Automated structural parameter estimation of semi-rigid complex joints in a benchmark laboratory steel grid by experimental modal analysis

Abstract

The mechanical properties of joints may impact a system’s static and dynamic behavior. Since connections are usually complex, modeling their geometric details could take time and effort. Thus, joints of structures are often overlooked in model creation and calibration. Therefore, reliable analytical modeling of in-service structures requires accurate and efficient parameter estimation of the connections in their simplified models. However, joints are physically small parts of a system, and parameter estimation techniques may not be sufficiently sensitive to the variations of connections’ mechanical properties. This paper examines the finite element model updating of a laboratory steel grid focusing on the structural parameter estimation of its complex connections using modal data. The mechanical properties of the joints are parametrized by added mass and reduced rigidity. Therefore, several modified models with different combinations of heavier semi-rigid joints are developed. Each model is updated using two modal-based error functions, and the most representative updated model is selected. The results demonstrate how the grid modal outputs are influenced by updating the mass and stiffness of its connections. Moreover, mass and stiffness interactions of the grid joints in the parameter estimation procedure are illustrated. The updated models can efficiently simulate the structural behavior of the grid with increased confidence and reliability.

Keywords

model updating parameter estimation modal data semi-rigid connection stiffness mass

1. Introduction

Since the transmission of forces throughout a structural or mechanical system depends on the function of the connections between its members, it is essential to evaluate the rigidity of joints. Despite being semi-rigid, most connections are generally treated as pinned or fixed to facilitate their analysis and design. This oversimplification could lead to uncertainties in developing reliable models. Calibrating stiffness parameters for complex connections can alleviate these problems. Moreover, the constituent elements utilized in steel connection zones, such as plates, stiffeners, and bolts, introduce additional localized mass, increasing the weight of the joints. Therefore, it is necessary to estimate both mass and stiffness parameters of connections appropriately.

The effects of the connections’ rigidity on the behavior of structures have been investigated by many researchers (Mehrkash and Santini-Bell, 2018). Wu and Li (2006) adopted weighted least squares and Bayesian estimation methods to identify the connection stiffness of beam-column joints. Sanayei et al. (2006) utilized their proposed combined multiple parameter estimation algorithms to update the finite element model of a laboratory grid structure with bolted joints. Santini-Bell et al. (2007) conducted similar research using the test data of the same laboratory grid structure. Cunha et al. (2008) identified the stiffness of semi-rigid bolted joints in framed structures formed by pultruded GFRP profiles. Turker et al. (2009) evaluated the effects of semi-rigid connections on responses of steel frame structures by determining the actual connections. Altunisik et al. (2010) presented a finite element model updating procedure for an arch-type steel laboratory bridge model with semi-rigid connections. Basaga et al. (2011) updated the stiffness of connections in the analytical models of two laboratory structures using the design points for unknown structural properties. Davoodi et al. (2012) determined the axial force-displacement relationship of a ball joint system by model updating of a double-layer grid. Zapico-Valle et al. (2012) proposed two models comprising beams to reproduce the dynamic behavior of a beam-column bolted moment connection. Wang (2014) extended the cross-model cross-mode method for model updating by incorporating connection flexibility. Using the method proposed by Sanayei et al. (2006) for the parametrization of joint models, Sanayei et al. (2015) estimated stiffness and mass values for the connection zones of the UCF benchmark laboratory steel grid. Yuan et al. (2016) presented a nonlinear joint model updating method with static structural data. Adel et al. (2017) updated the model of bolted joints in hybrid aluminum/composite structures using modal test data. Yuan et al. (2019) used the hysteresis Iwan model to describe the nonlinear mechanical behaviors of the contact surfaces. Ming et al. (2019) proposed a bottom-up model updating method based on modal and strain Frequency Response Function (FRF) and verified the method by an L-shaped joint. Dai (2020) proposed a model updating technique based on uniform design and applied it to a traditional beam-column connection. Mehrkash and Santini-Bell (2020) updated the model of the UCF Grid using experimental modal data by estimating the stiffness values of the grid joints. Omar et al. (2020) proposed an appropriate finite element representation of bolted joints. Aziz Shah et al. (2021) presented stochastic analytical modeling of a bolt-jointed structure. Basri et al. (2021) investigated using a model updating method to identify the invalid assumptions and uncertainties in adhesive jointed composite structures. Golea et al. (2023) proposed a generalized mechanical model for the characterization of structural joints to address the group effects and the actual deformability of the column web panel zone, disregarded in classical spring models. Wang et al. (2023) proposed an iterative joint identification method based on substructure decoupling for improving the accuracy and robustness in the case of many interface degrees of freedom and large noise. Kreutz et al. (2023) presented a robust procedure for the identification of bolted joints using frequency-based substructuring.

