Roman symbols

A

sectional aero

E

elastic modulus

f

natural frequency

G

shear modulus

H

the thickness of the simplified cylindrical shell

I

sectional moment of inertia

k

shear coefficient

l

axial length of the stiffened cylindrical shell

N

the quantity of stiffeners

R

mid-radius of the stiffened cylindrical shell

R _j

the distance between the jth stiffener and the x-axis of the cross section

x

the direction along the axis of beam

y

the lateral displacement of beam

Greek symbols

δ

thickness of the stiffened cylindrical shell

ε

the error of similarity

λ

scale factor

ν

Poisson’s ratio

ρ

density

Subscripts

()₀

referring to a single stiffener

()₁

referring to the shell

()₂

referring to all stiffeners

() _m

referring to the model

()_pre

referring to the predicted value of the scaled model for the prototype

()_pro

referring to the prototype

()_sim

referring to the simplified cylindrical shell

3.1 Process of equivalence

To achieve the equivalence of global bending mode, cylindrical shell without stiffeners is chosen as simplified object (simplified prototype and simplified scaled model). Thus, it is necessary to establish the equivalent criteria from the stiffened cylindrical shell to a cylindrical shell without stiffeners. The transversal free vibrational differential equation of cylindrical shell based on the hypothesis of Timoshenko Beam is^{( 26 )}

(…3) $${\rm \rrho }A{{\partial ^{2} y(x,t)} \over {\partial t^{2} }}{\plus}EI{{\partial ^{4} y(x,t)} \over {\partial x^{4} }}{\minus}{\rm \rrho }I\left[ {1{\plus}{{(EI)({\rm \rrho }A)} \over {k(GA)({\rm \rrho }I)}}} \right]{{\partial ^{4} y(x,t)} \over {\partial x^{2} \partial t^{2} }}{\plus}{{({\rm \rrho }A)({\rm \rrho }I)} \over {k(GA)}}{{\partial ^{2} y(x,t)} \over {\partial t^{4} }}{\equals}0$$

where x donates the direction along the axis of beam, y is the lateral displacement of beam and k is the shear coefficient.

From Equation (3), we know that ρA, EI, ρI and GA are the ‘equivalent parameters’. Then, the four equivalent parameters are analysed as follows.

In terms of the equivalent parameter EI, the relationship between the stiffened cylindrical shell and the simplified cylindrical shell can be represented as

(…4) $$E_{{{\rm sim}}} I_{{{\rm sim}}} {\equals}E_{1} I_{1} {\plus}E_{2} I_{2}$$

or

(…5) $$E_{{{\rm sim}}} {\rm \rpi }HR_{{{\rm sim}}}^{3} {\equals}E_{1} {\rm \rpi \rdelta }R^{3} {\plus}E_{2} I_{2}$$

where the subscript ‘sim’ refers to the simplified cylindrical shell and H represents the thickness of the simplified cylindrical shell. It is assumed that the simplified cylindrical shell has the same elastic modulus and midface radius as the shell of the stiffened cylindrical shell, that is

(…6) $$\left\{ \matrix{ E_{{{\rm sim}}} {\rm {\equals}}E_{1} \hfill \cr R_{{{\rm sim}}} {\rm {\equals}}R \hfill \cr} \right.$$

By substituting Equation (6) into Equation (5), the expression of H is obtained

(…7) $$H{\equals}{\rm \rdelta }{\plus}{{E_{2} I_{2} } \over {E_{1} I_{1} }}{\rm \rdelta }$$

In terms of the equivalent parameter ρA, the relationship between the stiffened cylindrical shell and the simplified cylindrical shell can be represented as

(…8) $${\rm \rrho }_{{{\rm sim}}} A_{{{\rm sim}}} {\equals}{\rm \rrho }_{1} A_{1} {\plus}{\rm \rrho }_{2} A_{2}$$

where the cross-sectional areas can be further expressed as

(…9) $$\left\{ \matrix{ A_{{{\rm sim}}} {\rm {\equals}}2{\rm \rpi }HR \hfill \cr A_{1} {\equals}2{\rm \rpi \rdelta }R \hfill \cr A_{2} {\equals}NA_{0} \hfill \cr} \right.$$

Then by simplifying Equation (8) with Equation (9), the density of the simplified cylindrical shell ρ_sim can be expressed as

(…10) $${\rm \rrho }_{{{\rm sim}}} {\equals}{{2{\rm \rpi \rdelta }R{\rm \rrho }_{1} {\plus}NA_{0} {\rm \rrho }_{2} } \over {2{\rm \rpi }HR}}$$

In terms of the equivalent parameter ρI, the relationship between the stiffened cylindrical shell and the simplified cylindrical shell can be represented as

