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Minimum stiffness criteria for ring frame stiffeners of space launch vehicles

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Abstract

Frame stringer-stiffened shell structures show high load carrying capacity in conjunction with low structural mass and are for this reason frequently used as primary structures of aerospace applications. Due to the great number of design variables, deriving suitable stiffening configurations is a demanding task and needs to be realized using efficient analysis methods. The structural design of ring frame stringer-stiffened shells can be subdivided into two steps. One, the design of a shell section between two ring frames. Two, the structural design of the ring frames such that a general instability mode is avoided. For sizing stringer-stiffened shell sections, several methods were recently developed, but existing ring frame sizing methods are mainly based on empirical relations or on smeared models. These methods do not mandatorily lead to reliable designs and in some cases the lightweight design potential of stiffened shell structures can thus not be exploited. In this paper, the explicit physical behaviour of ring frame stiffeners of space launch vehicles at the onset of panel instability is described using mechanical substitute models. Ring frame stiffeners of a stiffened shell structure are sized applying existing methods and the method suggested in this paper. To verify the suggested method and to demonstrate its potential, geometrically non-linear finite element analyses are performed using detailed finite element models.

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Abbreviations

a :

Length of a shell section, ring frame spacing

\({a_e}\) :

Length of a beam element

\({A_{ij}}\) :

Entries of the in-plane stiffness matrix

\({A_\mathrm{r}}\) :

Section area of ring frame

\({A_\mathrm{str}}\) :

Section area of stringer

\({b_{1\ldots 3}}\) :

Geometrical properties of stringer-stiffened shell

\({b_{\mathrm{r},F/S}}\) :

Flange width of ring frame according to Friedrich/Shanley

\({D_{ij}}\) :

Entries of the plate stiffness matrix

E :

Young’s modulus

\({F_\mathrm{R}}\) :

Radial unit force

G :

Shear modulus

\({I_\mathrm{T}}\) :

Torsional moment of inertia of ring frame

\({I_{y,\mathrm{r}}}\) :

Second moment of inertia about the frame’s \({y_\mathrm{r}}\)-axis

\({I_{z,\mathrm{r}}}\) :

Second moment of inertia about the frame’s \({z_\mathrm{r}}\)-axis

\({I_{y,\mathrm{str}}}\) :

Second moment of inertia about the shell’s y-axis of the stringer and the effective width

\({I_\mathrm{str}}\) :

Second moment of inertia about the shell’s y-axis of the stringer

\({k_{el.,fou.}}\) :

Stiffness of an elastic foundation

\({k_\mathrm{lin}}\) :

Translational stiffness of substitute model

\({k_\mathrm{lin,min}}\) :

Minimum translational stiffness

\({k_{lin,o}}\) :

Translational stiffness of load introduction frames

\({k_\mathrm{rot}}\) :

Rotational stiffness of substitute model

\({k_\mathrm{rot,min}}\) :

Minimum rotational stiffness

\({k_{rot,o}}\) :

Rotational stiffness of load introduction frames

l :

Length of the substitute model

L :

Length of the shell

\({M_R}\) :

Unit bending Moment acting about the ring frame’s circumferential direction

M :

Bending moment subjected to shell structure

\({m_{tot}}\) :

Total structural mass

\({\nu }\) :

Poisson’s ratio

\({n_{max}}\) :

Maximum number of coefficients used

\({n_{r}}\) :

Number of ring frames

N :

Axial load per unit length

\({n_\mathrm{str}}\) :

Number of stringer

P :

Axial compression

\({P_\mathrm{crit}}\) :

Critical axial compression load causing column buckling of one stringer and its effective width

\({\Pi _{tot}}\) :

Total potential

\({\Pi _{i}}\) :

Inner potential

\({\Pi _{e}}\) :

External potential

R :

Radius of the shell

\({t_\mathrm{skin}}\) :

Wall thickness of the shell

\({t_\mathrm{str}}\) :

Wall thickness of the stringer

\({t_{r}}\) :

Wall thickness of the ring frame

u :

displacement in axial direction of the shell

w :

Radial displacement

\({W_{s,j}}\) :

Coefficients of the ansatz function—sinus terms

\({W_{c,j}}\) :

Coefficients of the ansatz function—cosinus terms

x, y, z :

Coordinates of the cylindrical shell

\({x_\mathrm{r},y_\mathrm{r},z_\mathrm{r}}\) :

Coordinates of the ring frame stiffener

\({\varphi }\) :

Rotation of the ring frame about its \({x\mathrm{_r}}\)-axis

\({z_\mathrm{r}}\) :

Distance between the shell’s mid-plane and the center of area of the ring frame

\({z_\mathrm{str}}\) :

Distance between the shell’s mid-plane and the center of area of the stringer

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Acknowledgments

The authors acknowledge the computing time grant provided by the IT Center of RWTH Aachen University, which allowed us to perform the numerical studies on the high-performance computing cluster of RWTH Aachen University.

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Authors and Affiliations

  1. Institute of Structural Mechanics and Lightweight Design, RWTH Aachen University, Wüllnerstrasse 7, 52062, Aachen, Germany

    Linus Friedrich & Kai-Uwe Schröder

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  1. Linus Friedrich
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Correspondence to Linus Friedrich.

Additional information

This paper is based on a presentation on the German Aerospace Congress, Sept. 22–24, 2015, Rostock, Germany.

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Friedrich, L., Schröder, KU. Minimum stiffness criteria for ring frame stiffeners of space launch vehicles. CEAS Space J 8, 269–290 (2016). https://doi.org/10.1007/s12567-016-0126-4

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