Influences of foundation elasticity on the vibration and buckling characteristics of arches

Abstract

The effect of foundation deformation has been studied in detail with reference to the buckling of elastically supported and uniformly compressed inextensible circular arches. The numerical results obtained indicate that this effect can be quite significant for thicker arches, and more so in the case of arches of smaller semi central angles. The final expressions for the problem of buckling of elastically supported and uniformly compressed inextensible cycloidal, catenary and parabolic arches, whose cross sections vary as

I=I0cos I = I_0 \cos \thetaI=I0 cos , I=I0cos 1/2 I = I_0 \cos^{-1/2}\thetaI=I0 cos 1/2 , and I=I0/cos 2 I = I_0 / \cos^2\thetaI=I0 /cos2 , respectively,

are derived, and the numerical results are tabulated for the combinations of various values of central angle, foundation deformability index, and slenderness of the arch in Tables 4.1 to 4.4. It has been pointed out that the value of axial compressive force is a function of central angle, foundation deformability index, and slenderness of the arch for arches of shapes other than the circular that are supported on elastic abutments. An extension of the study of the cycloidal, catenary and parabolic arches of variable cross section to arches of uniform cross section is desirable. Furthermore, detailed study of the variation of the axial compressive force for cycloidal, catenary and parabolic arches supported on elastic abutments, with respect to the three parameters-central angle, slenderness, and foundation deformability index-is desirable. As pointed out earlier, the study presented herein is concerned only with inextensible arches. A study covering extensible (i.e., physically more realistic) arches would be desirable.

5.3 Conclusions The investigations carried out seem to justify the following conclusions: a. The elastic buckling load of slender symmetrical arches supported on elastic abutments may be predicted adequately by applying the principles of Vogt抯 artifice and Vogt抯 theory of foundation deformation to the differential equation of the deflection curve of the buckled arch. The critical pressure of an arch may be obtained easily using the numerical results presented in Tables 4.1 to 4.4. b. The foundation deformability index (XXX) reduces the ultimate buckling load of the arches by a considerable amount. Both Vogt抯 artifice and Vogt抯 theory confirm the reduction in critical pressure of arches on elastic abutments. However, they do not give identical results in the case of elastic abutments. Vogt抯 artifice is simpler to use, but in view of the results obtained, it is preferable to base the critical pressure studies of arches by this method only for slender arches on elastic abutments. The critical pressure is not very much affected by XXX in the case of wide and slender circular arches resting on elastic abutments. c. The reduction in critical pressure with central angle and slenderness is highly dependent on the actual values of these parameters and on the foundation deformability index.

The influence of central angle on the critical pressure is quite significant in the case of circular arches resting on elastic abutments where XXX is very small. When XXX is not very small, this influence of central angle on the critical pressure is quite significant for slender arches. The influence of slenderness on the critical pressure is significant in the case of circular arches resting on elastic abutments for all values of central angles and foundation deformability index.