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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 584901, 10 pages

http://dx.doi.org/10.1155/2013/584901

## Damage Identification of Trusses with Elastic Supports Using FEM and Genetic Algorithm

Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Republic of Korea

Received 25 June 2013; Accepted 6 August 2013

Academic Editor: Magd Abdel Wahab

Copyright © 2013 Nam-Il Kim and Jaehong Lee. 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

The computationally efficient damage identification technique for truss structures with elastic supports is proposed based on the force method. To transform the truss with supports into the equivalent free-standing model without supports, the novel zero-length dummy members are employed. General equilibrium equations and kinematic relations, in which the reaction forces and the displacements at the elastic supports are taken into account, are clearly formulated. The compatibility equations, in terms of forces in which the flexibilities of elastic supports are considered, are explicitly presented using the singular value decomposition (SVD) technique. Both member and reaction forces are simultaneously and directly obtained. Then, all nodal displacements including constrained nodes are back calculated from the member and reaction forces. Next, the microgenetic algorithm (MGA) is used to properly identify the site and the extent of multiple damages in truss structures. In order to verify the superiority of the current study, the numerical solutions are presented for the planar and space truss models with and without elastic supports. The numerical results indicate that the computational effort required by this study is found to be significantly lower than that of the displacement method.

#### 1. Introduction

The development of methods with higher computation efficiency is a crucial subject in engineering problems, and elimination of undue repetition in the analysis and the design procedures can lead to a considerable reduction in computation efficiency. The well-known two counterpart approaches have been usually carried out for the matrix structural analysis. The displacement method has been considered as a dominated structural analysis method due to its generality and simplicity in the computer implementation, and a lot of commercial softwares have been developed based on this method.

In comparison with the displacement method, the force method has its own advantages: (a) generating accurate stress and displacement results even for modest finite element models, since both equilibrium and compatibility conditions are simultaneously satisfied; (b) the properties of members of a structure most often depend on the member forces rather than the joint displacements; (c) simple formulation for optimization problems including stress constraints; and (d) easier basic concept. These advantages have made the force method as important part in the field of the structural engineering.

Up to now, considerable research efforts have been made by many researchers to solve the structural analysis problems using various types of force methods. Topological force methods were proposed for rigid-jointed skeletal structures using manual selection of the cycle bases of their graph models [1, 2]. Algebraic force methods were developed by Robinson [3], Kaneko et al. [4], and Pellegrino [5]. The general formulation of the force method, which was named as integrated force method, was suggested by Patnaik [6].

The damage identification of the system is an imperative issue in structural engineering and the functional age of the structure will increase after rehabilitating. To identify the site and the extent of damage in the structural systems, various methods have been introduced. Messina et al. [7] showed a new correlation coefficient termed the multiple damage location assurance criterion (MDLAC) to provide reliable information about the location and absolute size of damages. Begambre and Laier [8] proposed the particle swarm optimization combined with the simplex algorithm based on the damage identification procedure using frequency domain data. The principal advantages of this approach were the high reliability and stability of the algorithm and its independence from the initial estimate of heuristic parameters. Wang et al. [9] improved the damage signature matching technique through a proper definition of the measured damage signatures and predicted damage signatures to locate damage in the structure. Qian et al. [10] proposed a two-stage damage diagnosis approach in order to detect the structural damage in steel braced frame. This approach was comprised of the damage locating vectors method and eigensensitivity analysis. Among the variety of damage detection methods, the genetic algorithms (GAs) are one of the most successful search methods and they have been applied to the problems of the multiple structural damage identification [11–15]. The concept of residual force techniques have also been used to specify the objective function for the optimization procedure which was then implemented by using GAs [16–18]. However, the above mentioned damage detection techniques used the finite element approaches based on the displacement method.

From the previously cited references, it is noted that even though a significant amount of research has been conducted on the development of the damage detection methods in structural systems, there still has been no study reported of the damage detection of the structural system considering the effect of elastic supports using the force method. The truss supporting with elastic springs has received increasing attention due to its wide applicability in various engineering problems. The elastic springs can be used to investigate the behaviour of the partial connection at support, the subsoil, or the elastic foundation.

