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Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

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Summary

The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

Übersicht

Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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Abbreviations

ak, bk, ck, Ak, Bk, Ck :

coefficients used in series solutions

A, B:

constants in the nondimensional equation of the meridional curve

\(\bar n\) :

normal component of the load per unit area of the middle surface

\(\bar N_\varphi ,{\text{ }}\bar N_\theta \) :

meridional and circumferential forces per unit width

\(\bar p_r \) :

radial pressure per unit area of the middle surface, \(q_r = {{\bar p_r } \mathord{\left/ {\vphantom {{\bar p_r } {\left( {\bar \gamma ^{\bar t} 0} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar \gamma ^{\bar t} 0} \right)}}\)

\(\bar p_s \) :

skin weight per unit area of the middle surface, \(q_s = {{\bar p_s } \mathord{\left/ {\vphantom {{\bar p_s } {\left( {\bar \gamma ^{\bar t} {\text{0}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar \gamma ^{\bar t} {\text{0}}} \right)}}\)

\(\bar p_v \) :

vertical external load per unit horizontal area, \(q_v = {{\bar p_v } \mathord{\left/ {\vphantom {{\bar p_v } {\left( {\bar \gamma ^{\bar t} {\text{0}}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\bar \gamma ^{\bar t} {\text{0}}} \right)}}\)

\(\bar r\) :

base radius, \(r = {{\bar r\bar \gamma } \mathord{\left/ {\vphantom {{\bar r\bar \gamma } {\bar \sigma }}} \right. \kern-\nulldelimiterspace} {\bar \sigma }}_0 \)

R:

radius of convergence

s:

\({{\bar p_s } \mathord{\left/ {\vphantom {{\bar p_s } {\bar p_0 }}} \right. \kern-\nulldelimiterspace} {\bar p_0 }}\)

\(\bar t\) :

cupola thickness, \(t = {{\bar t} \mathord{\left/ {\vphantom {{\bar t} {\bar t_0 }}} \right. \kern-\nulldelimiterspace} {\bar t_0 }}\)

u, w:

subsidiary functions for quadratic cupolas

\(\bar v\) :

vertical component of the load per unit area of middle surface

\(\bar V\) :

resultant vertical force on a cupola segment

\(\overline W *\) :

structural weight of cupola, \(W* = {{\overline W *} \mathord{\left/ {\vphantom {{\overline W *} {\left( {\pi \bar r^2 \bar p_i } \right)\left( {i = s, v, r} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi \bar r^2 \bar p_i } \right)\left( {i = s, v, r} \right)}}\)

\(\overline W \) :

combined weight of cupola and skin, \(W = {{\overline W } \mathord{\left/ {\vphantom {{\overline W } {\left( {\pi \bar r^2 \bar p_s } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi \bar r^2 \bar p_s } \right)}}\)

\(\bar x\) :

distance from the axis of rotation, \({{x = \bar x\bar \gamma } \mathord{\left/ {\vphantom {{x = \bar x\bar \gamma } {\bar \sigma _0 }}} \right. \kern-\nulldelimiterspace} {\bar \sigma _0 }}\)

\(\bar y\) :

vertical distance from the shell apex, \(y = {{\bar y\bar \gamma } \mathord{\left/ {\vphantom {{\bar y\bar \gamma } {\bar \sigma _0 }}} \right. \kern-\nulldelimiterspace} {\bar \sigma _0 }}\)

z:

auxiliary variable in series solutions

\(\bar \gamma \) :

specific weight of structural material of cupola

\(\bar \varrho \) :

radius of the middle surface, \(\varrho = {{\bar \gamma \bar \varrho } \mathord{\left/ {\vphantom {{\bar \gamma \bar \varrho } {\bar \sigma _0 }}} \right. \kern-\nulldelimiterspace} {\bar \sigma _0 }}\)

\(\bar \sigma _0 \) :

uniaxial yield stress

\(\bar \sigma _\varphi \) :

meridional stress, \(\sigma _\varphi = {{\bar \sigma _\varphi } \mathord{\left/ {\vphantom {{\bar \sigma _\varphi } {\bar \sigma _0 }}} \right. \kern-\nulldelimiterspace} {\bar \sigma _0 }}\)

\(\bar \sigma _\theta \) :

circumferential stress, \(\sigma _\theta = {{\bar \sigma _\theta } \mathord{\left/ {\vphantom {{\bar \sigma _\theta } {\bar \sigma _0 }}} \right. \kern-\nulldelimiterspace} {\bar \sigma _0 }}\)

ηa, ηb, ηc, ηd, ηe :

subsidiary variables used in evaluating the meridional stress

ω:

auxiliary function used in series solutions

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Authors and Affiliations

  1. Dept. of Civil Engineering, Monash University, 3168, Clayton, Victoria, Australia

    M. Dow,  H. Nakamura &  G. I. N. Rozvany

Authors
  1. M. Dow
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  2. H. Nakamura
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  3. G. I. N. Rozvany
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Additional information

This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager

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Dow, M., Nakamura, H. & Rozvany, G.I.N. Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells. Ing. arch 52, 335–353 (1982). https://doi.org/10.1007/BF00537193

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