**7. References**


Chai, H. (1998) The Post-Buckling Behavior of a Bilaterally Constrained Column. Journal of the Mechanics and Physics of Solids 46, 1155–1181.

62 Advances in Computational Stability Analysis

**6. Conclusions** 

**Author details** 

**7. References** 

Jen-San Chen and Wei-Chia Ro

of fracture 46, 237-256.

whole lower branch is unstable, as in the inset of Figure 2. When *H* increases by a small amount, the load-deflection curves degenerates into two branches veering away from each other. When *H* <sup>=</sup> <sup>5</sup> 5 10 , the stable range on the upper branch shrinks. When *H* continues to increase to <sup>4</sup> 2 10 , the stable range on the upper branch disappears altogether. For the lower branch, there exists a limit point. The locus with positive slope before the limit point is stable.

Figure 13(b) shows another scenario when *H* varies from 0 to <sup>5</sup> 5 10 , and <sup>4</sup> 2 10 . It is observed that for a negative *H* , the sharp corner degenerates into two smooth curves crossing each other. The one emerging from the lower part has a stable range which ends at

In this chapter we introduce a vibration method which is suitable to analyze the stability of a constrained elastica. A planar elastica constrained by a space-fixed point constraint is used to demonstrate the method. Generally speaking, static analysis allows one to find all the possible equilibrium configurations of a constrained elastica. In order to predict how the elastica behaves in reality, the stability of these equilibrium configurations needs to be determined. The key of the vibration method is to take into account the sliding between the elastica and the unilateral constraint during vibration. In order to accomplish this, Eulerian coordinates are defined to specify the positions of the material points on the elastica. After transforming the governing equations and the boundary conditions from the Lagrangian description to the Eulerian one, the natural frequencies and the vibration mode shapes of the constrained elastica can be calculated. The vibration method is applied to an elastica constrained by a point constraint in this chapter. The same principles can be extended to other similar problems as well, for instance; multiple point

the peak of the curve. The other curve emerging from the top is unstable all the way.

constraints (Chen et al., 2010) and plane constraints (Ro et al., 2010).

International Journal of Mechanical Sciences 28, 153–162.

Constraints. Journal of Applied Mechanics 61, 764–772.

*Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan* 

Adams, G.G., Benson R.C. (1986) Postbuckling of an Elastic Plate in a Rigid Channel.

Adan, N., Sheinman, I., Altus, E. (1994) Post-Buckling Behavior of Beams Under Contact

Chai, H. (1990) Three-Dimensional Analysis of Thin-Film Debonding. International Journal


Santillan, S.T., Virgin, L.N., Plaut, R.H. (2006) Post-Buckling and Vibration of Heavy Beam on Horizontal or Inclined Rigid Foundation. Journal of Applied Mechanics 73, 664- 671.

**Chapter 4** 

© 2012 Thai, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

**Advanced Analysis of Space Steel Frames** 

This chapter presents advanced analysis methods for space steel frames which consider both geometric and material nonlinearities. The geometric nonlinearities come from second-order

nonlinearities are due to gradual yielding associated with residual stresses and flexure. The *P* effect results from the axial force acting through the relative displacement of the ends of the member, so it is referred to as a member chord rotation effect. The *P* effect is accounted in the second-order analysis by updating the configuration of the structure during

displacement of the member relative to its chord, so it is referred to as a member curvature

functions are derived from the closed-form solution of a beam-column subjected to end forces,

Geometric imperfections result from unavoidable errors during the fabrication or erection. There are three methods to model the geometric imperfections: (1) the explicit imperfection modeling, (2) the equivalent notional load, and (3) the further reduced tangent modulus. The explicit imperfection modeling for braced and unbraced members is illustrated in Fig. 2(a). For braced members, out-of-straightness is used instead of out-of-plumbness. This is due to the fact that the *P* effect due to the out-of-plumbness is vanished by braces. The limitation of this method is that it requires the determination of the direction of geometric imperfections which is often difficult in a large structural system. In the equivalent notional load method, the geometric imperfections are replaced by equivalent notional lateral loads in proportion to the gravity loads acting on the story as described in Fig. 2(b). The drawback of this method is that the gravity loads must be known in advance to determine the notional loads before analysis. Another way to account for the geometric imperfections is to further reduce the tangent modulus. The advantage of this method over the explicit imperfection modeling and

effects (see Fig. 1.) as well as geometric imperfections, while the material

effect can be captured by using stability functions. Since the stability

effect without using stability functions is to divide the member into

effect is caused by the axial force acting through the lateral

effect by using only one element per member. Another

effect is transformed to the *P* effect.

Additional information is available at the end of the chapter

Huu-Tai Thai

http://dx.doi.org/10.5772/45808

the analysis process. The *P*

they can accurately capture the *P*

many elements, and consequently, the *P*

**1. Introduction** 

*P* and *P*

effect. The *P*

way to capture the *P*

Vaillette, D.P., Adams, G.G. (1983) An Elastic Beam Contained in a Frictionless Channel. ASME Journal of Applied Mechanics 50, 693–694.