In this paper, a computationally efficient numerical procedure is proposed for the damage detection of the truss structures accounting for the flexibilities of the elastic supports through the force method. To transform the truss with supports into an equivalent free-standing model without supports, the novel zero-length dummy members are utilized. General equilibrium equations and general kinematic relations in which the reaction forces and the displacements at the elastic supports are, respectively, taken into account are formulated. A direct approach using the singular value decomposition (SVD) technique is employed on the general equilibrium matrix instead of the kinematic matrix to generate the compatibility conditions for the indeterminate trusses without the need to select any consistent redundant members as the classical force method [19]. The compatibility equations in terms of forces, in which the flexibilities of the elastic supports are considered, are clearly given. The analysis governing equations with unknown reaction forces, initial elongations, and elastic supports are developed. Both member and reaction forces are simultaneously and directly obtained and then all nodal displacements including constrained nodes are back computed based on the obtained member and reaction forces. Next, the microgenetic algorithm (MGA) is utilized to identify the multiple structural damages in truss structures. The numerical solutions are presented for the planar and space truss models in order to show the superiority of the proposed methodology.

#### 2. Structural Analysis Based on Force Method

##### 2.1. General Equilibrium Equations

For a -dimensional ( or 3) truss structure with members, free nodes and constrained nodes, the coordinate difference vectors , , and of members in -, - and -directions, respectively, are given by where and are the nodal coordinate vectors of the free and constrained nodes, respectively, in -, -, and -directions; () is the length vector; , , , and are the diagonal square matrices of the coordinate difference vectors and the member length vector, respectively; and denote the connectivities of members to the free and constrained nodes, respectively. The equilibrium equations of the free nodes in each direction of a truss structure can be stated as where () is the equilibrium matrix that transforms the vector of member forces of the system to the vector of external loads of the free nodes, defined by It is noted that the equilibrium equations given in (2) do not include any boundary constraints.

Let , , and denote the number of external, internal, and total indeterminacies, respectively. The external indeterminacy is related to the boundary constraints while the internal indeterminacy is associated with the number of bars in the truss. They are defined, respectively, as follows: where and are the numbers of boundary constraints and independent rigid-body motions, respectively; is the total number of nodes of the truss. The number of total statically indeterminacy is given by where is the sum of the number of members and boundary constraints.

Let () denote the unknown vector of reaction forces employed to remove all boundary constraints. The general equilibrium equations for all nodes of the discrete truss can be written considering the reaction forces as external nodal loads as follows: where is the unknown member and reaction force vector, and () is the external load vector of all nodes; () is the general equilibrium matrix; () is the coefficient matrix of unknown reaction forces in which its columns correspond to the last unknown components of . The columns of the matrix are vectors with all of their entries being zero except the entries in the rows corresponding to the constrained degrees of freedom having the negative unit value. If , the number of the unknowns of member and reaction forces exceed the number of equations, that is, the general equilibrium matrix is rectangular. This means that the equilibrium equations themselves are not sufficient to find the member and reaction forces. Therefore, additional equations of compatibility conditions are needed to determine the unknown force vector.

Next, as an example, consider the 3-bar planar truss supported by three hinges with two springs with the stiffness and as shown in Figure 1(a). Based on its free body diagram in Figure 1(b), the equilibrium equations in (7) for all nodes after moving the unknown reaction forces to the left hand side can be written as

In this study, the novel dummy members are introduced to replace the elastic supports, which enable the analysis of the trusses accounting for the flexibilities of elastic supports. Figure 1(c) shows the six novel dummy members 4–9 (which are represented by thick lines) having zero lengths (i.e., , ) and the stiffness –. Thus, the truss with elastic supports can be converted into the equivalent free-standing structure without supports. After implementation of structural analysis using the proposed method, the novel zero-length dummy members 4–9 will be removed to transform the nodes 2–4 back to the supports.

##### 2.2. Kinematic Relations

The element deformation vector is related to the displacements of all nodes by the following kinematic relation: where where is the kinematic matrix; and are the elongation vectors of the members and the novel dummy members, respectively; similarly, and are the displacement vectors of the free nodes and the constrained nodes, respectively.

