Complexity

Volume 2018, Article ID 2167326, 16 pages

https://doi.org/10.1155/2018/2167326

## Numerical Study on Crack Distributions of the Single-Layer Building under Seismic Waves

^{1}Department of Bridge Engineering, Tongji University, Shanghai 200092, China^{2}Department of Civil Engineering, Shanxi University, Taiyuan Shanxi 030013, China^{3}State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Zhipeng Zhong; nc.ude.uxs@6002_zpz

Received 20 March 2018; Revised 17 April 2018; Accepted 22 April 2018; Published 15 May 2018

Academic Editor: Changzhi Wu

Copyright © 2018 Fenghui Dong 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

This paper conducts a numerical simulation of the antiseismic performance for single-layer masonry structures, completes a study on crack distributions and detailed characteristics of masonry structures, and finally verifies the correctness of the numerical model by experimental tests. This paper also provides a reinforced proposal to improve the antiseismic performance of single-layer masonry structures. Results prove that the original model suffers more serious damage than the reinforced model; in particular, longitudinal cracks appear on bottoms of two longitudinal walls in the original model, while these cracks appear later in the reinforced model; a lot of cracks appear on the door hole of the original model, and no crack appears in the reinforced model till the end of seismic waves; seismic damage of walls in the reinforced model is obviously lighter than that in the original model; dynamic responses at all observed points of the reinforced masonry are obviously less than those of the original model. Strains at all positions of the reinforced model are obviously smaller than those of the original model. From macroscopic and microscopic perspectives, the computational results prove that the reinforced proposal proposed in this paper can effectively improve the antiseismic performance of the masonry structure.

#### 1. Introduction

As a traditional structure form, masonry structures are widely applied in middle and small cities. Materials used by masonry structures, such as clay, sand, and stones, are local materials. Therefore, important materials including steel, cement, and timber can be saved, and engineering cost can be reduced. Masonry materials are featured by high durability and high fire resistance and do not require special technical equipment in construction [1–5]. Because of these advantages, masonry structures have been applied widely since ancient times. At present, the proportion of masonry structures in wall structures is more than 90% [6]. Masonry structures are the most utilized structure form in construction engineering. However, the traditional structure form is deficient in poor antiseismic performance, high weight, bad tension, and poor ductility. When an earthquake takes place, the structure is often damaged due to the serious displacement outside the complete wall plane [7–10]. Earthquakes have caused so many tragedies, involving house damage, wall cracks, and resultant living risks, as well as large destructions such as direct collapse, economic losses, and casualties.

Therefore, a lot of researches have been conducted on the antiseismic performance of masonry buildings at present. Zheng et al. [11] analyzed the mechanical performance of multilayer masonry structures under horizontal seismic waves, while the mechanical performance included force bearing characteristics, deformations, and damage forms as well as dynamic characteristics and antiseismic capacities of the structure under different force stages. Jia et al. [12] discussed common seismic safety problems of masonry walls, analyzed seismic damage of walls for masonry buildings in different intensity regions, obtained damage forms of houses in the intensity regions, and also proposed some suggestions on reducing seismic damage of masonry buildings. Liu and Tong [13] studied houses with a frame masonry structure, established a finite element model, applied seismic waves to the structure in order to conduct the elastoplasticity analysis, and analyzed changing trends of structural cracks before and after adding antiseismic walls. In order to study impacts of front longitudinal columns in walls on antiseismic performance of a multilayer masonry building, Liang et al. [14] conducted an experimental test on a masonry building using the vibration table and analyzed damage processes of this model as well as parameters including acceleration amplification coefficients and typical position strains. In order to study and prevent seismic damage of masonry structures, Liu et al. [15] used LS-DYNA to simulate collapse processes of a masonry structure, verified the correctness of the numerical model by experiments, and found weak positions in this structure under strong seismic waves, which proposed a powerful support to improve the antiseismic performance of masonry buildings. The macroelement technique for modeling the nonlinear response of masonry panels is particularly efficient and suitable for the analysis of the seismic behavior of complex walls and buildings. Therefore, Penna et al. [16] have established a macroelement model specifically developed for simulating the response of masonry walls, with possible applications in nonlinear static and dynamic analysis of masonry structures under seismic waves. Betti et al. [17] have conducted a comparison between different methods and numerical models to estimate the seismic behavior of unreinforced masonry buildings, and the model is able to predict the damaged areas and the incipient collapse mechanism, as well as the collapse load.

In these reports, detailed experimental tests and numerical simulation are mainly conducted on the antiseismic performance of multilayer masonry structures and some achievements are obtained. However, single-layer masonry structures in rural regions are rarely reported. Single-layer masonry structures are the main buildings in some regions. Residents do not have strong consciousness for antiseismic performance and prevention of these buildings. Therefore, it is very necessary to study and design the antiseismic performance. Dong et al. [18] conducted the numerical simulation of the antiseismic performance for single-layer masonry structures and proposed some reinforced measures, but failed to report the distribution, expanding processes, and detailed characteristics of building cracks. Lou et al. [19] completed the numerical simulation on the collapse of single-layer masonry structures and also conducted comparative verification between computational results and experimental results, but failed to propose any reinforced measure for improving the antiseismic performance of single-layer masonry structures. Aimed at this current status, this paper conducts the numerical simulation of the antiseismic performance for single-layer masonry structures, completes a systematic study on crack distribution and detailed characteristics of masonry structures, and finally verifies the correctness of the numerical model by experiments. This paper provides a technology support for the antiseismic performance of single-layer masonry structures.