(…11) $${\rm \rrho '}_{{{\rm sim}}} I_{{{\rm sim}}} {\equals}{\rm \rrho }_{1} I_{1} {\plus}{\rm \rrho }_{2} I_{2}$$

where ρ′_sim represents the equivalent density of the simplified cylindrical shell through the equivalent parameter ρI. Meanwhile, ρI of the shell and the stiffeners can be expressed, respectively, as⁽ Reference Xing, Pan and Yang ^{27 )}

(…12) $${\rm \rrho }_{1} I_{1} {\rm {\equals}\rrho }_{1} A_{1} R^{2} \,/\,2$$

(…13) $${\rm \rrho }_{2} I_{2} {\rm {\equals}\rrho }_{2} \left( {NI_{0} {\plus}A_{0} \mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right)\,\approx\,{\rm \rrho }_{2} A_{0} \mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } $$

where R _j represents the distance between the j ^th stiffener and the x-axis of the cross section, NI ₀ is smaller than $$A_{0} \mathop{\sum}\nolimits_{j{\equals}1}^N {R_{j}^{2} } $$ and can be ignored. When enough stiffeners are arranged equidistantly along the circumference, Equation (13) can be simplified as

(…14) $${\rm \rrho }_{2} I_{2} \,\approx\,{\rm \rrho }_{2} A_{0} \mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } {\rm {\equals}\rrho }_{2} A_{0} NR^{2} \,/\,2$$

By substituting Equations (12) and (14) into Equation (11), the expression of ρ′_sim is obtained

(…15) $${\rm \rrho '}_{{{\rm sim}}} {\rm {\equals}}\;{\rm \rrho }_{{{\rm sim}}} $$

Equation (15) shows that the equivalent density derived from the equivalent parameters ρA and ρI is the same, that is, the two equivalent parameters are equivalent.

In terms of the equivalent parameter GA, the relationship between the stiffened cylindrical shell and the simplified cylindrical shell can be represented as

(…16) $$G_{{{\rm sim}}} A_{{{\rm sim}}} {\equals}G_{1} A_{1} {\plus}G_{2} A_{2} $$

By substituting Equation (8) into Equation (16), next Equation (17) is obtained

(…17) $$G_{{{\rm sim}}} {\equals}G_{1} {{{\rm \rdelta }{\plus}{{G_{2} } \over {G_{1} }}{{NA_{0} } \over {A_{1} }}{\rm \rdelta }} \over H}$$

Meanwhile, according to Equations (12) and (14), $${{I_{1} } \over {I_{2} }}$$ is represented as

(…18) $${{I_{1} } \over {I_{2} }}{\equals}{{A_{1} R^{2} \,/\,2} \over {NA_{0} R^{2} \,/\,2}}$$

that is,

(…19) $${{I_{1} } \over {I_{2} }}{\equals}{{A_{1} } \over {NA_{0} }}$$

By substituting Equation (19) into Equation (17), the shear modulus of the simplified cylindrical shell G _sim can be expressed as

(…20) $$G_{{{\rm sim}}} {\equals}G_{1} {{{\rm \rdelta }{\plus}{{G_{2} E_{1} } \over {G_{1} E_{2} }}{{E_{2} I_{2} } \over {E_{1} I_{1} }}{\rm \rdelta }} \over H}$$

Then, by substituting Equation (7) into Equation (20), G _sim is simplified as

(…21) $$G_{{{\rm sim}}} {\equals}G_{1} {{{\rm \rdelta }{\plus}{{G_{2} E_{1} } \over {G_{1} E_{2} }}\left( {H{\minus}{\rm \rdelta }} \right)} \over H}$$

At last, synthesising the above derivation obtains the simplified cylindrical shell with axial length l, mid-radius R, thickness H, Young’s modulus of elasticity E ₁, shear modulus G _sim and density ρ_sim. Equations (7), (10) and (21) are the expression of H, ρ_sim and G _sim, respectively.

3.2 Calculation of scaling laws

The next step is to derive the scaling laws of complete similarity. By applying the equational analysis to Equation (3), the sub-items, including $${\rm \rrho }A{{\partial ^{2} y(x,t)} \over {\partial t^{2} }}$$ , $$EI{{\partial ^{4} y(x,t)} \over {\partial x^{4} }}$$ , $${\rm \rrho }I{{\partial ^{4} y(x,t)} \over {\partial x^{2} \partial t^{2} }}$$ , $${{{\rm \rrho }I(EI)({\rm \rrho }A)\partial ^{4} y(x,t)} \over {k(GA)({\rm \rrho }I)\partial x^{2} \partial t^{2} }}$$ and $${{({\rm \rrho }A)({\rm \rrho }I)\partial ^{2} y(x,t)} \over {k(GA)\partial t^{4} }}$$ , should have the same value of scale factor, that is,