The displacement-deformation relationship in the field of small displacements for the elastic conservative systems can be obtained by equating the external work and the internal strain energy as By substituting from (7) and in (9) into (11), the following relation is obtained: Based on the principle of virtual work, the above equation can be ensured to be true only if the following relation holds Hence, the kinematic relation in (9) can be rewritten as The expression given by (14) is the general form of the deformation-displacement relationship of the discrete system. In (14), there are compatibility equations which can be achieved through the elimination of the nodal displacements from the elongations. The components of the member and reaction force vector must satisfy the equilibrium equations in (7) along with compatibility equations obtained from (14).

##### 2.3. Compatibility Equations

In this study, the singular value decomposition (SVD) method is adopted as an effective and numerically stable algorithm in order to extract the compatibility conditions from (14). The SVD is performed on the general equilibrium matrix instead of the kinematic matrix (i.e., the transpose of the general equilibrium matrix ) due to the static-kinematic duality where and are the orthogonal matrices; is a diagonal matrix with nonnegative singular values of .

The vector space bases of compatibility equations of any discrete truss structures are calculated from the transpose of the null space of the general equilibrium matrix . In this case, the matrix in (15) can be expressed as follow:

For any displacement vector , the element deformation vector must be in the subspace spanned by the column vectors of from (14) and (15). Hence, it has to be orthogonal to the columns of ; that is, the transpose of the last vectors () in corresponding to the last zero singular values in forms a set of compatibility equations as follows: where is the compatibility matrix in terms of elongations defined by The compatibility equations in (18) assure that the elongation vector is constrained such that the deformed elements fit together to form the structure free of voids and discontinuities.

##### 2.4. Determination of Member Forces and Nodal Displacements

Similarly, the compatibility matrix in terms of elongations can be decomposed into two parts as where and are two compatibility submatrices in terms of elongations of the members and the dummy ones of the equivalent system, respectively. Assuming linear elastic material, elongation-force relation is given as where () and () are the initial elongation vector (due to initial stresses, imperfections, and temperature effects) and the diagonal flexibility matrix, respectively. The diagonal flexibility matrix can also be decomposed into where and are the diagonal flexibility matrices of the members and the dummy ones of the equivalent system, respectively. Inserting (20) into (17) yields the following expression:

By substituting (19) and (21) into the left hand side of (22), the compatibility equations expressed in terms of forces are obtained as follows: where ) is the compatibility matrix in terms of forces of all members; and are two compatibility submatrices in terms of forces of the members and the dummy ones, respectively; is a virtual force vector in the indeterminate truss caused by member initial elongations . , , and are respectively given by

Combining (7) and (23) yields or where is the external actions coupling the external loads and the virtual forces due to member initial elongations (or imperfections and temperature effects); is a square full rank matrix coupling the general equilibrium matrix and the compatibility matrix in terms of forces , defined by If all constrained nodes are fully rigid (i.e., there exists no spring support in the structure), all the diagonal components of must be zero (i.e., ), which leads to in (25). Similarly, if there is no initial elongation (i.e., ), which leads to in (26). Then (29) becomes

Regarding indeterminate trusses, however, it is not sufficient to conversely obtain the nodal displacements using the member elongations since the general equilibrium matrix is rectangular and cannot be inverted. This implies that the compatibility equations should be melted into the general equilibrium equations. For this reason, the nodal displacements must be calculated using in (28) instead of as where is given by

#### 3. Implementation of Microgenetic Algorithm

Genetic algorithms (GAs) are the numerical search algorithms based on Darwin’s theory of evolution and survival of the fittest. The information regarding an optimization problem, such as design variable, is coded into a genetic string known as an individual. Each of these individuals has an associated fitness value, which is determined by the objective function to be maximized or minimized. The genetic algorithms proceed by taking a population, which is comprised of different individuals with different fitness. The best individuals of the current generation are selected to produce the new generation which has better fitness by crossover and mutation operators [20]. These GAs have been shown to be capable of solving various optimization problems via some basic concepts and operators. However, it has been revealed from some research works [12, 13, 20] that the simple GA cannot solve the damage identification problems properly.

