**Author details**

Zissimos P. Mourelatos1\*, Dimitris Angelis2 and John Skarakis3

\*Address all correspondence to: mourelat@oakland.edu

1 Mechanical Engineering Department, Oakland University, U. S. A.

2 Beta CAE Systems S. A., Greece

3 Beta CAE Systems, U.S.A., Inc.

## **References**


[4] Yasui, Y. (1998). Direct Coupled Load Verification of Modified Structural Compo‐ nent. *AIAA Journal*, 36(1), 94-101.

material properties, etc) the FEA analysis must be repeated many times in order to obtain the optimum design. Also in probabilistic analysis where parameter uncertainties are present, the FEA analysis must be repeated for a large number of sample points. In such cas‐

To drastically reduce the computational cost without compromising accuracy beyond an ac‐ ceptable level, we developed and used various reanalysis methods in conjunction with re‐ duced-order modeling, in optimization of vibratory systems. Reanalysis methods are intended to efficiently calculate the structural response of a modified structure without solv‐ ing the complete set of modified analysis equations. We presented a variety of reanalysis methods including the CDH/VAO method, the Combined Approximations (CA) and Modi‐ fied Combined Approximations (MCA) method, and the Parametric Reduced-Order Model‐ ing (PROM) method. Their advantages and limitations were fully described and

Future work will concentrate on developing reanalysis methodologies for shape and topolo‐ gy optimization of vibratory systems and extend the presented work in optimization under uncertainty where efficient deterministic reanalysis methods will be combined with efficient

and John Skarakis3

[1] Abu Kasim, A. M., & Topping, B. H. V. (1987). Topping, Static Reanalysis: A Review.

[2] Arora, J. S. (1976). Survey of Structural Reanalysis Techniques. *ASCE J. Str. Div*, 102,

[3] Barthelemy, J. -F M., & Haftka, R. T. (1993). Approximation Concepts for Optimum

Structural Design- A Review. *Structural Optimization*, 5, 129-144.

es, the computational cost is very high, if not prohibitive.

demonstrated with practical examples.

178 Advances in Vibration Engineering and Structural Dynamics

Zissimos P. Mourelatos1\*, Dimitris Angelis2

*ASCE J. Str. Div*, 113, 1029-1045.

\*Address all correspondence to: mourelat@oakland.edu

1 Mechanical Engineering Department, Oakland University, U. S. A.

probabilistic reanalysis methods.

2 Beta CAE Systems S. A., Greece

3 Beta CAE Systems, U.S.A., Inc.

**Author details**

**References**

783-802.


[19] Rong, F., et al. (2003). Structural Modal Reanalysis for Topological Modificsatioms with Extended Kirsch Method. *Comp. Meth. in Appl. Mech. and Engrg.*, 192, 697-707.

**Chapter 8**

**Vibration Analysis of Cracked Beams Using the Finite**

Most of the members of engineering structures operate under loading conditions, which may cause damages or cracks in overstressed zones. The presence of cracks in a structural member, such as a beam, causes local variations in stiffness, the magnitude of which mainly depends on the location and depth of the cracks. These variations, in turn, have a significant effect on the vibrational behavior of the entire structure. To ensure the safe operation of structures, it is extremely important to know whether their members are free of cracks, and should any be present, to assess their extent. The procedures often used for detection are di‐ rect procedures such as ultrasound, X-rays, etc. However, these methods have proven to be inoperative and unsuitable in certain cases, since they require expensive and minutely de‐ tailed inspections [1]. To avoid these disadvantages, in recent decades, researchers have fo‐ cused on more efficient procedures in crack detection using vibration-based methods [2].

The majority of published studies assume that the crack in a structural member always re‐ mains open during vibration [3-7]. However, this assumption may not be valid when dy‐ namic loadings are dominant. In this case, the crack breathes (opens and closes) regularly during vibration, inducing variations in the structural stiffness. These variations cause the structure to exhibit non-linear dynamic behavior [8]. The main distinctive feature of this be‐ havior is the presence of higher harmonic components. In particular, a beam with a breath‐ ing crack shows natural frequencies between those of a non-cracked beam and those of a faulty beam with an open crack. Therefore, in these cases, vibration-based methods should employ breathing crack models to provide accurate conclusions regarding the state of dam‐ age. Several researchers [9-11] have developed breathing crack models considering only the

> © 2012 Bouboulas et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 Bouboulas et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Element Method**

http://dx.doi.org/10.5772/51173

N. K. Anifantis

**1. Introduction**

A. S. Bouboulas, S. K. Georgantzinos and

Additional information is available at the end of the chapter

Modelling of a crack is an important aspect of these methods.