Most studies focused on the stiffness of joints, while the connections’ mass may considerably influence the dynamic behavior of structures. Hence, this study proposes an applied and efficient procedure for the selection of unknown parameters and input data. Also, simplified modeling and characterization of complex connections are discussed. Finally, the stiffness and mass of the complex joints in a benchmark laboratory steel grid are estimated using its experimental modal data. The studied structure is the benchmark steel grid, known as the University of Central Florida (UCF) Grid. The authors estimated the stiffness of the grid connections by updating its simplified analytical model (Mehrkash and Santini-Bell, 2020). In the current research, the proposed structural parameter estimation method is developed by incorporating the mass of joints. Therefore, it is shown how the interaction of mass and stiffness of the connections can alter the model updating procedure. First, the grid’s experimentally measured frequency response functions are used for modal extraction. Then the modal parameters are employed to update the joints’ stiffness and mass. It is assumed that the connections of the structure are semi-rigid. Their semi-rigidity is modeled by incorporating partial fixities at the connection zones. The rotational stiffness of the connections is estimated through an optimization algorithm. This stiffness as the main observable degree of freedom of the connection is selected based on the performance of the grid structure and the vertical excitation induced by the impact hammer. Different parametrizations and groupings are considered for the stiffness and mass properties of the connections. The parametrization regards selection of some structural parameters, for example, stiffness and mass, to be estimated, while grouping refers to constraining a set of structural parameters to have the same estimated value in the updated model. Then, structural parameter estimations are performed using modal-based error functions. Finally, the most representative model of the grid is selected based on the connection characteristics.

2. Methodology

2.1. Modeling connections and finite element model updating

The high-fidelity simulation of joints may not be conducted due to the required time and effort, and modeling limitations. Thus, updating the simplified model to make a trade-off between the simplicity and accuracy of analyses is essential. For the simplified modeling of joints, the most straightforward technique is constraining all the kinematics at the end of connecting members. This constraint builds a rigid joint with seemingly no structural parameters to be estimated. Alternatively, models may introduce semi-rigid connections to represent lower degrees of stiffness for less rigid joint configurations. Incorporating joint properties into models presents an additional challenge caused by the complexity of joints, which results in high uncertainties. Analysis and quantification of these uncertainties are critical to ensure structures’ safety, reliability, and performance and have been the topic of much research in recent years. For instance, Meggitt (2022) proposed an interval sensitivity analysis methodology to identify the uncertainty contribution of individual joints within a complex built-up structure. Connections are small parts of a structure, limiting their impact on the system’s global characteristics, such as natural frequencies. Analytical models often yield predictions different from experimental results. Using idealized connections in modeling is one factor leading to disparities between the model and reality. These discrepancies should be minimized through model updating. The modal-based methods can be employed for estimating the structural parameters of joints; however, numerical difficulties may be posed during the solution procedure. Structural connections are geometrically small parts of a structure, limiting their impact on the system’s global characteristics, such as natural frequencies. Therefore, the global modal characteristics may not exhibit observable sensitivity to joint parameters, while the structural performance can be affected.

2.2. Formulation of the optimization problem for structural parameter estimation

The optimization determines estimated values for the structural parameters corresponding to the objective function’s minimum. In the modal-based methods, the experimentally measured modal properties of the actual structure are used as the given information of the system. The problem can be linearized, and the optimization is performed iteratively.

2.2.1. Gauss-Newton algorithm

For finding a minimum of a nonlinear function $F (θ) : R^{n} \to R$ by linearization based on Newton’s method, the following relation is used (Aster et al., 2019)

Δ θ = - H^{- 1} \nabla F

(1)

where

Δ θ

\nabla F \in R^{n}

, and

H \in R^{n \times n}

are the vector of change in optimization variables, the Jacobian vector, and the Hessian matrix of the function, respectively. Also, sub/superscript

n

is the number of optimization variables. The Jacobian vector of the function and the function’s Hessian matrix are given by equations (2) and (3), respectively

\nabla F (θ) = {[\begin{array}{l} \frac{\partial F (θ)}{\partial θ_{1}} & \dots & \frac{\partial F (θ)}{\partial θ_{n}} \end{array}]}^{T}

(2)

H (θ) = [\begin{array}{l} \frac{\partial^{2} F (θ)}{\partial θ_{1}^{2}} & \dots & \frac{\partial^{2} F (θ)}{\partial θ_{1} \partial θ_{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial^{2} F (θ)}{\partial θ_{n} \partial θ_{1}} & \dots & \frac{\partial^{2} F (θ)}{\partial θ_{n}^{2}} \end{array}]