#### 2. The Theoretical Basis of the Numerical Computation

##### 2.1. Modal Solution Theories

Modal analysis is used to determine the natural vibration characteristics of the structure, which is the natural frequency and mode shape of the structure. It is an important parameter to study the elastoplastic analysis of vertical earthquakes and also the basic premise of computing structural dynamics. Under the external load, the dynamic equation of the structural system at any moment is as follows: Among them, , , are the inertial force vector, damping force vector, and elastic force vector of the structure, respectively, and is the external load vector acting on the structure. Inertial force vector, damping force vector, and elastic force vector can be represented by and its reciprocal, as follows: Among them, , , are the mass matrix, damping matrix, and stiffness matrix of the structure, respectively. , , are the displacement, velocity, and acceleration vectors of the structure, respectively. The dynamic equation of the structural system at any time becomes as follows if formula (2) is substituted into formula (1): When the structural system is in the free vibration state, the vibration direction of the structural system becomes When the structural system is in the free vibration state, assuming that the displacement of the structure in free vibration is , the homogeneous equations can be obtained as follows when substituting the former into formula (4): Among them, is not all , so the value of the determinant of the coefficient matrix must be all 0; that is,

According to formula (6), the mass matrix and stiffness matrix of the structural system at any time can be solved, and then the natural frequency and mode shape of the structure can be extracted.

##### 2.2. The Constitutive Relation of the Solved Masonry

The constitutive relation of the solved masonry is one of the most important mechanical properties of the masonry structure and also the basic parameter that the model must input when solving masonry structure. At present, the constitutive relations of the solved masonry have mainly the following three kinds.

###### 2.2.1. Single-Segment Constitutive Relations

The constitutive relation of masonry under a short-term load is usually determined by the axial compression test. When the stress of the masonry structure reaches a maximum value, the masonry structure will suddenly collapse under normal circumstances. According to a great deal of experience, the constitutive relation of masonry structure material is put forward as follows [20]:

###### 2.2.2. Two-Segment Constitutive Relations

Based on the experimental results, Zhu and Dong give a constitutive relationship for the expression of a two-stage full-curve with ascending and descending segments [21]:

Formula (8) is relatively simple; although it can reflect the mechanical performance of masonry structures in the stress drop stage, the formula is not derivable at , indicating that the stress-strain curve is discontinuous. Zhuang conducted an experimental study of the masonry model and also gave the ascending and descending constitutive relations [22]:

###### 2.2.3. Polynomial Constitutive Relations

The stress-strain relation curve proposed by Turnserk and Cacovic is not only consistent with experimental results, but also smooth and continuous. Therefore, this paper selects the constitutive relation of the polynomial, and its constitutive expression is as follows [23]: Among them, is the peak stress, and is the strain corresponding to the peak stress.

#### 3. The Numerical Computation for Dynamics of Masonry Structures under Seismic Waves

##### 3.1. The Numerical Computation Model

In this paper, a single-layer masonry building with a relatively simple structure is selected. Related dimensions of this model are obtained through the real investigation, as shown in Figure 1(a). It is shown in this figure that doors and windows are set on the front longitudinal wall, and no hole is set on other three walls. Actual dimensions of the window hole are as follows: height is 1700 mm and width is 2150 mm. Actual dimensions of the door hole are as follows: height is 2800 mm and width is 1320 mm. No structural antiseismic measures including column and beam are applied to this model. In order to simplify the computation, all the surfaces of this model are smooth and even planes. Dimensions of the experimental model cannot be too large; in order to conduct the experimental verification on the correctness of the numerical model, a numerical model with scaled ratio 1 : 2 is used for the modeling. The finite element model is shown in Figure 1(b). The degree of freedom in the bottom of the masonry structure was constrained in 6 directions to simulate the actual condition. In order to realize the constraint, the degree of freedom of the element in the bottom of the finite element model was constrained. In respect of the large size of masonry models in this paper, if the finite element model of a masonry structure is established based on a discrete model, it will be limited by the computational software and computer performance. Besides, this paper is focused on the numerical simulation on a macroscopic damage type of a masonry structure, so that an integral model with easily establishing the finite element model is adopted. In this model, the integral modeling is conducted on masonry and concrete, and components are connected by co-nodes [24–28]. In addition, with regard to roofs of rear houses, a clay layer will be paved on the roof board. In view of features of the clay layer, its rigidity contribution is not considered in the finite element model, its mass is uniformly distributed on the roof board, and reinforced steel bars are simulated using rod elements with dual-directional forces. Solid65 elements are used for the masonry structure. In view of tension failure and breakdown failure, element size of walls and roofs is 0.15 m. All the materials are deemed as isotropic. Parameters of the masonry are as follows: compressive strength is 1.5 × 10^{6} Pa, tension strength is 0.14 × 10^{6} Pa, elasticity modulus is 2.4 × 10^{9} Pa, Poisson’s ratio is 0.15, and density is 1900 kg/m^{3}.