In order to acquire better identification results and high efficiency, in this study, the microgenetic algorithm (MGA) proposed by Krishnakumar [21] is used to properly identify the site and the extent of multiple damages in truss structures. The MGA utilizes a relatively smaller population size than the simple GA resulting in less computation time. The index used as the objective function to be minimized in the optimization process is demonstrated as follows: where where is the strain of the th member for the damaged truss and is the one that can be predicted from an analytic model; is the strain of the member at which the minimum axial force occurs.

Now, some basic concepts and operations of the current algorithm are introduced.

*(i) Coding*. The coding of variables that describe the problem are an essential characteristic of the genetic algorithm. A binary coding method is used in this study. This method is to transform variables to a binary string of specific length.

*(ii) Selection Operator*. The selection procedure used by current study is based on the fitness of each individual. There are number of reproduction schemes commonly used in the genetic algorithm. These include tournament selection, ranking selection, steady-state selection, greedy over selection, and proportionate reproduction. In this study, the tournament selection is used.

*(iii) Crossover*. Crossover is the operator that produces new individuals by exchanging some bits of a couple of randomly selected individuals. The two-point crossover is utilized since the two-point crossover performs much better among the multipoint crossover techniques [22]. In addition, the crossover rate is set as 1.0; therefore, all populations must perform a crossover operation at every generation.

*(iv) Mutation and Restart Operation*. There is no use of mutation operation because 1.0 crossover rate provide adequate variability. In addition, we do not use the restart operation in this study.

#### 4. Numerical Examples

To show the accuracy and the superiority of the proposed damage detection method, two types of indeterminate truss structures are solved. Through numerical examples, all members are assumed to have the same elastic modulus = 70 MPa and the area of cross-section = 0.01 m^{2}, and the initial elongation vector is assumed to be zero. The damage of truss structures is simulated via the reduction of the elastic modulus in each element. The parameters of the current genetic algorithm are selected here from the experience of our work and a trial and error method as follows: the maximum number of generation is 500, the population size is 500, and the 30 bits are used to represent each optimization variable. The convergence of the algorithm is met when the maximum number of generation is attained.

##### 4.1. 16-Bar Planar Truss

Figure 2 shows the simply supported 16-bar planar truss with the elastic springs at node 4. The truss is modelled using 16 consistent finite elements and subjected to a vertical force −1 kN acting at node 7. First, in case of the simply supported boundary condition at node 4, the flexibility matrix of the novel dummy members is zero (i.e., ) in (21) since the elastic supports are fully rigid. The number of external indeterminacy in (4) is , while the number of internal indeterminacy ; thus, the total number of redundancy in (6) is . In case of truss with elastic supports, this study uses experimentally determined values for the soil modulus of elasticity and the Poisson ratio . If the soil is the loose sand with = 1750 kN/cm^{−1} and = 0.28, the application of the Vallabhan-Das method [23] produces the coefficient of subgrade reaction N/cm^{3}. For a truss member width = 10 cm and height = 10 cm, the Winkler foundation moduli are = 9.946 N/cm^{2} and = 9.946 N/cm^{2}.

The member and reaction forces and all nodal displacements for the planar trusses with simple support and elastic supports at node 4 are presented in Tables 1 and 2, respectively. It can be found from Tables 1 and 2 that the results obtained from the prosed method are well compared with those from SAP2000 [24] program.

Next, for the truss with elastic supports, the damage is introduced in element 9 by reducing its stiffness to 30% of the initial value as the first scenario. The measurement noise is simulated and entered into the analysis. The identified damage expressed in the ratio of elastic modulus reduction and the convergence history of the objective function value with generations are depicted in Figures 3 and 4, respectively. For comparison, the results from the finite element model based on the displacement method (DM) is presented together. It can be seen from Figures 3 and 4 that the two algorithms based on the force and displacement methods combined with MGA are properly achieved to the site and the extent of the hypothetical damage. The CPU time calculated for analyses by using the force and displacement methods are presented in Table 3. It is seen that the required computational time for the FM is approximately one fifth of that required by the DM, illustrating the efficiency of the FM over the DM for the analysis of planar truss structure.

