Elsevier

Engineering Structures

Volume 56, November 2013, Pages 1880-1892
Engineering Structures

Average acceleration discrete algorithm for force identification in state space

https://doi.org/10.1016/j.engstruct.2013.08.004Get rights and content

Highlights

  • A new average acceleration discrete algorithm is used for force identification.

  • Force identification results are better compared with ZOH or FOH discrete algorithms.

  • Experimental results from a planar steel frame are used for validation.

  • A shear wall building with steel frame is used for substructure force estimation.

Abstract

A discrete force identification method based on average acceleration discrete algorithm is proposed in this paper. The method is formulated in state space and the external excitation acting on a structure is estimated with regularization method. A three-dimensional three-storey frame structure subject to an impact force and random excitations is studied respectively with numerical simulations. Uncertainties such as measurement noise, model error and unexpected environmental disturbances are included in the investigation of the accuracy and robustness of the proposed method. Experimental results from a seven-storey planar frame structure in laboratory are also used for the validation. The above results are also compared with those from two existing force identification methods, which are based on the Zeroth-Order-Hold (ZOH) discrete algorithm and the First-Order-Hold (FOH) discrete algorithm. Model of a fourteen-storey concrete shear wall building is studied experimentally with shaking table tests to further validate the proposed method. The shear wall structure has a two-storey steel frame on top with base isolation. The interface force in the isolation at the bottom of the steel frame during the seismic excitation is estimated with the proposed force identification method.

Results from both numerical simulations and laboratory tests indicate that the proposed method can be used to identify external excitations and interface forces effectively based on the structural acceleration responses from only a few accelerometers with accurate results. The proposed method is capable to identify the dynamic load fairly accurately with measurement noise, model error and environmental disturbances.

Introduction

The dynamic load environment is an important component of design, condition assessment and health monitoring of structure. The external excitation estimation methods can be classified into two categories of direct methods and indirect methods. Force transducers are installed at locations where the forces are applied in the direct method. Though numerous types of force transducers have been developed, it is impossible to measure all the excitations on a structure directly due to the lack of access to the loading position. The number of required sensors may also be very large. The indirect method is an alternative tool for the evaluation of external forces acting on a structure [1], [2].

The load environment assessment is an inverse problem with ill-conditioning, and regularization method is usually adopted [3] for a solution. A lot of indirect force re-construction or force identification methods have been developed [3], [4], [5], [6]. They are usually based on the finite element model of the structure which is often inaccurate [4]. One of the most common errors is with the properties of materials which affect the accuracy of both the forward and backward analysis.

The force identification algorithm is usually formulated in the state space. The discretization of the continuous state space equation will, however, affect the accuracy and stability of computation. It would subsequently affect the estimation result in the inverse analysis. The formulation for force identification in state space has been solved directly with regularization method based on the Zeroth-Order-Hold (ZOH) Discrete method [7], [8], [9], [10]. Some researchers employed the ZOH discrete method for the first-order regularization [9], [10] in force identification based on the dynamic programming method, but the computational is expensive. Many methods have also been proposed for the identification of external excitations, including deterministic forces [11], stochastic forces [9] and methods based on artificial intelligence [12]. However, the accuracy of methods based on different algorithm of discretizing the continuous function, such as ZOH discrete method and First-Order-Hold (FOH) discrete method is rarely compared and investigated.

In this paper, a new force identification method based on average acceleration discrete algorithm is proposed in state space. Few literature reports on this kind of discrete method for the force identification. The method is formulated recursively in state space. A three-dimensional three-storey steel frame is firstly studied numerically with single excitation. In addition to measurement noise in the responses, model error in the structural material and unexpected environmental excitation are included in the study to investigate the robustness of the proposed method. Results obtained are compared with those from the force identification methods based on ZOH discrete method and FOH discrete method. For practical application purpose [13], a scenario of multiple random excitations identification is also numerically investigated with the three-dimensional steel frame structure.

