CALIBRATION UNDER UNCERTAINTY FOR FINITE ELEMENT MODELS OF MASONRY MONUMENTS

ABSTRACT

Historical unreinforced masonry buildings often include features such as load bearing unreinforced masonry vaults, and their supporting framework of piers, fill, buttresses, and walls.  The masonry vaults of such buildings are among the most vulnerable structural components and certainly among the most challenging to analyze. The versatility of finite element (FE) analyses in incorporating various constitutive laws, as well as practically all geometric configurations, has resulted in the widespread use of FE method for the analysis of complex unreinforced masonry structures over the last three decades. However, an FE model is only as accurate as its input parameters, and there are two fundamental challenges while defining FE model input parameters: (1) material properties and (2) support conditions. The difficulties in defining these two aspects of the FE model arise from the lack of knowledge in the common engineering understanding of masonry behaviour. As a result, engineers are unable to define these FE model input parameters with certainty, and inevitably uncertainties are introduced to the FE model.

As the complexity of the building increases, as is the case for historical unreinforced masonry buildings, the errors and uncertainties in the analysis also increase. In the presence of high and numerous uncertainties originating from multiple sources, deterministic approaches in which parameters are defined as constant values assumed to be known with certainty cannot be implemented reliably. Probabilistic methods, however, provide a rigorous and rational means in treating the uncertainty present in the FE analysis of historical unreinforced masonry buildings. The way in which uncertainty in

 

historical unreinforced masonry construction is treated is one of the novel and main contributions of this dissertation.

While building FE models, sometimes it is advantageous to model only a smaller portion of a larger structure. This substructure modelling approach not only reduces the computational time of FE analysis but also reduces required preliminary work for the model development. In this dissertation, substructure FE models of vaulted sections of two Gothic churches are calibrated using a Bayesian statistics-based procedure against physical evidence collected through experimental modal analysis.  During calibration both the FE calculations and experimental measurements are treated probabilistically. The probabilistic nature of the FE calculations stems from the fact that several FE model parameters which are determined to introduce significant analysis uncertainty, are treated probabilistically. The probabilistic nature of experimental measurements stems from the fact that a large number of repeated experiments are compiled in order to determine experimental uncertainty.  The fact that uncertainty in both numerical calculations and experimental measurements are accounted for is one of the novelties of this dissertation. The modal parameters measured on the vault are statistically compared to the predictions of the FE model during calibration.  According to the automated Bayesian statistics based calibration procedure, the posterior distributions for the appropriately selected calibration parameters, such as modulus of elasticity of the vault material, and support spring constants of the vaults, are obtained. This stochastic procedure is applied to the substructure FE models of the choir vaults of the National Cathedral, Washington, DC.  and to the nave vaults of Beverley Minster, Beverley, UK.

TABLE OF CONTENTS

LIST OF FIGURES ……………………………………………………………………………………….. viii

LIST OF TABLES …………………………………………………………………………………………. xii

ACKNOWLEDGEMENTS …………………………………………………………………………….. xiv

Chapter 1  INTRODUCTION ………………………………………………………………………….. 1

1.1  Introduction to the Problem ………………………………………………………………… 1

1.2 Problem Statement and Objectives ……………………………………………………….. 7

1.3 Research Hypothesis …………………………………………………………………………… 9

1.4 Model Verification, Validation and Calibration ……………………………………… 9

1.5 Scope of the Research …………………………………………………………………………. 11

Chapter 2  LITERATURE REVIEW ………………………………………………………………… 12

2.1 Introduction ……………………………………………………………………………………….. 12

2.2 Model Correlation ………………………………………………………………………………. 14

2.2.1 Visual Methods for Model Correlation ………………………………………… 14

2.2.2 Static Methods of Correlation …………………………………………………….. 15

2.3 Deterministic Model Calibration ………………………………………………………….. 17

2.3.1 Dynamic Tests ………………………………………………………………………….. 17

2.3.1.1 Scaled Laboratory Models ………………………………………………… 19

2.3.1.2 Existing Structures …………………………………………………………… 22

2.3.2 Calibration Studies ……………………………………………………………………. 25

2.3.2.1 Manual Model Calibration ………………………………………………… 25

2.3.2.2 Automated Model Calibration ……………………………………………. 28

2.4 Stochastic Model Calibration ………………………………………………………………. 312.5 Discussions and Conclusions ……………………………………………………………….. 34

