Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 6092479, 16 pages

https://doi.org/10.1155/2018/6092479

## Nonlinear Analysis for the Crack Control of SMA Smart Concrete Beam Based on a Bidirectional B-Spline QR Method

^{1}College of Civil Engineering and Architecture, Guangxi University, Nanning 530004, China^{2}The Key Laboratory of Disaster Prevention and Structural Safety of the Education Ministry, Guangxi University, Nanning 530004, China^{3}Electric Power Research Institute of Guangxi Power Grid Co., Ltd., Nanning 530023, China^{4}Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Shuang-bei Li

Received 1 August 2017; Revised 10 November 2017; Accepted 16 November 2017; Published 11 January 2018

Academic Editor: Fabrizio Greco

Copyright © 2018 Yan Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A bidirectional B-spline QR method (BB-sQRM) for the study on the crack control of the reinforced concrete (RC) beam embedded with shape memory alloy (SMA) wires is presented. In the proposed method, the discretization is performed with a set of spline nodes in two directions of the plane model, and structural displacement fields are constructed by the linear combination of the products of cubic B-spline interpolation functions. To derive the elastoplastic stiffness equation of the RC beam, an explicit form is utilized to express the elastoplastic constitutive law of concrete materials. The proposed model is compared with the ANSYS model in several numerical examples. The results not only show that the solutions given by the BB-sQRM are very close to those given by the finite element method (FEM) but also prove the high efficiency and low computational cost of the BB-sQRM. Meanwhile, the five parameters, such as depth-span ratio, thickness of concrete cover, reinforcement ratio, prestrain, and eccentricity of SMA wires, are investigated to learn their effects on the crack control. The results show that depth-span ratio of RC beams and prestrain and eccentricity of SMA wires have a significant influence on the control performance of beam cracks.

#### 1. Introduction

Cracking is one of the main nonlinear characteristics of reinforced concrete (RC) structures. Once tiny cracks are created, they may expand and lead to accelerated corrosion of steel bars, which reduce structural reliability and durability. And the large cracks that exceed the limit may cause the structural failure [1]. Therefore, the durability and carrying capacity of structures can be effectively improved by controlling the development of cracks. With the continuous progress of material science, smart concrete provides a new and effective method to solve this problem. From the 1980s till now, many special functions of smart concrete are derived, including self-test, self-adjustment, self-cleaning, and self-healing [2–5]. The concrete added with new intelligent materials can effectively control crack expansion and even cause cracks to close, so as to improve the service life of the concrete.

Shape Memory Alloys (SMAs), a unique smart material, has been researched and applied more and more in intelligent structures and essential functional components [6–9]. There are two major characteristics simultaneously with the SMAs because of the reversibility of its phase transition. One is the shape memory effect (SME). In the martensite state, applying external force to produce deformation on the SMAs, the deformation will be recovered gradually when the SMAs are heated up to the transformation temperature of the austenite [10]. Depending on the feature above, the SMAs can be used to produce SMA smart concrete, which can improve the mechanical behaviors of concrete members, repair and control the deformation of the structure, and extend its service life.

At present, research of the SMA smart concrete is primarily concentrated on experiments. Numerous results show that concrete structures embedded with SMA wires are effective in deformation and crack control. Choi’s team [11] buried SMA wires after cold drawing in the tensile zone of the reinforced mortar beam and carried out a three-point bend test. And Li et al. [12] buried SMA wires in the reinforced concrete beam. Both of their results showed that the cracks could be closed with a great restoring force generated by heating the built-in SMA wires. Considering the possible complex situations in practical applications, some scholars have studied the related parameters that affect the driving effect of SMA wires and the controlling effect of structural deformation [13, 14]. Many test results show that the quantity, diameter, prestrain, excitation mode, and volume of the SMA wires have a certain influence on the deformation recovery of SMA concrete beam. In addition to the built-in method, SMA wires also can be arranged outside the structure, which has the effect of controlling deformation as well [15–17]. However, due to the complexity of mechanical behaviors of RC structures, there is less numerical analysis than experiment of the SMA concrete structure. Most of them are modeled and analyzed by software of the finite element method (FEM), such as ANSYS, ABAQUS, and SeismoStruct. Based on the superelastic constitutive law of the SMAs, Abdulridha et al. [18] established a finite element (FE) model to simulate the bond interface between SMA wires and concrete with a bond slip element and then obtained the load-displacement relationship of the beam. Khaloo’s team [19] used the ANSYS to establish a FE model of a cantilever RC beam embedded with SMA wires. The analysis results showed that, with the increase of the ratio of SMA reinforcement, the shear force and displacement hysteresis curve area of concrete beams decrease; the residual deformation of beams and the cross-sectional stiffness decrease as well. Alam et al. [20] established the FE model of SMA beam-column joints and analyzed its nonlinear behaviors by the SeismoStruct software. Chen and Andrawes [21] carried on the static analysis for the concrete column with SMA wires lateral restraint by the ABQUS software. When using the FEM to analyze the effect of the SMAs on the structure, it is crucial for the treatment of the SMA recovery force. There are some methods, mainly including pseudo temperature load method [22], negative thermal expansion coefficient method [23], equivalent eccentric force method [24], and equivalent reinforcement stress method [25, 26].