(3)

where superscript

T

denotes the transpose of a vector or matrix. Equation (1) is obtained by applying Newton’s method for solving

\nabla F (θ) = 0

As a specialized version of Newton’s method, the Gauss-Newton method is employed to minimize a nonlinear least squares problem, while a similar approach is used with a different Hessian. Hence, the objective function for an overdetermined system of equations is expressed in the following least squares form

F (θ) = \frac{1}{2} {‖ E (θ) ‖}_{2}^{2} = \frac{1}{2} {(E (θ))}^{T} E (θ)

(4)

where

E (θ)

is the stacked error function as given below,

{‖ . ‖}_{2}

indicates the Euclidean norm, and

m

denotes the number of measured modes

E (θ) = {[\begin{array}{l} E_{1} (θ) & \dots & E_{m} (θ) \end{array}]}^{T}

(5)

As equation (1) shows, the Jacobian and the Hessian of the objective function are required. The gradient of equation (4) is written as

\nabla F (θ) = {(\nabla E (θ))}^{T} E (θ)

(6)

where

\nabla E (θ) \in R^{m \times n}

is the Jacobian of the error function, that is, the sensitivity matrix, which is

\nabla E (θ) = [\begin{array}{l} \frac{\partial E_{1} (θ)}{\partial θ_{1}} & \dots & \frac{\partial E_{1} (θ)}{\partial θ_{n}} \\ ⋮ & ⋱ & ⋮ \\ \frac{\partial E_{m} (θ)}{\partial θ_{1}} & \dots & \frac{\partial E_{m} (θ)}{\partial θ_{n}} \end{array}]

(7)

Also, the Hessian of the objective function is

\nabla^{2} F (θ) = {(\nabla E (θ))}^{T} \nabla E (θ) + \nabla^{2} E (θ) E (θ)

(8)

here

\nabla^{2} E (θ) \in R^{n \times n \times m}

is the Hessian of the error function, as defined by

\nabla^{2} E (θ) = {[\begin{array}{l} \nabla^{2} E_{1} & \nabla^{2} E_{2} & \dots & \nabla^{2} E_{m} \end{array}]}^{T}

(9)

where

\nabla^{2} E_{i} (θ) = [\begin{array}{l} \frac{\partial^{2} E_{i} (θ)}{\partial {θ_{1}}^{2}} & \frac{\partial^{2} E_{i} (θ)}{\partial θ_{1} \partial θ_{2}} & \dots & \frac{\partial^{2} E_{i} (θ)}{\partial θ_{1} \partial θ_{n}} \\ ⋮ & ⋮ & ⋮ \\ \frac{\partial^{2} E_{i} (θ)}{\partial θ_{n} \partial θ_{1}} & \frac{\partial^{2} E_{i} (θ)}{\partial θ_{n} \partial θ_{2}} & \dots & \frac{\partial^{2} E_{i} (θ)}{\partial {θ_{n}}^{2}} \end{array}]

(10)

Also, $i = 1, \dots, m$ is the mode number. In the Gauss–Newton method, the second term of equation (8) is dropped. Hence, the approximate Hessian of the error function is simplified in the following form

\nabla^{2} F (θ) = {(\nabla E (θ))}^{T} \nabla E (θ)

(11)

Using equation (1) and showing the Jacobian of error with $S (θ)$ as the sensitivity matrix, the change in parameters is given by

Δ θ = - {(S {(θ)}^{T} S (θ))}^{- 1} S {(θ)}^{T} E (θ)

(12)

When the vector of change in parameters is computed in each iteration, the updated error is found by the following linearized relation

E (θ + Δ θ) = E (θ) + S (θ) Δ θ

(13)

2.2.2. MATLAB optimization toolbox™

Constrained optimization pertains to the task of minimizing an objective function $F (θ) : R^{n} \to R$ subject to one or more of the following constraints

\begin{array}{l} g_{j} (θ) = 0 & j = 1, \dots, p_{e} \\ g_{j} (θ) \leq 0 & j = p_{e} + 1, \dots, p \\ θ_{l} \leq θ \leq θ_{u} \end{array}

(14)

where

g_{j} (θ)

is a function containing the values of the equality and inequality constraints evaluated at

θ

, while

θ_{l}

and

θ_{u}

are the lower and upper bounds of parameters, respectively. For translating the constrained problem to a basic unconstrained problem, it is a good practice to solve Karush–Kuhn–Tucker equations. The following constraints, as the Kuhn–Tucker equations, are added to the original constraints in equation (14)

\begin{array}{l} \nabla F (θ^{*}) + \sum_{j = 1}^{p} λ_{j} \cdot \nabla g_{j} (θ^{*}) = 0 \\ λ_{j} \cdot \nabla g_{j} (θ^{*}) = 0 & j = 1, \dots, p_{e} \\ λ_{j} \geq 0 & j = p_{e} + 1, \dots, p \end{array}

(15)

where

λ_{j}, j = 1, \dots, p

are Lagrange multipliers. The algorithms employed to solve these equations aim to compute the Lagrange multipliers directly. Sequential quadratic programming methods are commonly used for constrained quasi-Newton optimization. These procedures are utilized in the active-set algorithm of fmincon in the MATLAB Optimization Toolbox™ (MATLAB, MathWorks, 2023).