In the second scenario, the damages are induced in members 5 and 13. Damaged members are assumed to have 40 and 30% of their initial stiffnesses, respectively. The identified damages and the convergence history are presented in Figures 5 and 6, respectively. It is seen that the present study detects the location and the extent of the damage with good accuracy. It can also be found in Table 3 that the computational time for the FM is significantly lower than that required by the DM.

##### 4.2. 36-Bar Space Truss

The 36-bar space truss with 12 nodes as shown in Figure 7, subjected to horizontal forces = = 1 kN and vertical force kN, is considered. The truss has mixed statically indeterminacy of twelve in which six are externally statically indeterminacy, and the others are internally generated. Therefore, twelve equations of compatibility conditions are required to determine the unknown member and reaction force vector which can be straightforwardly obtained by the analysis governing equations in (27). For trusses with simple support and elastic supports with = = = 9.946 N/cm^{2} at node 9, the obtained member and reaction forces , and all nodal displacements , as given in Tables 4 and 5, respectively, coincide exactly with those of SAP2000 [24] program.

As a first damage scenario, the member 13 is damaged by reduction of its stiffness to 40% of the initial value for the truss considering elastic supports. Figures 8 and 9 show the identified damages and the convergence history, respectively, obtained from the force and displacement methods. In addition, the CPU time for analyses by two methods is given in Table 6. It is observed from Figures 8 and 9 and Table 6 that the current MGA can identify the locations as well as the extent of the damaged space truss in both cases, whereas the FM shows the superior efficiency on the CPU time required when compared with the DM as can be seen in Table 6.

In our final scenario, the stiffnesses of members 1 and 21 are reduced to 30 and 50% of their initial values, respectively. The identified damages and the convergence history are plotted in Figures 10 and 11, respectively. From the results shown in Figures 10 and 11, it is observed that the present damage detection method achieves to the site and the extent of the induced damages truthfully. It can also be found from Table 6 that the computational time for the FM is significantly lower than that required by the DM, again pointing out the efficiency of the FM.

#### 5. Conclusions

An efficient force method combined with the microgenetic algorithm (MGA) for identifying the location and the extent of multiple damages in the truss structures with elastic supports has been proposed. The general equilibrium equations considering the effect of elastic supports and the kinematic relations have been formulated, and the compatibility equations in terms of forces using the singular value decomposition technique are presented. Then, the optimization problem has been solved using the MGA to identify the actual damage. In order to assess the performance of the proposed method, two illustrated test examples are considered. Throughout numerical examples, the relative performance of the force and displacement methods in the damage detection analysis of the truss structures is also investigated. The numerical results indicate that the combination of the force method and MGA can provide a reliable tool to accurately identify the multiple damages of the truss structures and is computationally far more efficient than the displacement method combined with MGA.

#### Acknowledgment

The support of the research reported here by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0019373 and 2012R1A2A1A01007405) is gratefully acknowledged.

