore

simulation of the calibrated roof model under the experimentally measured blast loading. Comparisons can be made with

shock acceleration and reflected overpressure measurements taken on the double-tee joist roof during the blast as well as

with the post-blast modal parameter estimates to evaluate the fidelity of the simulation and relative impact of the structural

identification on accuracy.

Acknowledgements The authors would like to acknowledge the support of Applied Science International, LLC, in providing licensing for the

Extreme Loading® for Structures software. Additionally, the authors would like to specifically acknowledge the technical support provided by

Michael Hahn and Ismael Mohamed of Applied Science International, LLC with modeling the structure in the Extreme Loading for Structures

software environment and exporting structural matrices for the eigenproblem. The authors would also like to acknowledge the assistance provided

by Corey Rice, Mike Moss, and Special Agent Yvonne Becker for assistance during the field testing of the structure.

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