## 1 Introduction  
The ability to extract key parameters of existing structures using smart sensors attracted growing attention in past decades due to these parameters accurately reflecting structural in-service conditions. Such information is vital for decision-making in terms of maintenance and retrofitting. Among these performance parameters, deflection is a commonly used index. In the design of civil-engineering structures (bridges, buildings, tunnels, etc. ), the deflection limit is usually utilized as a control index of structures when subjected to potential external loads. The deflection limit for structures is a mandatory index in most design codes to ensure structural safety. It is essential to obtain deflection for the performance assessment of structures in operation. The methods to measure deflection are classified into direct and indirect methods. Displacement transducers are well-known tools for the direct measurement of structural deflection. Although this is a convenient method to measure deflection in a laboratory experiment due to perfect test conditions, it is hard to utilize this kind of sensor in practical engineering. In order to arrange this kind of sensor, the reference point is required, but it is difficult to find such reference point in real structures especially for long-span bridges and buildings. The global positioning system (GPS) is another direct method for measuring displacement and can be utilized in real structures [1]. Compared to flexural structures, the accuracy of GPS in a rigid structure which has relatively smaller deflection is still a concern. Recently, advanced techniques in other direct measurement tools have been developed. A new technique was developed by [2] for the displacement measurement of a flexible bridge based on the digital image technique. The feasibility of vertical displacement measurement was investigated by [3] for bridges using charge-coupled device (CCD) cameras. In addition, [4] developed a method for the displacement measurement of buildings based on webcams. These image-processing methods are confronted with the problem of the influence of light when applied in on-line monitoring. Indirect methods have also attracted growing attention. Among these techniques, deflection estimation using strain responses has been investigated by many researchers. The advantages of this concept are summarized as: (1) the reference point is not required; (2) various goals can be achieved using only one kind of sensor and costs can be reduced. According to the relationship of double integration between deflection and curvature, deflection can be estimated by curvature or strain [5]. In addition, the authors in [6; 7] presented curvature function expressed by polynomial, and the coefficients in polynomial can be determined by regression analysis of experiment data. Once the curvature function is achieved, deflection can be estimated by double integral. Other relevant studies about deflection estimation using curvature can be found in [8; 9; 10]. For structures subjected to dynamic load, dynamic deflection can be estimated using dynamic strain. Dynamic strain is the numerical summation of the results of strain mode shapes multiplied by modal coordinates, while dynamic deflection is the numerical summation of the results of displacement mode shapes multiplied by modal coordinates. Hence dynamic deflection can be obtained by strain mode shapes [11; 12; 13; 14; 15]. Moreover [16], investigated the displacement estimation of a two-dimensional plate based on the modal approach. The objective of this paper is to develop an improved moment-area method for the deflection estimation of engineering structures.