Intermediate Elastic Supports Research Articles

Buckling of Bars with Variably Positioned Supports

In considering the instability of members subjected to axial compressive loads, several classical examples of support conditions are usually mentioned, hinged-hinged, clamped-free, clamped-clamped, and clamped-hinged. The corresponding problems of members that are transversely supported at arbitrary points and that are evidently encountered in practice, while not representing any particular analytic difficulty, appear infrequently in the literature. A typical case, presented is that of a simply supported rod with an intermediate elastic spring support at an arbitrary point.

Buckling of Continuous Circular Plates

The general theoretical development given in this paper considers the material to be orthotropic. Single Frobenius series solutions and double series solutions are obtained for orthotropic continuous plates. These solutions reduce to exact known solutions in the case of isotropic plates. For the purpose of numerical calculations, two cases of isotropic continuous plates—one over an intermediate rigid ring support and the other over an intermediate elastic ring support—are considered. For the case of a free plate over a rigid intermediate support, buckling loads increase as the support moves towards the center. In the case of an elastically supported fixed plate, the support elasticity considerably influences the buckling values, when the coefficient of support elasticity is more than 100. For lower modes a value of K 1 equal to 10,000 gives an almost fixed support. When the support is at distance of 0.355 times the radius of the plate the buckling resistance is found to be a maximum.

Optimum Design of Columns Supported by Tension Ties

A rational procedure for the optimization, on a weight basis, of columns supported by tension ties is developed. The tension ties provide a degree of intermediate elastic support which renders these systems more efficient than simple tubular columns in regions of low structural index P/L2. Two different configurations are considered. The first has one intermediate support-plane, and is stable up to a compressive load at which the second buckling mode occurs. The second has three intermediate support-planes and remains stable up to a compressive load corresponding to the fourth mode shape. Results are presented in such a fashion that conditions under which tie supported columns are more efficient than simple columns are readily apparent, and such that all design parameters are obtainable with a minimum of computational effort. Examples are included. Results indicate that, in favorable circumstances, columns supported by tension ties may be constructed with approximately half the weight required for a simple column capable of transmitting the same load over the same distance.

2033 圧縮材の支点の補剛について(構造,昭和38年度(仙台)大会学術講演要旨集)

2033 圧縮材の支点の補剛について(構造,昭和38年度(仙台)大会学術講演要旨集)

Free Vibrations of Constrained Beams

Abstract A general expression is developed from which the frequency equation for the vibration of a constrained beam with any combination of intermediate elastic or rigid supports, concentrated masses, and sprung masses can be found readily. The method also is extended to the case where the constraint is a continuous elastic foundation or uniformly distributed load of any length. This method requires only the knowledge of the natural frequencies and natural modes of the beam supported at the ends in the same manner as the constrained beam but not subjected to any of the constraints between the ends. The frequency equation is obtained easily and can be solved to any desired degree of approximation for any number of modes of vibration in a quick and simple manner. Numerical examples are given for a beam with one concentrated mass, for a beam with one sprung mass, and a continuous beam with one sprung mass.

Buckling Loads of Beams or Plates on Continuous Supports

A method is developed for finding the buckling loads of beams or plates over continuous rigid or elastic supports. These loads are found as the roots of characteristic equations tha t are developed in terms of the known characteristic modes of buckling and corresponding characteristic buckling loads of the beam or plate without the rigid or elastic supports. The method is an application of the direct method of the Calculus of Variations and consists of minimizing the potential energy of the system subject to the constraints tha t are the inner supports, rigid or elastic. The object of this paper is to develop a method whereby the buckling loads of beams or plates, having simply supported, fixed, or free boundaries, and subject to inner constraints in the form of intermediate rigid or elastic supports, may readily be calculated. These buckling loads are the roots of the characteristic equations tha t arise from the condition tha t the potential energy of the system shall be a minimum subject to the constraints. By developing the buckled shape of the system as a Generalized Fourier Series in terms of the characteristic (or eigen) functions of the unrestrained system, in many cases known or easily found, it is possible to express the potential energy in a simple and compact form from which there is no difficulty in deriving the characteristic equation. Few terms of these equations need be used to obtain reasonably accurate results in most problems. The beam has been previously solved, but the present treatment will be found to have advantages in some cases over the existing method. This is particularly evident if the beam is uniform. In the case of the plate results believed to be new are found.

The Behavior of Bowed Columns With Intermediate Elastic Supports

The Behavior of Bowed Columns With Intermediate Elastic Supports