Chapter 3  METHODOLOGY …………………………………………………………………………. 37

3.1 Introduction ……………………………………………………………………………………….. 37

3.2 Finite Element (FE) Analysis ………………………………………………………………. 39

3.3 Experimental Modal Analysis ……………………………………………………………… 44

3.4 Selection of Comparative Features ……………………………………………………….. 51

3.4.1 Comparative Features in Linear Dynamics …………………………………… 52

3.4.2 Comparative Feature Dimensionality …………………………………………… 53

3.5 Selection of Calibration Parameters ……………………………………………………… 55

3.5.1 Parameter Uncertainty ……………………………………………………………….. 55

3.5.2 Parameter Sensitivity …………………………………………………………………. 57

3.6 Test Analysis Correlation ……………………………………………………………………. 63

3.7 Bayesian Model Calibration under Uncertainty ……………………………………… 68

3.7.1 Mathematical Formulation of Calibration Algorithm …………………….. 69

3.7.1.1 Surrogate Model – η(x, t) ………………………………………………….. 71

3.7.1.2 Discrepancy Model – δ(x) …………………………………………………. 73

3.7.1.3 Experimental Errors – ε(x) ………………………………………………… 74

3.7.2 Propagation of Uncertainty …………………………………………………………. 75

3.8 Special Considerations for Masonry Structures ……………………………………… 79

3.8.1  FE Model Development for Masonry Monuments ……………………….. 80

3.8.1.1 Geometry ………………………………………………………………………… 81

3.8.1.2 Element Type Selection ……………………………………………………. 82

3.8.1.3 Meshing ………………………………………………………………………….. 85

3.8.1.4 Material Properties …………………………………………………………… 89

3.8.1.5 Boundary Conditions ………………………………………………………… 91

3.8.1.6 Loads ……………………………………………………………………………… 93

3.8.2 Dynamic Experiments on Masonry Monuments ……………………………. 94

3.8.2.1 Instrumentation ………………………………………………………………… 95

3.8.2.2 Data Acquisition ………………………………………………………………. 98

3.9 Concluding Remarks ………………………………………………………………………….. 99

Chapter 4  WASHINGTON NATIONAL CATHEDRAL …………………………………… 101

4.1 Introduction ……………………………………………………………………………………….. 1014.2 Description of the Structural System …………………………………………………….. 1024.3 Finite Element Model Development and Parameterization ………………………. 1044.4 Dynamic Experiments ………………………………………………………………………… 1124.5 Selection of Comparative Features ……………………………………………………….. 1184.6 Selection of Calibration Parameters ……………………………………………………… 1214.7 Test-Analysis Correlation ……………………………………………………………………. 1274.8 Characterization of Modeling Parameters ……………………………………………… 133

4.9 Discussions and Results ………………………………………………………………………. 136

4.9.1 Posterior Distributions of Calibration Parameters ………………………….. 137

4.9.2 Validation of the Calibrated FE Model ………………………………………… 139

4.9.3 Stability of Calibration ………………………………………………………………. 144

4.10 Concluding Remarks ………………………………………………………………………… 146

Chapter 5  BEVERLEY MINSTER …………………………………………………………………. 148

5.1 Introduction ……………………………………………………………………………………….. 1485.2 Description of the Structural System …………………………………………………….. 1495.3 Finite Element Model Development ……………………………………………………… 1535.4 Dynamic Experiments ………………………………………………………………………… 1625.5 Selection of Comparative Features ……………………………………………………….. 1715.6 Selection of Calibration Parameters ……………………………………………………… 175

5.7 Test Analysis Correlation ……………………………………………………………………. 177

5.8 Characterization of Modeling Parameters ……………………………………………… 179

5.9 Concluding Remarks ………………………………………………………………………….. 182

Chapter 6  DISCUSSION AND CONCLUSIONS ……………………………………………… 184

6.1 Summary of the Research Program ………………………………………………………. 184

6.2 Findings of the Presented Research ………………………………………………………. 188

6.3 Remaining Technical Issues ………………………………………………………………… 1936.4 Recommendations for Future Work ……………………………………………………… 195

6.5 Concluding Remarks ………………………………………………………………………….. 197

Bibliography …………………………………………………………………………………………………. 199

Chapter 1

 

INTRODUCTION

A computer lets you make more mistakes faster than any invention in human history—with the possible exceptions of handguns and tequila. 

