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.
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 [
Kirsch and Liu [
Liu et al. [
Three possible modifications of substructures are considered as follows [ Case in which the number of DOFs is reduced due to deletion of members and joints Case in which the number of DOFs is increased due to addition of members and joints 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 [
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.
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
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
The second term in the right-hand side of (
Equation (
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
Synthesis of an original structure and
The original structure and substructures are modeled by using finite elements. It is necessary for the static response due to the addition of substructures to be continuous at the interface DOFs. Reanalysis involves predicting displacement variation caused by interactive forces.
The original structure and
The equilibrium equation is condensed by using the Guyan method to reduce the computational operation. Displacements corresponding to internal DOFs from the first equation at each simultaneous equilibrium equation of (
With respect to unstable substructures, it is not possible to derive the displacements in (
Furthermore,
Original truss structure.
Modified truss structure adding a roller support.
The equilibrium equation of the original truss structure is expressed as follows:
The constraint condition to restrict the vertical displacement at node 4 is expressed as follows:
This section considers the reanalysis of a modified structure due to the addition of rank-deficient floating substructures. The condensation method and the generalized inverse method assume a positive-definite full-rank stiffness matrix. The generalized inverse method is modified to obtain a constrained response due to the rank-deficient stiffness matrix.
It is assumed that the initial structure B in Figure
Coupling and decoupling of substructures: (a) substructure A; (b) initial structure B; (c) entire structure.
The equations are divided into equations corresponding to the internal DOFs and interface DOFs as follows:
The Guyan condensation method is used to derive the equilibrium equation at the boundary DOFs as follows:
The compatibility conditions at the interface nodes are expressed as follows:
Equations (
Equation (
Coupling and decoupling of a substructure composed of three elements. (a) Initial and modified truss structure. The dashed line represents the added substructure. (b) Added floating substructure.
The removal of substructures from a structure leads to a decrease in the initial DOFs. The removal indicates the release of compatibility conditions with respect to the interface nodes, and the structure is separated. A decoupling process is performed as a reversal of the coupling process. The removal of five elements connected at node
Substructure A is divided into an internal node and five boundary nodes, and the Guyan condensation method is applied to express this as follows:
The equilibrium equations at the 16 boundary nodes of the structure B are expressed as follows:
The constraints of compatibility conditions at free interface DOFs are established at the interfaces and expressed as follows:
Equations (
Equation (
The decoupling indicates the release of DOFs at the interface nodes. The boundary DOFs of the substructure B include the DOFs at nodes
The section considers the reanalysis of the modified structure due to changes of design variables. The number of DOFs is unchanged in this case. The approach commences with the assumption that the original structure is additionally constrained by elements of changed design variables. The changed elements are reinforced because of additional constraints.
A structure is divided into substructures composed of
Equations (
In this case, it is assumed that the element is additionally constrained by an element of the cross section corresponding to
The study provides exact methods of structural reanalysis given the addition and removal of substructures and changes in design variables. The method adopts a constraint concept and commences with a generalized inverse method to describe constrained displacements and the Guyan condensation method. The generalized inverse method is modified to describe the coupling of substructures and the decoupling of substructures from an entire structure based on a stable or unstable substructure. The validity of the proposed method is illustrated via five examples to examine the reanalysis of a truss structure.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by the Chung-Ang University research grant in 2017. This research is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09918011).