2.2.3. Modal-based error functions

The minimization of an objective function implies the reduction of discrepancies between the measured response of the actual structure and its model predictions. If the measured modal properties are utilized, modal-based error functions can be formulated. The stiffness- and flexibility-based error functions are adopted in this research.

2.2.3.1. The modal stiffness-based error function

E_ms (θ) is based on the residual modal elastic and inertia forces predicted at a subset of degrees of freedom, as given by equation (16) for each mode (Sanayei et al., 1999)

E_{ms} (θ) = [K (θ) - ω^{2} M (θ)] φ

(16)

where

ω

is natural frequency,

φ

is mode shape vector, and

K (θ)

and

M (θ)

are stiffness and mass matrices, respectively. This characteristic equation is partitioned in terms of mode shapes at measured and unmeasured degrees of freedom for each measured mode

([\begin{array}{l} K_{aa} & ⋮ & K_{ab} \\ \dots & \dots \\ K_{ba} & ⋮ & K_{bb} \end{array}] - λ [\begin{array}{l} Μ_{aa} & ⋮ & Μ_{ab} \\ \dots & \dots \\ Μ_{ba} & ⋮ & Μ_{bb} \end{array}]) {\begin{cases} φ_{a} \\ \dots \\ φ_{b} \end{cases}} = 0

(17)

where subscripts

a

and

b

denote the measured and unmeasured degrees of freedom and

ω^{2}

has been shown by

λ

. This equation is expanded, and the unmeasured mode shapes are condensed out. Hence, equation (16) is written in the following form

E_{ms} = [(K_{aa} - λ M_{aa}) - (K_{ab} - λ M_{ab}) {(K_{bb} - λ M_{bb})}^{- 1} (K_{ba} - λ M_{ba})] φ_{a}

(18)

2.2.3.2. The modal flexibility-based error function

E_mf (θ) is based on the residual modal displacements predicted at a subset of degrees of freedom. This function can be stated by equation (19) for each mode (Sanayei et al., 2001)

E_{mf} (θ) = [λ K^{- 1} (θ) M (θ) - I] φ

(19)

where

I

is the identity matrix. If

K^{- 1} M

is denoted by

D

as the dynamic matrix, equation (19) is rewritten as

E_{mf} (θ) = [λ D (θ) - I] φ

(20)

The dynamic matrix and the mode shapes are partitioned based on the measured and unmeasured degrees of freedom in each mode

[λ [\begin{array}{l} D_{aa} & ⋮ & D_{ab} \\ \dots & \dots \\ D_{ba} & ⋮ & D_{bb} \end{array}] - I] {\begin{cases} φ_{a} \\ \dots \\ φ_{b} \end{cases}} = 0

(21)

By condensing out the unmeasured degrees of freedom for each mode, equation (19) is derived as follows

E_{mf} = [λ^{2} D_{ab} {(I - λ D_{bb})}^{- 1} D_{ba} + λ D_{aa} - I] φ_{a}

(22)

3. Test setup

3.1. The UCF grid

The structure studied in this research is a steel grid tested at the University of Central Florida (UCF). The response properties of the structure conform to the anticipated standards for bridges that cover distances of short to medium range. The structure is 18 ft. (5.49 m) long and 6 ft. (1.83 m) wide. Figure 1 displays the grid and also shows one connection, in which 30 bolts are utilized to connect the cross member with the girders using two clip angles and two plates. More detailed information about the structure is available in Catbas et al. (2008).

3.2. Instrumentation, impact test, and system identification

The grid was instrumented with eight PCB 393C accelerometers oriented vertically at Nodes 2, 3, 5, 6, 9, 10, 12, and 13, and excitation was applied on the grid using an IPC 086D20 impact hammer. The layout of the accelerometers and the location of one of the impacts are depicted in Figure 2. The test generated a multi-input multi-output dataset comprising four input and eight output signals. The frequency response functions were obtained from this dataset using five averaging trials for each input location.