Mitch Radcliffe

1.1  Introduction to the Problem

Growing interest in the preservation of architectural heritage has created a need for tools capable of reliably analyzing unreinforced masonry structures. The versatility of FE analyses in incorporating various constitutive laws, as well as practically all geometric configurations, has made the FE analysis a more generally applicable method for masonry systems compared to graphical or semi-graphical analysis methods initially proposed by Heyman (1966). Over the last three decades, FE methods became a widely applied tool for the analysis of unreinforced masonry structures. However, the success of FE model depends on the accuracy of its input parameters.

As the complexity of the problem increases, as is the case for historic masonry structures, the ability to fully incorporate the physical reality in the FE model decreases. The difficulties routinely faced during the FE model development of unreinforced masonry structures are primarily in obtaining physical dimensions and material properties. While defining these two aspects, uncertainty and error arise from numerous sources:

• Aside from natural variability between masonry units, the variable and time-dependent properties of mortar add uncertainty to the analysis. Even in cases where material coupons or spare stone units can be obtained from the structure, the limited number of tests provides statistically insignificant information. Even when these tests are considered representative, the properties of stone units alone are not sufficient to define the material behavior, as the behavior of masonry heavily relies on mortar properties (De Stefano 2007).
• Determining material properties of mortar is also problematic because extraction of an intact mortar specimen from an existing structure is a very challenging task. On the other hand, tests to measure mortar properties of young laboratory mortar specimens yield unrealistic results due to the agedependent hardening of mortar.
• The mechanical properties of a homogenized masonry assembly are strictly anisotropic due to the presence of mortar joints. However, these anisotropic material properties are difficult to determine due to the highly variable mortar joint thickness, hidden material defects, non-uniform dimensions of the stone units, and irregular layout of units and joints.
• The interior constitution of masonry construction, especially historic construction, often includes empty or roughly filled volumes and material discontinuity. Although an inspection of the interior constitution may be possible through thermal or radar-based methods, incorporating this information into an FE model is not straightforward.

 

• The geometry of masonry construction is almost always imperfect, even when built in laboratory conditions; for an example see the arch specimen of Ramos (2007). Moreover, the out-of-plane rotation of vertical members due to lateral loads, the flattening of arches and vaults due the formation of cracks, and the geometric deformation due to the movement of supports induce further variability to the geometry of these structures. Typically, in the FE analysis, the geometry is idealized. This aspect unavoidably introduces uncertainties in the analysis.
• The environmental conditions, such as temperature, are known to affect the behavior of masonry structures. Ramos (2007) noted another very important but less obvious environmental factor: the effect of moisture on a masonry system. Absorbed moisture increases the mass of stone units and reduces the stiffness of mortar joints. As a results, an increase in moisture results in a decrease in natural frequencies. As seen, environmental effects must be included in the analysis. However, unless the FE analysis incorporates probabilistic methods, it is difficult to include environmental variability.
• Effects of accumulated structural damage and past repairs or interventions on a historic masonry structure are often poorly documented. These aspects increase the number of unknown factors, and likewise increase the complexity of FE modeling.
• The effect of workmanship on the masonry structural behavior is known to be an important factor. However, it is very difficult to quantify the effects

of this factor for a large-scale historic structure and even more difficult to incorporate in the FE model.

To reduce the problem to a manageable size, it is crucial to establish appropriate assumptions and simplifications for each of these aspects related to the material behavior and physical geometry of masonry construction. Moreover, further uncertainties are introduced to the analysis while representing the support conditions in the FE model due to the complicated soil-structure interaction at the base of the structure. Accurate boundary condition representation also becomes a problem when the FE model is built to analyze a substructure of the entire system. Substructure modeling is feasible when (1) structural analysis is necessary only for a small portion of a larger structure, for instance when analyzing one of the spans of a multi-span system (Brencich and Sabia, 2008), (2) the structure of interest has a complex interaction with an adjacent structure which is not of interest, for instance when analyzing a tower that has a common wall with an adjacent building (Gentile and Saisi, 2007, Bayraktar et al. 2008 and Júlio et al. 2008), or (3) the structure has self repetitive components, in which analysis of one will be sufficient, for instance when analyzing a church with multiple nominally identical vaults (Erdogmus 2004, and Atamturktur 2006).

