Advances in Civil Engineering

Volume 2018, Article ID 3084078, 9 pages

https://doi.org/10.1155/2018/3084078

## Reanalysis of Modified Structures by Adding or Removing Substructures

^{1}Department of Architectural Engineering, Chung-Ang University, Seoul, Republic of Korea^{2}Department of Architectural Engineering, Kangwon National University, Chuncheon, Republic of Korea

Correspondence should be addressed to Hee-Chang Eun; rk.ca.nowgnak@gnahceeh

Received 22 June 2017; Accepted 3 October 2017; Published 8 April 2018

Academic Editor: Gianmarco de Felice

Copyright © 2018 Yong-Su Kim and Hee-Chang Eun. 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 study considers structural reanalysis owing to the modification of structural elements including (1) addition of substructures, (2) removal of substructures, and (3) changes in design variables. Coupling and decoupling reanalysis methods proposed in the study are performed by using the concept of compatibility conditions at interface nodes between the substructures or between the original structure and the substructures. Subsequently, a generalized inverse method to describe constrained responses is modified to obtain the reanalysis responses. In this study, constrained equilibrium equations are modified to consider a reanalysis of a structure with the addition and removal of statically stable or unstable substructures. The proposed reanalysis method is examined by using five examples of handling coupling and decoupling reanalysis of a truss structure.

#### 1. Introduction

The design stage of complex structures may include the addition of substructures with respect to the initial structure or the removal of substructures from the original structure; this requires further analysis. Structural reanalysis involves predicting structural responses of a modified structure owing to changes in structural members by using initial information without repeatedly solving a complete set of modified simultaneous equations.

The reanalysis method is categorized into direct methods (exact methods) and iterative methods (approximate methods). A direct approach is efficient given changes in a few structural members. In contrast, an approximate approach is suitable given changes in several members. Exact approaches were first proposed in studies by Sherman and Morrison [1] and Woodbury [2]. Nair [3] mentioned that various reanalysis methods were only valid for small perturbations in structural parameters. The aforementioned studies considered large perturbations in structural parameters. Hager [4] reviewed the formulae proposed by Sherman–Morrison and Woodbury that relate the inverse of a matrix after a small-rank perturbation to the inverse of the original matrix. Akgun et al. [5] extended low-cost linear reanalysis in the spirit of the Sherman–Morrison and Woodbury formulae to update the inverse of a matrix.

Kirsch and Liu [6, 7] provided first-order reduced basis expressions for exact displacements and stresses in terms of cross-sectional variables. The concept of structural rigid body motion eigenvectors and a generalized structural compliance matrix was adopted by Huang and Verchery [8] to consider an exact structural static reanalysis method for locally modified structures. Cheng et al. [9] considered a structural static reanalysis due to added DOFs (degrees of freedom) by using a Guyan condensation method and an extrapolation method.

Liu et al. [10] presented a direct reanalysis method with added supports. Liu et al. [11] proposed a static reanalysis method given the modification of supports by using modified master stiffness matrices, a rank-one decomposition of the corresponding incremental stiffness matrix, and a sparse Cholesky rank-one update/downdate algorithm. Liu and Yue [12] considered static reanalysis problem with modification of deleting some supports using the initial information and preserving the ease of implementation. The virtual distortion method (VDM) has been extensively developed and proved to be a versatile reanalysis tool in various applications, including structures and truss-like systems. Kolakowski et al. [13] demonstrated the capabilities of the VDM both in statics and dynamics. Liu et al. [14] provided a superelement-based VDM to improve the efficiency of the FEM updating of large-scaled bridges by using static information. Garcia de Jalon and Viadero [15] provided a linear static reanalysis method of structures based on the displacement method with respect to the addition, elimination, or substructuring of one or more elements. Cheikh and Coorevits [16] presented a direct method to introduce the concept of a reflexive inverse and a decomposition method for a stiffness matrix to avoid its inversion.

Three possible modifications of substructures are considered as follows [7]:(1)Case in which the number of DOFs is reduced due to deletion of members and joints(2)Case in which the number of DOFs is increased due to addition of members and joints(3)Case in which the numerical values of variables are modified and the number of DOFs is unchanged

The dimensions of a stiffness matrix can increase or decrease corresponding to the addition or deletion of substructures, respectively. The addition and deletion of substructures are considered as the restriction and release, respectively, of a constrained condition with respect to existing responses. Thus, reanalysis commences with a concept of describing constrained responses. The constrained response with respect to the satisfying constraints uses a generalized inverse method [17] that is explicitly calculated. Response variation along the constrained path that deviates from the initial response path is estimated by constraint forces such that it is executed on the initial structure. Response variation of structural members is estimated to predict the response of the remaining members by using the Guyan condensation method.

This study considers direct reanalysis methods to predict modified responses on the above three modification cases in a truss structure. The basic concept in the study originated from the extension of constraints of the compatibility conditions at free interfaces between structures and substructures. The study derives modification forms of a constrained equilibrium equation by using a generalized inverse method to describe the resulting responses due to the addition and deletion of stable or unstable substructures. Five different case studies of coupling and decoupling of substructures in a truss structure are considered, and the validity of the proposed methods is illustrated.

#### 2. Formulation

##### 2.1. Generalized Inverse Method

The reanalysis of an original structure due to the attachment or removal of substructures is performed by using constraints. Reanalysis is performed by restricting or releasing constraint conditions by attaching or removing structural members. The generalized inverse method for constrained responses utilized in the study is summarized in the following section.

The equilibrium equation of *n* DOF original structure is expressed as follows:where and denote the symmetric and positive-definite stiffness matrix and displacement vector at the original state, respectively, and corresponds to the load vector.

With respect to the process of structural synthesis, it is necessary to consider compatibility conditions at interface nodes between the original structure and the substructures as constraints. It is assumed that the structural response is subjected to linear constraints as follows:where denotes an coefficient matrix, denotes an constrained displacement vector, and denotes an vector. All elements of the vector **b** should correspond to zeros when compatibility conditions at interface nodes in synthesizing substructures are considered. The basic property of the Moore–Penrose inverse and the least square method is used, and (1) and (2) are combined to derive a resulting constrained equilibrium equation as follows:where “+” denotes the Moore–Penrose inverse and .

The second term in the right-hand side of (3) denotes the variation in the displacement vector due to the existence of constraints. Both sides of (3) are premultiplied by , and thus the second term in the right-hand side of the resulting equation corresponds to the constraint forces required to satisfy the constraints. The constraint forces are expressed as follows:

Equation (3) is modified to describe the constrained response for establishing structural reanalysis, and reanalysis methods of several types of modified structures are thereby provided. The study presents coupling and decoupling methods to describe resultant responses due to the attachment and removal of substructures, respectively.

##### 2.2. Attachment of Substructures

It is necessary for a static response to be continuous at the end nodes of a finite element model for structural members. It indicates that the compatibility conditions at interface nodes between the adjacent members should be satisfied. Structural synthesis is an analytical process to assemble substructures as shown in Figure 1(a). Substructures are assembled into the entire structure as shown in Figure 1(b) by using a compatibility condition. The assembled substructures correspond to stable substructures of a full-ranked stiffness matrix or the floating substructures of a rank-deficient stiffness matrix.