Experimental natural frequencies and mode shapes are required to use modal-based parameter estimation methods. Accelerometers can measure structural acceleration, but time histories of these accelerations must be processed to obtain modal properties. Modal parameters can be extracted using the peak-picking technique in impact testing if the excitation source is known (Ewins, 2000). Signal processing techniques that utilize discrete Fourier transform (or power spectral density functions) can obtain frequency spectra of input and output signals, which are then used to create frequency response function plots. The peaks of these graphs correspond to the natural frequencies, while the imaginary parts of the peaks yield the mode shape components of each mode. This technique is based on vibrating mode contributions dominating FRF in resonance vicinity, while other modes have limited effects.

Nondestructive impact tests were conducted on the grid in the Civil Infrastructure Technologies for Resilience and Safety (CITRS) laboratory in the Civil, Environmental, and Construction Engineering Department at the University of Central Florida (UCF). In this study, the acceleration data corresponding to the impact applied at Node 10 (AC6) are used, as shown in Figure 2. Nodes 2, 3, 5, 6, 9, 10, 12, and 13 were instrumented by the accelerometers, so the time histories of their vertical accelerations were recorded during the impact test. Figure 3 shows the frequency response function of the accelerometer located at Node 2 (AC1). The spikes in plots of each FRF’s absolute and imaginary parts denote the grid’s natural frequencies and mode shapes, respectively. It is observed that the FRF peak values mentioned for the first 12 modes are detected clearly, which are considered for updating the analytical model of the structure; however, Modes 7, 8, and 9 are not captured by this instrumentation, as they correspond to the local vibration of Members 13, 16, and 19. Hence, the mentioned three modes are discarded from the estimation procedure.

4. Analytical model and joint parametrization

The analytical model is developed by beam elements in the SAP2000^® (SAP2000, Computers and Structures, 2023). Figure 4 shows the model and its predicted mode shapes. In the model, the grid members are divided into smaller elements to obtain more accurate predictions. In the initial model, all connections are modeled as fixed joints. This assumption is based on the fact that during the impact test, the bolts were tight. The bolt looseness in bolted joints complicates the connection modeling (Barhorst, 2008). The numerous holes drilled in the connection zones create uncertainties about the rigidity of the joints. Also, no additional mass is considered in the joint regions. Each accelerometer weighs 1.95 lb. (0.88 kg). Partial fixities are incorporated at the ends of the members as the rotational springs, while the additional mass is assigned to the joints. The analytical and experimental natural frequencies of the structure for the first 12 modes are given in Table 1.

Table 1.

Experimental and analytical natural frequencies of the UCF grid.

Mode number	Natural frequencies (Hz)
Mode number	Experimental	Analytical
1	22.3	22.6
2	26.8	28.4
3	33.3	33.9
4	40.6	43.8
5	64.6	62.4
6	67.6	65.6
7	—	73.2
8	—	73.2
9	—	73.4
10	94.1	96.3
11	96.4	99.5
12	102.4	109.6

In the structural parameter estimation, initial values for the mass and stiffness parameters of the connection zones must be specified. Unlike the mass values, variations of the stiffness ones have asymptotic behavior, that is, the rigidities greater than a threshold can be considered complete fixity. Therefore, it is a good practice to make a graphical comparison between the modal outputs of the grid with fixed connections and the ones corresponding to the structure with less rigid joints. Figure 5 shows the rotational springs at the ends of the grid members as the partial fixities in the plan of the model, while the same value is assigned to all the fixities. No springs are considered at the ends of Members 13, 16, and 19 because they do not contribute to the global modes of the structure. Variations of the natural frequency of each mode with the stiffness values of the partial fixities are displayed in Figure 6. This graph may give a rough insight into the initial stiffness values of the semi-rigid connections.

5. Results and discussion

Six different analytical cases are assessed based on the parametrization and combination of the joints’ mechanical properties, as shown in Table 2. The parametrization of Cases IV, V, and VI is similar, however, their order of parameter estimation is different, as described in Sections 5.4, 5.5, and 5.6.

Table 2.

Different scenarios for updating the parameters of the joints.

Case number	Longitudinal springs	Transverse springs	Mass
I	✓	✓	×
II	✓	×	×
III	×	×	✓
IV	✓	×	✓
V	✓(1st)	×	✓(2nd)
VI	✓(2nd)	×	✓(1st)

5.1. Case I: longitudinal and transverse springs considered

Here, the role of the transverse springs in the structure’s modal behavior is evaluated. Two analytical models are compared; first, all the grid connections are fixed, while in the second model, the transverse beams are modeled as simply supported. It was observed that the differences between the values of the two cases are negligible. Thus, it appears that the rotational rigidity of the transverse beams marginally impacts the modal characteristics of the steel grid. Since the behavior of these springs is not observable for the available structural response data, either individually or accompanied with other mass and stiffness parameters, they should be excluded from the structural parametrization of the grid.

