Stability and vibrations of cylindrical shells discretely reinforced with rings

Abstract

The linearized system of homogeneous equations of motion of cylindrical shells discretely reinforced with quasiregular sets of ring ribs is solved. Approximate formulas for the critical stresses and natural frequencies are presented Introduction. The stability and vibrations of cylindrical shells discretely reinforced with unidirectional (longitudinal or ring) ribs were studied in detail in [1, 7, 8, etc.], where closed shells were mainly considered. Quite simple analytic solutions for open shells reinforced with longitudinal ribs were obtained in few publications [2,3,5,7,8]. These solutions have the form of trigonometric series in a coordinate orthogonal to the ribs. Such solutions allow the critical stresses and natural frequencies to be determined by finding the roots of simple transcendental equations, which are convenient for both design of shells and analysis of their behavior without tedious calculations as well as for the derivation of approximate formulas.Below, we will find solutions for the critical stresses and natural frequencies in a similar form for shells discretely reinforced with quasiregular sets of ring ribs. Unlike shells reinforced with a regular set of ring ribs [3], these solutions are obtained without additional assumptions. A set of ribs will be regular if all the geometrical and mechanical characteristics of all ribs are equal, the ribs are equally spaced, and the distance from the extreme ribs to the edges of the shell are equal to the distance between ribs (if the set of ribs is quasiregular, then the distance from the extreme ribs to the edges of the shell are equal to half the distance between ribs). The quasiregular arrangement of ribs is widely used in thin-walled structures in aircraft and rocket construction.The monomial approximation of the displacement components was used in [2] to approximately solve the stability problem for closed cylindrical shells reinforced with quasiregular sets of ring ribs. Some problems for shells reinforced with quasiregular sets of longitudinal ribs and related problems were addressed in [9][10][11][12][13].1. Equations of Motion. The linearized equations of motion for circular cylindrical shells reinforced with discretely arranged ring ribs were obtained by generalizing the equations of motion derived in [3] to shells subjected to static loads (longitudinal compressive or tensile forces and external or internal pressure) and assuming that the prebuckling state of the shells is momentless. Applying the hypotheses of the engineering theory of shells, we simplify the equations as follows [4,6]:

Effect of the stiffness of ribs on the vibrations and stability of open cylindrical shells

“…Table 2 shows that when s k b = = < 1 4 1 ( ), 2.5 and the ribs are on the inside surface, c min in the first special case of deformation is lower than in the general case (as indicated above, c min > 0.7 for other values of s). This paradox was earlier noticed in [4]. The eccentricity of the ribs has a strong effect on c min in the general case of deformation as well.…”

“…Here, we will vary it to analyze in more detail the effect of ribs on the stress-strain state of the shell. The equations to be used to determine the minimum natural frequencies and critical stresses are borrowed from [3,4].We will show that the phenomenon (first discovered in [4]) of significant decrease in the minimum natural frequencies for certain vibration modes and number of ribs is independent of their stiffness.Similar problems of statics, dynamics, and static stability of shells were solved in [5][6][7][8]. …”

“…The exact solution of the equations of motion of such shells was found in [3]. This solution was used in [4] to examine the influence of the number and arrangement of ribs on the natural frequencies and critical stresses in such shells under axial compression.The stiffness of ribs was kept constant in [4]. Here, we will vary it to analyze in more detail the effect of ribs on the stress-strain state of the shell.…”

“…The stiffness of ribs was kept constant in [4]. Here, we will vary it to analyze in more detail the effect of ribs on the stress-strain state of the shell.…”

“…Here, we will vary it to analyze in more detail the effect of ribs on the stress-strain state of the shell. The equations to be used to determine the minimum natural frequencies and critical stresses are borrowed from [3,4].…”