When building a substructure model, boundary conditions between components involve factors depending on contact pressure, surface friction, existing cracks, and load path, as well as the elastic behavior of each masonry unit and mortar. However, the connectivity options in general-purpose FE packages typically include translational and rotational restraints without providing any options to implement the more complex underlying physics such as joint friction, inelastic deformation, rigid body motion, etc.

On the other hand, an attempt to include these relevant physical phenomena further complicates the problem due to the unknown parameters of these phenomena. For macromodels, this additional complication is hardly justified. Thus, implementing the admittedly approximate boundary conditions available in the FE package still remains the option commonly selected by the engineer.

Many similar instances routinely experienced during FE model development of an existing masonry system limit the analysis capabilities to represent the physical reality. As a result, the burden of appropriate implementation of FE tools lies entirely on the skill and intuition of the engineer. When called upon to analyze an existing masonry structure, engineers are also confronted with a lack of analysis guidelines. Therefore, engineers are forced to choose an FE model, which according to their best engineering judgment will yield satisfactory results. An example of this common confusion regarding masonry behavior was recently reported subsequent to the Catoctin Creek Aqueduct elliptical arch restoration. The consulting engineers reported that the numerical model resulted in unrealistically high stresses within the stone arch (Biemiller L., 2006). When developing masonry structure FE models, particularly for historic structures, there are numerous opportunities to misinterpret the actual system, to build an unsuitable model, and to obtain erroneous solutions.

Over the last three decades, progress has been made in correlating FE solutions with physical evidence for civil structures with corresponding measurements (i.e., bridges, frame buildings, towers, stadiums, etc.), a procedure commonly known as model correlation. As the need for structural assessment of historic buildings increased, the model correlation concept has been applied to the masonry structure analysis such as masonry towers (e.g., bell towers, minarets), buildings (e.g., residential, public), and monuments (e.g., churches, mosques, basilicas, arch bridges,). Typically, when the FE solutions compare favorably with the corresponding measurements, this is accepted as a sign of accuracy of the model. However, if the comparison does not yield an acceptable match, the discrepancy is attributed to the deficiencies in the model due either to imprecise model parameters or due to erroneous modeling decisions.

Following the advancements in model correlation, researchers in other fields investigated the use of physical evidence to reduce FE model deficiencies, a process commonly known as model calibration. During the calibration process uncertain parameters are either manually or automatically adjusted until the resulting FE model reproduces acceptable agreement with the physical evidence. In this context, physical evidence is obtained through experimental measurements that are relevant to the identified deficiencies in the model. The relevancy of physical evidence to the model deficiencies is typically decided based on engineering judgment.

Calibration of masonry structure FE models requires considerations about the large uncertainty in masonry construction. This topic has not been fully addressed in the pertinent literature. To address this topic, this dissertation brings together the aspects of model calibration under uncertainty and outlines a probabilistic framework applicable to historic masonry structures. The study ultimately aims to obtain calibrated FE models with calculated uncertainty bounds on the input parameters. Such models will provide engineers the ability to predict masonry monument structural behavior with increased confidence where experimental technology is not readily available.

1.2 Problem Statement and Objectives

This study formulates a Bayesian calibration approach suitable for complex vaulted historic masonry structures and probabilistically characterizes the poorly known FE model input parameters.  The choice of the structure type is motivated by the high uncertainties associated with historic masonry systems as discussed in the previous section. The procedure outlined below may ultimately be applied to analysis of other civil engineering structures with high parameter uncertainties originating from numerous sources. Specific objectives of the study are outlined below:

Objective 1: Develop FE Models of Historic Masonry Structures

A model intended for calibration must be parameterized appropriately. The first objective is to present an FE modeling approach suitable for calibration activities.

The FE models in this study are developed based on observed geometry and construction of the selected case study structures. The models are representations of a substructure of the overall building and they include the ribs and webbing of a masonry vault, the adjacent nave walls and the fill. The boundary conditions, representing the structural interaction between the modeled elements and those that are excluded from the model, are to be abbreviated in the FE model. The boundary conditions that are difficult to represent through fixed or hinged connections are defined with linear elastic springs. This study is devoted to the determination of appropriate boundary conditions and material property values to be used in analysis.