5.2. Case II: longitudinal springs considered

For Case II, the release of the restraints is limited to the longitudinal girders. Based on the graph shown in Figure 6, k = 8000 kN.m/rad is taken as the initial value for the stiffness of the partial fixities. The lower and upper bounds of the parameter are specified as k = 2000 kN.m/rad and k = 18,000 kN.m/rad, respectively. The experimental modal data of the first 12 modes of the structure are contributed, while the local Modes 7, 8, and 9 are excluded. After updating the model, very large MAC values are observed for Modes 11 and 12, which indicates that the grid instrumentation is unsuitable for capturing these two modes. Therefore, Modes 7, 8, 9, 10, and 11 are excluded from the analyses for the remainder of this study. The graphs of the stiffness- and the flexibility-based error functions are shown in Figure 7, which are enlarged near the estimated value to highlight the objective function variations in its flat region. Unlike the stiffness-based function, there are some local minima, far from the global minimum, in the objective function plot of the flexibility-based function for the smaller stiffness values. The stiffness values estimated by the fmincon are 11,306 kN.m/rad and 9813 kN.m/rad for the stiffness- and flexibility-based error functions, respectively.

5.3. Case III: joints’ mass considered

In this section, it is assumed that the only parameter is the mass of the connections, which is estimated using the Gauss–Newton algorithm and the MATLAB Optimization Toolbox^™. The mass is defined as point assignment to the structural nodes of the instrumented joints. Assuming all eight nodes have the same mass, they are put together in one group. No additional mass is considered at the nodes above the columns, as their contribution to the structure’s vibration is insignificant due to the large axial stiffness of the columns. The objective functions plotted in Figure 8 show that a minimum exists around the accelerometer mass (m = 0.88 kg); Hence, 1 kg is chosen as the initial value for the mass. The lower and upper bounds are selected as 0.1 kg and 4 kg, respectively. The fmincon and the Gauss–Newton method yield identical results. The estimated mass (weight) values are 1.3 kg and 1.1 kg, for the stiffness- and the flexibility-based methods, respectively. The Gauss–Newton method converges after three and two iterations for the stiffness- and the flexibility-based functions, respectively.

5.4. Case IV: longitudinal springs and joints’ mass considered

In this case, a combination of stiffness and mass parameters of the grid’s connections is considered. The stiffness-based objective function is displayed in Figure 9. A similar plot can be shown for the flexibility-based objective function. The longitudinal stiffness asymptotically converges to a very large value, implying rigid connections exist. Hence, when the joints’ mass values are estimated, one could assume fixed connections instead of partial fixities, that is, Case IV might be merged with Case III.

5.5. Case V: estimate longitudinal springs first, and then estimate joints’ mass values

In Case V, the longitudinal springs group is first considered the only parameter, while no additional mass is assigned to the connections. The model with estimated partial fixities is updated as the mass of the joints is estimated. Hence, a two-step updating procedure is applied, whose steps are independent. The same initial values and lower and upper bounds are set for the stiffness and mass parameters. The graphs of the objective functions for this case are shown in Figure 10. The initial value and the lower and upper bounds of the parameter are the same as in Case III. The stiffness values estimated by the fmincon are 0.00069 kN.m/rad and 0.00029 kN.m/rad for the stiffness- and the flexibility-based error functions, respectively.

The estimated mass of the joints is significantly smaller than the ones estimated in Case III. These results are consistent with the expected dynamics of the system because, due to the introduced semi-rigidities, the structure becomes less stiff, so the grid should be lighter to restore the system’s natural frequencies. The discrepancy between the estimated joints’ weights is more substantial than in Case III.

5.6. Case VI: estimate joints’ mass values first, and then estimate longitudinal springs

In this case, first, the mass of the joints and then the partial fixities are estimated. If the stiffness- and the flexibility-based objective function graphs are plotted, one can see that the minimum occurs for very large values of the partial fixities. This observation is consistent with the dynamics of the problem because the mass increases, so the structure needs to become stiffer to restore the expected natural frequencies. The asymptotic nature of the stiffness variations simulates fixed connections at the end of the longitudinal girders, so this case can be merged with Case III.

5.7. Model selection

Any of the six introduced grid representations based on the structural parametrization of their connections may be used as an updated simplified model. However, a comparison can be made to examine which one more accurately describes the modal properties of the structure. The two-parameter Cases I, IV, and VI exhibited parametrization redundancy so that one of the parameters could be removed, and only Cases II, III, and V are compared here. The estimated parameters of these cases are summarized in Table 3.

Table 3.

Summary of the estimated structural parameters of the connections for the retained cases.

