## Abstract

Ring-stiffened cylinders under hydrostatic pressure have been analyzed for more than a century. The contribution of Salerno and Pulos in the fifties of the previous century can be considered as a standard that nowadays still can be found in the rules and guidelines of classification societies. Their work comprises stresses and deformation in a perfectly circular cylindrical shell with ring frames, and solutions are presented midbay between the frames and at frame position. It is remarkable that the formula for the radial deflection midbay is much simpler than the one for frame position. This brief note shows a significant simplification of the latter one.

## 1 Introduction

Salerno and Pulos in their seminal paper [1] solve the problem of the axisymmetric ring-stiffened cylinder under hydrostatic load by first solving the fourth-order differential equation for the radial deflections. They were the first to include the beam-column effect. However, in the derivation [1], (Appendix A) they seem to introduce the effect twice but still arrive at the correct differential equation. This is rather confusing.

They rightly assume that normally their readers are only interested in the stresses, so while they present their expression for radial deflections [1], (Eqs. (62) and (63)), the emphasis is to arrive at explicit expressions for the critical axial and hoop membrane stresses [1], (Eqs. (64) and (65)). However, when working backward from the latter, to arrive at the radial deflection, one discovers an anomaly in the radial deflection at the frame location in that the equation found is much simpler than shown in Ref. [1], (Eq. (62)), raising the question whether both results are equal or not.

In order to avoid future confusion both points will be cleared up in this brief technical note. In particular, the simple equation for radial deflection at frame location could find its way to the rules and guidelines of classification societies and probably the software code of designers.

## 2 Governing Equation

In the derivation of the fourth-order differential equation, Salerno and Pulos introduced the beam-column effect directly in the radial force equilibrium [1], (Eq. (A1)). However, when deriving moment equilibrium, they introduced the beam-column effect again [1], (Eq. (A3)) and here also made a sign error.

Figure 1(a) is a derivative of Fig. 15 in Ref. [1] and shows the forces per unit length acting on the shell with length dx.

Essential for this approach is the rotation of the shell. The sides of the shell part remain perpendicular to the deformed shell. The small difference in angle between the two sides of the element *dx* causes a small resultant of the normal force *N _{x}* in the radial direction.

The situation in Fig. 1(a) has an alternative by considering the element *dx* with sides in the radial direction. This is illustrated in Fig. 1(b). In this situation, the axial force *N _{x}* does not contribute to the equilibrium of forces in the radial direction [1], (Eq. (A1)). However, the force

*N*has a lever and contributes to the equilibrium of moments [1], (Eq. (A3)), but with a sign error.

_{x}Both options are valid, but not simultaneously. For a straight beam column, the difference is explained by Timoshenko and Gere [3]. In their final result [1], Eqs. (A5) and (A12), Salerno and Pulos seem to have ignored the beam-column effect in the moment equilibrium and therefore arrived at the correct result.

## 3 Radial Deflection as Published

The radial deflection of the ring-stiffened shell is presented by Salerno and Pulos [1], (Eq. (61)) as a function over the length *w*(*x*). The origin is located between the frames midbay *x* = 0, and at frame position *x* = ±*L*/2.

### 3.1 Radial Deflection Midbay.

*x*= 0 follows from Ref. [1], Eq. (61) the radial deflection midbay

*F*

_{1}and

*F*

_{2}are defined by

Salerno and Pulos presented this deflection in Ref. [1], (Eq. (62)), however, it must be noted that their formula contains a typing error: the {1 –} is missing in the equation.

### 3.2 Radial Deflection at Frame Location.

*x*= ±

*L*/2 is given, with the short-hand notation in Eq. (4), by

*C*is defined by

_{ADD}*F*

_{4}in

*C*is given by

_{ADD}The radial deflection in Eq. (6) was presented in Ref. [1], (Eq. (63)), and it must be noticed that the radial deflection at frame location is far more complicated than the one midbay (5).

*σ*is given by the stress for unstiffened cylinders

_{u}The mathematical proof of this finding is presented in the next section.

## 4 Some Algebraic Processing

*F*

_{2}[1], ((Eq. (73)) and

*F*

_{4}[1], ((Eq. (75)) are rewritten to

Note: Salerno and Pulos seem to have assumed a Poisson’s ratio *ν* = 0.3 to arrive at $1\u2212\nu 2=0.91$.

*η*

_{1}and

*η*

_{2}are rewritten to

The numerator in Eq. (22) consists of four parts and application of the definition for *A*_{24} and *B*_{24} gives for

*A*

_{24}

*B*

_{24}

*A*

_{24}

*B*

_{24}

^{2}(

*x*) + cos

^{2}(

*x*) = 1 is applied to the summation of the second (24) and third part (25)

^{2}(

*x*)−sinh

^{2}(

*x*) = 1 follows for the summation of the first (23) and the fourth part (26)

## 5 Revised Deflections and Conclusion

With the modifications presented above follows for the deflections:

In particular, the latter expression is much simpler than the one originally published and repeated amongst others the Det Norske Veritas rules [4].

## Acknowledgment

This Technical Brief is a spin-off of a research project supported by Nevesbu B.V.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

No data, models, or code were generated or used for this paper.

## Nomenclature

- $b$ =
faying width of the ring frame in contact with the shell

- $h$ =
shell thickness

- $p$ =
hydrostatic pressure

- $E$ =
Young's modulus of elasticity

- $R$ =
radius to the shell mid-fiber

- $Af$ =
actual cross-sectional area of the frame

- $Aeff$ =
effective cross-sectional area of the frame

- $Lf$ =
frame distance

- $L=Lf\u2212b$ =
unsupported length of the shell

- $F1,F2,F4$ =
load and geometry functions for the shell

- $\alpha =AeffLfh$ =
ratio of effective frame area to shell area

- $\beta =bLf$ =
ratio of faying width to frame distance

- $\gamma \u2261pp*=p3(1\u2212\nu 2)2E(h/R)2$ =
measure of the beam-column effect

- $\epsilon \phi $ =
strain in the circumferential direction

- $\eta 1=121\u2212\gamma $ =
parameter beam-column effect

- $\eta 2=121+\gamma $ =
parameter beam-column effect

- $\theta =3(1\u2212\nu 2)4\u22c5LRh$ =
shell flexibility parameter

- $\nu $ =
Poisson's ratio

- $\sigma \phi M$ =
membrane stress in the circumferential direction