There is no doubt that the FEM is the most widely used numerical method so far, in which the whole structure is discretized with meshes. Due to the dependence of mesh generation, there are some drawbacks such as complicated solution, extensive calculation, and being time consuming for the FEM inevitably [27], while the meshless method can overcome these limitations when analyzing regular structures. In the meshless method, a set of points is used to discretize the solution region and construct approximation functions, which can eliminate meshes completely or partly and need no initiation and repetition of meshes. Thus, the calculation accuracy of the meshless method is ensured and the calculation difficulty is reduced. In the literature [28–30], the mechanical analysis of various structures is performed with different meshless methods, and a series of effective conclusions are obtained. Thereinto, the spline function method, which is semianalytical and semidiscrete, utilizes spline functions to construct displacement interpolation functions and has a widespread applicability and feasibility in different structural analysis problems. Liu et al. [31] used B-spline finite point method (B-sFPM) to analyze the free transverse vibration of axially functionally graded tapered Euler-Bernoulli beams. Compared with the FEM, the B-sFPM has the advantages of high efficiency and low computational complexity. Based on the B-sFPM, Li et al. [32] proposed the bidirectional B-spline finite point method (BB-sFPM), namely, the spline meshless method, for the parameter identification of piezoelectric laminated plates. This method discretized the structure along two directions with spline nodes, which is more accurate and effective in calculation.

Combining the advantages of the sFPM and the FEM, Professor Qin [33] put forward the unidirectional B-spline QR method (B-sQRM) named after him, which has been successfully implemented in the dynamic, static, and stability analysis of various structures. The B-sQRM is different from the sFPM. In the B-sQRM, one of the directions of the integral structure is discretized by uniformly scattered spline nodes, and the other is discretized by displacement shape functions which can reflect the deformation regularity of the structure. And the structural displacement function is constituted by a linear combination of the product of displacement functions and cubic B-spline functions which are compact and of high order. Moreover, by means of element interpolation functions, element potential energy functions and element stiffness matrices of the FEM, the displacement of element nodes is expressed by the integral displacement of the B-sQRM. Then, the stiffness equation of the entire structure can be established by the total potential energy function and the principle of minimum potential energy. Although the B-spline QR method depends on the discrete mesh and element accuracy of the FEM, the number of unknowns of the B-sQRM is only related to the number of spline nodes and the series of orthogonal polynomials; also it has nothing to do with the number of elements and the total number of node displacements. Therefore, the B-sQRM is of simplicity and convenience in calculation. Based on the above methods, the bidirectional B-spline QR method (BB-sQRM) is proposed in this paper. The principle of the BB-sQRM is the same as the B-sQRM, but the discretization is performed with spline nodes in both two directions of the structure. It has the characteristics of meshless method and can improve the calculation accuracy while the calculation efficiency is guaranteed.

In view of fact that the numerical analysis for nonlinear behaviors of SMA smart concrete beams is less and the employment of the FEM in modeling and analysis is more, therefore, in order to enrich the theoretical system of SMA concrete structures and to increase the diversity of methods, the BB-sQRM is adopted in this paper to perform a nonlinear analysis of the SMA smart concrete beam, which ensures the accuracy of the results while the computational scale is reduced and the efficiency of numerical analysis improved. Based on the classical incremental elastoplastic theory of the RC, the stress-strain relations of elements of concrete and concrete with steel bars under different stress conditions are given. By means of the element stiffness matrix of the FEM, the recovery force of SMA wires is equivalent to the eccentric load of spline node by the QR transformation, and the elastoplastic stiffness equation of the RC beam is deduced. Then, based on the iteration method of incremental initial stress, the calculation format for nonlinear analysis of the BB-sQRM is established. The nonlinear behavior such as cracking of the SMA concrete beam can be analyzed by controlling the temperature of SMA wires. Meanwhile, several parameters, such as depth-span ratio, thickness of concrete cover, reinforcement ratio, prestrain, and eccentricity of SMA wires, will be discussed to learn their influence on the capability of crack control, which can provide supplementary and further analytical means for test methods.

#### 2. Principles of the BB-sQRM

In the BB-sQRM, the two directions of the integral structure are discretized by uniformly scattered spline nodes, and the displacement field is constructed by the cubic B-spline functions corresponding to the boundary conditions. The most critical step in the BB-sQRM is the QR transformation presented in Section 2.2. Based on element stiffness matrices, the load vector, and transformed displacement vectors of element nodes, the conversion strategy of total potential energy functions of elements are presented. Thus, the total potential energy function of the entire structure can be obtained directly by adding potential energy functions of elements after the QR transformation. Also, the integration of the total stiffness matrix and the total load vector can be available from the sum of element items in the BB-sQRM, which need not expand before the superposition as the FEM. It means that the solution process is greatly simplified, and the calculation is more efficient and accurate. Moreover, the computer program would be easier to implement. When using the BB-sQRM to analyze nonlinear problems, it just needs to change the expression of element stiffness according to different nonlinear factors and then adjust element values in corresponding positions of the stiffness matrix after materials enter the plastic state.

##### 2.1. Construction of Integral Displacement Function of Structure

As can be seen from Figure 1, the plane structure is discretized along and directions. The discrete element is a rectangle element as shown in Figure 2.