Case number	Error function	Used modes	Stiffness (kN.m/rad)	Mass (kg)
II	Stiffness-based	1, 2, 3, 4, 5, 6, 12	11,306	—
II	Flexibility-based	1, 2, 3, 4, 5, 6, 12	9813	—
III	Stiffness-based	1, 2, 3, 4, 5, 6, 12	—	1.3
III	Flexibility-based	1, 2, 3, 4, 5, 6, 12	—	1.1
V	Stiffness-based	1, 2, 3, 4, 5, 6, 12	11,306	0.69
V	Flexibility-based	1, 2, 3, 4, 5, 6, 12	9813	0.29

The natural frequencies of the updated models are given in Table 4. The natural frequencies of the original model and those of the measurements are listed again for comparison. The errors of the predicted frequencies by the updated models are presented in Table 5. Letters s and f denote the stiffness- and the flexibility-based error functions, respectively. The Root Mean Square Error (RMSE) values of the models are also given in the lowest row of the table. For Modes 5 and 6, the errors are greater than the errors of the original model. These results are consistent with the structure’s mode shapes, as the two modes primarily correspond to the vibration of the transverse beams. The errors of the several modes, particularly Modes 2, 4, 10, 11, and 12, have decreased in all retained cases. This observation also aligns with the expectations because these modes mainly represent the vibration of the longitudinal girders. Therefore, each parametrization case works better only in some vibration modes. The updated models predict natural frequencies closer to the experimentally extracted ones for most modes. Particularly, the frequency errors of the updated models in all retained cases show a significant decrease for Modes 2, 4, and 12. Case III yields the lowest errors, particularly the errors of Modes 2 and 11 obtained by the stiffness-based error function is zero for this case. This result is confirmed by comparing the RMSE values. For Cases II and III, the stiffness-based error function gives more accurate results, while the RMSE of the flexibility-based error function for Case V is smaller than its stiffness-based counterpart. Eventually, Case III using the stiffness-based error function, which has the lowest RMSE, can be identified as the most representative model.

Table 4.

Natural frequencies of the updated grid models.

Mode no.	Natural frequencies (Hz)
Mode no.	Original model	Case II (s)	Case II (f)	Case III (s)	Case III (f)	Case V (s)	Case V (f)	Experimental
1	22.6	22.2	22.1	21.8	21.9	21.7	21.9	22.3
2	28.4	27.9	27.8	26.8	27.0	27	27.4	26.8
3	33.9	33.1	32.9	32.8	32.9	32.5	32.7	33.3
4	43.8	42.6	42.5	41.2	41.6	41.3	41.9	40.6
5	62.4	61.6	61.5	61.9	62.0	61.4	61.4	64.6
6	65.6	64.9	64.8	65.3	65.4	64.7	64.7	67.6
7	73.2	73.2	73.2	73.2	73.2	73.2	73.2	—
8	73.2	73.2	73.2	73.2	73.2	73.2	73.2	—
9	73.4	73.4	73.4	73.4	73.4	73.4	73.4	—
10	96.3	96.3	96.3	93.7	94.1	94.9	95.7	94.1
11	99.5	99.3	99.3	96.4	96.8	97.6	98.6	96.4
12	109.6	107.1	106.7	103.4	104.3	103.6	105.2	102.4

Table 5.

Percent errors and RMSE of the predicted natural frequencies of the initial and updated models.

Mode no.	Errors
Mode no.	Initial model	Case II (s)	Case II (f)	Case III (s)	Case III (f)	Case V (s)	Case V (f)
1	1.3	−0.4	−0.9	−2.2	−1.8	−2.7	−1.8
2	6.0	4.1	3.7	0.0	0.7	0.7	2.2
3	1.8	−0.6	−1.2	−1.5	−1.2	−2.4	−1.8
4	7.9	4.9	4.7	1.5	2.5	1.7	3.2
5	−3.4	−4.6	−4.8	−4.2	−4.0	−5.0	−5.0
6	−3.0	−4.0	−4.1	−3.4	−3.3	−4.3	−4.3
7	—	—	—	—	—	—	—
8	—	—	—	—	—	—	—
9	—	—	—	—	—	—	—
10	2.3	2.3	2.3	−0.4	0.0	0.9	1.7
11	3.2	3.0	3.0	0.0	0.4	1.2	2.3
12	7.0	4.6	4.2	1.0	1.9	1.2	2.7
RMSE	3.13	2.51	2.45	1.27	1.36	1.62	2.01

