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There are two basic methods to analyze the continuum elastic structural mechanics: One is by applying Newton’s law of motion to a free body element of a structure; the other is by the energy method or calculus of variations [23]*. Two examples of vibration of a bar are used in this chapter to illustrate these two methods as a review.

Considered are the vibrations of inclined bars with the bottom end flxed in space and the top end having a constraint which requires that this end can move only in the vertical direction.

The model established for the vibration of the inclined bars in the last chapter now is applied to two- and three-bar frames with inclined members.

Because of devastation of structures due to earthquake and wind force, increasing attention has been paid to dynamic responses of braced structures [13, 22, 24, 45]. Extension of the formulation for vibrations of the two- and three-member frames of exact solutions in Chap. 3 to an x-braced portal frame is made in this chapter.

The study of vibrations for the x-braced portal frames in the last chapter is extended to high-rise x-braced multi-story frames in this chapter. Frequencies for two- to five-story frames are given. Forced vibrations for three-story frames are analyzed. Investigations are also made for the effects of horizontal ground motion for three- to five-story frames. Comparison of the present approach with available frequencies of a three-story rectangular frame is made.

The effect of rotatory inertia and shear deformation on the vibration of the inclined bar with the end constraint studied in Chap. 2 is examined here. The basic equations are derived by the variation method. The present work may be considered as an extension of the investigation of that chapter; therefore, the same nomenclature are used herein with some specified additions.

The frame vibrations studied in Chaps. 3 to 5 are for in-plane vibrations. If a plane frame acted upon by a force or horizontal ground vibration in the perpendicular direction of the plane, the frame will experience out-of-plane vibrations as shown in a numerical example in [41].

The stability of a simple truss subjected to a single force, , as shown in Fig. 8.1 was investigated by Mises [40] as a basic model to illustrate some of the buckling phenomena of structures. This model is known as the Mises truss. Buckling of the truss has also been studied by Von Karman and Kerr [54] and Huang and Vahidi [27]. In these studies, either the bending or the axial thrust was neglected.

A two-bar truss subjected to lateral normal distributed loads as shown in Fig. 9.1 represents a double leaf miter-type gate used in navigation locks as shown in Fig. 9.2. The gate is subjected to hydraulic static pressure whose intensity depends on the level of depth. The gate is supported by horizontal girders. Both ends of the girder are hinged and flxed in space, and the center joint is also hinged but deformable along the central line.

The so called edge-zone or boundary-layer solutions are widely used in large dediection problems of thin elastic plates [20,35,36] and shells [31,35], but little experimental work has been seen in the literature to verify these approximate solutions. The boundary-layer solutions for the buckling of a rectangular plate are shown in [36]. An experiment was performed for this simple problem. The agreement between the analytical and experimental results are excellent.