Objective 2: Conduct In Situ Calibration Experiments of Historic Masonry

Structures

Inherent in their definition, calibration experiments are tied to the deficiencies in the FE model. As the aim of this study is to improve the quality of the FE model by calibrating parameters that are directly related to the stiffness and mass distribution in the system, nondestructive dynamic test results are used to obtain relevant physical evidence. This study devotes attention to the particular aspects of dynamic testing as applied to complex vaulted masonry monuments.

Objective 3: Calibrate the FE Models of Historic Masonry Structures Based on In Situ Dynamic Measurements

The spring constant and material property values, parameterized in Objective 1, are poorly known; therefore, they are calibrated with the help of physical evidence, obtained in Objective 2. The calibration procedure implemented in this study goes beyond a deterministic method that ignores the presence of uncertainty to one that relies on the definition and propagation of parameter uncertainty. With this step, an improved, quantitative knowledge is gained about the material property values for each structural component, as well as about the restraining forces applied by adjacent components to each other, for instance from buttresses to the nave walls.

Objective 4: Validate the Calibrated Model Parameters

The aim of this objective is to validate the results of the calibration study. In the absence of refined knowledge about the material properties, the probability distributions of the material property values obtained through calibration in Objective 3 cannot be validated. However, it is possible to estimate the boundary condition spring constants by modeling the remainder of the structure and to judge the acceptability of the calibration exercise completed in Objective 3.

1.3 Research Hypothesis

This study starts with the hypothesis that the FE solutions to predict phenomena of interest (A, in Figure 1-1) can be improved by calibrating the appropriately selected model parameters according to the physical evidence that is provided by experimental measurements (B, in Figure 1-1). When the calibration is completed, a separate and independent information set can be used to judge the calibrated model (C, in Figure 1-1).

Physical phenomena where estimates are available:  spring constant estimates via FE

Physical phenomena where measurements are available: nondestructive low amplitude dynamic tests

Physical phenomena of interest where measurements are not available: tensile stresses, maximum deformations

 

Figure 1-1:  Calibration of the imprecise input parameter of the numerical model by the use of comparative features.

1.4 Model Verification, Validation and Calibration

Also of far-reaching importance is defining model calibration in a larger context and emphasizing its role in relation to model verification and validation. The terms calibration, validation, and verification are used interchangeably in the literature, hindering the adequate communication of these principles. To provide clarity, this section describes what model calibration is and is not.  For this clarification, the factors to which the accuracy of the FE solutions is dependent are listed below:

1. the adequacy of the governing equations involved in the analysis, i.e., mathematical definitions for dynamic behavior of shells,
2. the precision of numerical solution, i.e., fineness of discretization,
3. the accuracy of the physical parameters, i.e., values for material properties and definitions for boundary conditions, and
4. the adequacy of the constitutive element models, i.e., assuming linearity only when the response is predominantly linear.

The first two factors are purely mathematical and are the topic of model verification. As Roache (1998) states, model verification aims to answer the question: “Are we solving the equations right?” When disagreement between model predictions and measurements are believed to be the result of inadequate mathematical representation or imprecise numerical solution, verification activities must be initiated. Verification is a prerequisite to validation activities. Although the crucial role of verification is acknowledged, only a very limited attention is paid to the verification procedures in this dissertation.

The last two factors are based on physical phenomena and the assessment relates to the model validation. According to Roache’s definition, model validation aims to answer the question: “Are we solving the right equations?” When an FE analysis reproduces a match to a set of physical evidence, the model is typically considered validated. However, when there is disagreement between model predictions and physical evidence, the numerical model can be calibrated as discussed earlier. When the first set of physical evidence is used to calibrate a model, a separate and independent set of physical evidence must be gained to validate the FE model. Therefore, calibration can be considered as a subcomponent of validation. An extensive discussion about the semantics of Verification and Validation has been provided by Trucano et al. 2006.

1.5 Scope of the Research

This study is confined to linearly elastic analyses based on the FE method. This is a necessary step that needs to be successfully completed before nonlinear and inelastic characteristics of masonry can be incorporated into an FE analysis.

The study will be accomplished through investigation of records of an experimental program and simulations of two monumental, unreinforced masonry buildings: The National Cathedral (Washington D.C., USA) and Beverley Minster (Beverley, UK). These buildings contain characteristics of stone masonry monuments,

e.g., piers, walls, buttresses, and vaults.

CALIBRATION UNDER UNCERTAINTY FOR FINITE ELEMENT MODELS OF MASONRY MONUMENTS

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