5.8. Closing remarks

Here, some important points are highlighted as guidelines for selecting more efficient and reliable approaches for dealing with complex joints in modeling tasks. The proposed technique has several advantages. There is no need to model the details of complex connections. No more elements or springs are required in the model to simulate additional mass and semi-rigidity of joints at the connection zones. This joint modeling technique saves time and effort in the low-fidelity beam modeling of the large infrastructure. Both modal-based objective functions used in this research can capture the variations of the modal characteristics of the system with the stiffness and mass of its connections. Overshadowing of one or more structural parameters is detected easily, straightforwardly, and reliably. Grouping the joints’ structural parameters reduces the computation intensiveness. Further, one can inspect the behavior of the objective function graphically to ensure potential local minima do not mislead the optimization solver. Any number of modes can be considered for the estimation process, and some modes may be excluded if their contribution is troublesome and makes the results less accurate. MAC plots can be utilized to specify the modes that may not be captured accurately by a particular instrumentation layout. The asymptotic behavior of the semi-rigid connections can also be discerned by considering partial fixities at the joint zones. The estimation results by the two modal-based error functions are in agreement; however, their difference may be somewhat substantial in the two-step cases. Also, unlike the flexibility-based error function, which included several local minima in some cases, the stiffness-based error function had a single minimum. Hence, the latter error function may be a more reliable option, if there is a lack of confidence about bypassing local minima by the optimization solver.

Some downsides can be identified for the joint structural parameter estimation technique proposed in this study. The partial fixities are suitable tools for simulating translational and rotational springs at the ends of structural members. Nonetheless, if the connection is stiffened by parts such as stiffener plates, the additional members in the joint zones cannot be modeled by the fixities. In fact, the limiting state of a semi-rigid connection is a fixed connection without any stiffeners. Moreover, fictitious springs considered by the partial fixities incorporate some nonlinear terms in the stiffness matrix of the system. Therefore, building the sensitivity matrix may not be straightforward if the sensitivity methods are used to update the analytical models of the systems with semi-rigid joints. For taking the mass of the joints into account, additional mass values are assigned to the structural nodes representing the connections. If the mass of the connecting members is among the updating parameters, the elemental mass is lumped at the end nodes to be incorporated into the system mass matrix. The modeler must be mindful of these two different mass values assigned to the end joints and distinguish them if required. In SAP2000^®, the partial fixities and the point mass of structural nodes are input as dimensional quantities instead of unitless multipliers. Therefore, a great difference exists between the magnitudes of the updating parameters, so an alternative system of units should be chosen, or some sort of normalization may need to be applied to avoid potential numerical issues. In this study, the optimization algorithm was robust enough to resist the considerable difference in the order of mass and stiffness values in Case IV. Sequential updating procedures, such as Cases V and VI, could be other alternative solutions. The flowchart of Figure 11 illustrates the overall steps of the proposed model updating strategy.

6. Conclusions

The finite element model of the UCF Grid was updated solely based on the structural parameter estimation of its complex connections. Six different mass and stiffness parametrizations were considered for the connections of the structure. The modal properties of the a priori finite element model were promising, and the discrepancies between the initial model and the actual structure were insignificant. Thus, the model updating problem was well-conditioned, and the solution algorithms estimated the parameters of interest efficiently and accurately. Estimating both mass and stiffness parameters of the structural joints for the grid simultaneously was not advantageous and was rejected as a viable parameter estimation scenario. This issue is mainly due to the asymptotic nature of the semi-rigid connections, making them less distinguishable from fixed ones when accompanied by the mass of the joints in the parametrization schemes. This phenomenon led to considering Cases V and VI as two-step procedures to estimate the connections’ stiffness and mass parameters sequentially.

The grouping of parameters simplified the structural parameter estimation considerably and facilitated the inspection of the objective functions variations with the updating parameters. Such visual inspection of the objective function graphs had two main advantages. It was ensured that the parameter estimation algorithms would not get stuck in local minima, particularly when using the flexibility-based error function. Moreover, it helped specify the lower and upper bounds of the parameters in the constrained optimization methods involved. Ultimately, it was determined that the model representing the grid most accurately could be the one with fixed connections and estimated additional mass in the joint zones, provided that the parameter estimation is restricted to the connections’ structural parameters. The proposed algorithm highlighted the importance of thoughtful parametrization, input data selection, and careful evaluation and modification of the structural parameter estimation process for complex connections through a managed and controlled low-fidelity model updating and selection protocol. An aspect of the future work associated with this paper will include a parameter estimation methodology to conduct probabilistic and data-driven model updating of the structures with complex connections.

Footnotes

Acknowledgements

The authors would like to thank Professor F. Necati Catbas and his research group at the University of Central Florida for access to the UCF Grid’s experimental response data.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This material is based upon work partially supported by the National Science Foundation under Grant No. (1430260). Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

ORCID iD

Erin Santini-Bell

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