#### References

- J. C. C. De Henderson and E. A. W. Maunder, “A problem in applied topology: on the selection of cycles for the flexibility analysis of skeletal structures,”
*Journal of the Institute of Mathematics and its Applications*, vol. 5, no. 2, pp. 254–269, 1969. View at Publisher · View at Google Scholar · View at Scopus - E. A. W. Maunder,
*Topological and linear analysis of skeletal structures [Ph.D. thesis]*, Imperial College, London, UK, 1971. - J. Robinson,
*Integrated Theory of Finite Element Methods*, Wiley, New York, NY, USA, 1973. - I. Kaneko, M. Lawo, and G. Thierauf, “On computational procedures for the force methods,”
*International Journal for Numerical Methods in Engineering*, vol. 18, no. 10, pp. 1469–1495, 1982. - S. Pellegrino, “Structural computations with the singular value decomposition of the equilibrium matrix,”
*International Journal of Solids and Structures*, vol. 30, no. 21, pp. 3025–3035, 1993. View at Scopus - S. N. Patnaik, “An integrated force method for discrete analysis,”
*International Journal for Numerical Methods in Engineering*, vol. 6, no. 2, pp. 236–251, 1973. - A. Messina, E. J. Williams, and T. Contursi, “Structural damage detection by a sensitivity and statistical-based method,”
*Journal of Sound and Vibration*, vol. 216, no. 5, pp. 791–808, 1998. View at Scopus - O. Begambre and J. E. Laier, “A hybrid particle swarm optimization-simplex algorithm (PSOS) for structural damage identification,”
*Advances in Engineering Software*, vol. 40, no. 9, pp. 883–891, 2009. View at Publisher · View at Google Scholar · View at Scopus - X. Wang, N. Hu, H. Fukunaga, and Z. H. Yao, “Structural damage identification using static test data and changes in frequencies,”
*Engineering Structures*, vol. 23, no. 6, pp. 610–621, 2001. View at Publisher · View at Google Scholar · View at Scopus - J. Qian, X. Ji, and Y. Xu, “Two-stage damage diagnosis approach for steel braced space frame structures,”
*Engineering Structures*, vol. 29, no. 12, pp. 3277–3292, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. D. Villalba and J. E. Laier, “Localising and quantifying damage by means of a multi-chromosome genetic algorithm,”
*Advances in Engineering Software*, vol. 50, pp. 150–157, 2012. View at Publisher · View at Google Scholar · View at Scopus - H. M. Gomes and N. R. S. Silva, “Some comparisons for damage detection on structures using genetic algorithms and modal sensitivity method,”
*Applied Mathematical Modelling*, vol. 32, no. 11, pp. 2216–2232, 2008. View at Publisher · View at Google Scholar · View at Scopus - B. H. Koh and S. J. Dyke, “Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data,”
*Computers and Structures*, vol. 85, no. 3-4, pp. 117–130, 2007. View at Publisher · View at Google Scholar · View at Scopus - R. S. He and S. F. Hwang, “Damage detection by a hybrid real-parameter genetic algorithm under the assistance of grey relation analysis,”
*Engineering Applications of Artificial Intelligence*, vol. 20, no. 7, pp. 980–992, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. H. Chou and J. Ghaboussi, “Genetic algorithm in structural damage detection,”
*Computers and Structures*, vol. 79, no. 14, pp. 1335–1353, 2001. View at Publisher · View at Google Scholar · View at Scopus - M. A. Rao, J. Srinivas, and B. S. N. Murthy, “Damage detection in vibrating bodies using genetic algorithms,”
*Computers and Structures*, vol. 82, no. 11-12, pp. 963–968, 2004. View at Publisher · View at Google Scholar · View at Scopus - K. Moslem and R. Nafaspour, “Structural damage detection by genetic algorithms,”
*AIAA Journal*, vol. 40, no. 7, pp. 1395–1401, 2002. View at Scopus - C. Mares and C. Surace, “An application of genetic algorithms to identify damage in elastic structures,”
*Journal of Sound and Vibration*, vol. 195, no. 2, pp. 195–215, 1996. View at Publisher · View at Google Scholar · View at Scopus - M. Domaszewski and A. Borkowski, “On automatic selection of redundancies,”
*Computers and Structures*, vol. 10, no. 4, pp. 577–582, 1979. View at Scopus - M. Nobahari and S. M. Seyedpoor, “Structural damage detection using an efficient correlation-based index and a modified genetic algorithm,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 1798–1809, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in
*Proceedings of the SPIE Intelligent Control and Adaptive Systems*, vol. 1196, pp. 282–296, 1989. - T. Y. Chen and C. J. Chen, “Improvements of simple genetic algorithm in structural design,”
*International Journal for Numerical Methods in Engineering*, vol. 40, no. 7, pp. 1323–1334, 1997. View at Scopus - C. V. G. Vallabhan and Y. C. Das, “A refined model for beams on elastic foundations,”
*International Journal of Solids and Structures*, vol. 27, no. 5, pp. 629–637, 1991. View at Scopus - Computers and Structures,
*SAP2000, Version 14.1.0*, Computers and Structures, Berkeley, Calif, USA, 2010.