A planar seven-storey steel frame constructed in the laboratory of the Hong Kong Polytechnic University was experimentally investigated to validate the proposed method. The impact force on the seven-storey is identified with the ZOH discrete method, FOH discrete method and the proposed average acceleration discrete method. Scaled model of a 14-storey concrete shear wall building was test on a shaking table in the laboratory of the Institute of Engineering Mechanics, China Earthquake Administration, for further investigation of the proposed method with substructural external force identification. The shear wall building made of reinforced concrete has a two-storey steel frame fixed at the roof with rubber base isolation. The property of this isolation could not be evaluated directly during the shaking table test. However, the interface shear force in the isolator between the steel frame and the shear wall building can be identified satisfactorily with the proposed method.

Section snippets

Dynamic responses of a structural system

The equation of motion of a N DOFs damped structural system subjected to external excitation can be represented asMx¨+Cẋ+Kx=LFwhere M, C and K are the mass, damping and stiffness matrices of the structural system respectively. F is the vector of external excitation forces on the structure and L is the mapping matrix for the input forces. x¨, ẋ and x are vectors of acceleration, velocity and displacement of the structural system respectively. Rayleigh damping model is assumed withC=a1·M+a2·K

Discrete equation in state space

The equation of motion of the structural system shown in Eq. (1) can be expressed continuously in the state space asż=ACz+BCL·Fwherez=xẋ,AC=0I-M-1K-M-1CandBC=0M-1.

The superscript C denotes matrices for the continuous system. Vector y(t) is assumed to represent the output of the structural system and it is assembled from the measurements withy=Rax¨+Rvẋ+Rdxwhere Ra, Rv and Rd  Rm×ndof are the output influence matrices for the measured acceleration, velocity and displacement respectively, m is

Force identification with iterative regularization method

The force identification based on the ZOH discrete method, FOH discrete method or the average acceleration algorithm as shown in Eqs. (A.3), (B.15), (17) respectively can all be written in a general form asY=HLF

It is noted that Eq. (18) is ill-posed for the identification when there is measurement noise in the system. A straightforward least-squares solution will produce unbounded solution. Regularization method would provide an improved solution to the ill-posed problem. The damped

Implementation procedure

  • Step 1: Obtain the mass, damping and stiffness matrices of the target structure.

  • Step 2: Conduct dynamic measurement on the structure. In the case of simulation studies, compute the responses of the structure under excitation using the state space method as the “measured” responses.

  • Step 3: Obtain the matrix of system Markov parameters from the finite element model of the structure based on the corresponding discrete method from Eqs. (17).

  • Step 4: Solve Eq. (18) and identify the forces acting on

Numerical studies

A numerical model of a three-dimensional steel frame structure as shown in Fig. 1 is investigated to validate the proposed force identification method with the average acceleration algorithm. The midpoints and two ends of the beams and columns are modeled as nodes in the finite element model. Nodes 1–16 and some important nodes useful for the simulation are shown in the figure. Dimensions and properties of the frame members are shown in Table 1 and the material properties are shown in Table 2.

Laboratory validation

The proposed method with average acceleration discrete algorithm will be validated with the experimental studies of a two-dimensional planar frame and a scaled model of a concrete shear wall building as follows.

Conclusions

A new method based on the average acceleration discrete algorithm is proposed for the inverse identification of external force acting on a structure. The force identification methods based on the equation in state space with ZOH and FOH discrete methods are also studied and compared with the proposed method. Iterative Tikhonov regularization is adopted for the force identification. Numerical studies with a steel frame and experimental validation with a steel frame and a scaled model of a

Acknowledgements

The work described in this paper was supported by a grant from the Niche Area Research Funding of the Hong Kong Polytechnic University Project No. 1-BB6F, Project No. 01319406 Supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, Projects No. 51161120360, No. 51308160 and No. 91315301 of National Natural Science Foundation of China and Beijing Institute of Architecture Design. Also thank for the support of Prof. Li Hui in Harbin Institute of

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