(…22) $${{{\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{A} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{t}^{2} }}{\equals}{{{\rm \rlambda }_{E} {\rm \rlambda }_{I} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{l}^{4} }}{\equals}{{{\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{I} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{l}^{2} {\rm \rlambda }_{t}^{2} }}{\equals}{{{\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{I} ({\rm \rlambda }_{E} {\rm \rlambda }_{I} )({\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{A} ){\rm \rlambda }_{y} } \over {{\rm \rlambda }_{k} ({\rm \rlambda }_{G} {\rm \rlambda }_{A} )({\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{I} ){\rm \rlambda }_{l}^{2} {\rm \rlambda }_{t}^{2} }}{\equals}{{({\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{A} )({\rm \rlambda }_{{\rm \rrho }} {\rm \rlambda }_{I} ){\rm \rlambda }_{y} } \over {{\rm \rlambda }_{k} ({\rm \rlambda }_{G} {\rm \rlambda }_{A} ){\rm \rlambda }_{t}^{4} }}$$

By substituting the relevant parameters of the simplified cylindrical shell into Equation (22), the following equations are obtained:

(…23) $${{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{R} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{t}^{2} }}{\equals}{{{\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{R}^{3} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{l}^{4} }}$$

(…24) $${{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{R} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{t}^{2} }}{\equals}{{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{R}^{3} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{l}^{2} {\rm \rlambda }_{t}^{2} }}$$

(…25) $${{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{R}^{3} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{l}^{2} {\rm \rlambda }_{t}^{2} }}{\rm {\equals}}{{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{R}^{3} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{k} {\rm \rlambda }_{{G_{{{\rm sim}}} }} {\rm \rlambda }_{l}^{2} {\rm \rlambda }_{t}^{2} }}$$

(…26) $${{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} {\rm \rlambda }_{R} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{t}^{2} }}{\rm {\equals}}{{{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }}^{2} {\rm \rlambda }_{R}^{3} {\rm \rlambda }_{H} {\rm \rlambda }_{y} } \over {{\rm \rlambda }_{k} {\rm \rlambda }_{{G_{{{\rm sim}}} }} {\rm \rlambda }_{t}^{4} }}$$

By simplifying Equation (24), next Equation (22) is obtained

(…27) $${\rm \rlambda }_{R} {\rm {\equals}}\;{\rm \rlambda }_{l} $$

Further, substituting Equation (27) into Equation (23) obtains

(…28) $${\rm \rlambda }_{f} {\equals}{\rm \rlambda }_{t}^{{{\minus}1}} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} $$

After simplifying Equation (25), next Equation (29) is obtained

(…29) $${{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{k} {\rm \rlambda }_{{G_{{{\rm sim}}} }} }}{\equals}1$$

As the cross section of the simplified prototype and that of the simplified scaled model are both circular rings, therefore, λ _k = 1, Equation (29) can be simplified to obtain

(…30) $${\rm \rlambda }_{{E_{1} }} {\rm {\equals}}\;{\rm \rlambda }_{{G_{{{\rm sim}}} }} $$

Meanwhile, Poisson’s ratio of the simplified cylindrical shell is represented as

(…31) $${\rm \nu }_{{{\rm sim}}} {\equals}{{E_{1} {\rm {\minus}}2G_{{{\rm sim}}} } \over {2G_{{{\rm sim}}} }}$$

The scale factor of ν_sim is expressed as

(…32) $${\rm \rlambda }_{{{\rm \nu }_{{{\rm sim}}} }} {\equals}{\rm \rlambda }_{{{{E_{1} {\rm {\minus}}2G_{{{\rm sim}}} } \over {2G_{{{\rm sim}}} }}}} {\rm {\equals}}{{{\rm \rlambda }_{{E_{1} {\rm {\minus}}2G_{{{\rm sim}}} }} } \over {{\rm \rlambda }_{{2G_{{{\rm sim}}} }} }}$$

Then, according to similitude theory^{( 28 )}, when Equation (30) is satisfied, next Equation (33) is obtained

(…33) $${\rm \rlambda }_{{E_{1} {\rm {\minus}}2G_{{{\rm sim}}} }} {\rm {\equals}\rlambda }_{{E_{1} }} {\rm {\equals}\rlambda }_{{2G_{{{\rm sim}}} }} $$

By substituting Equation (33) into Equation (32), $${\rm \rlambda }_{{{\rm \nu }_{{{\rm sim}}} }} $$ can be simplified as

(…34) $${\rm \rlambda }_{{{\rm \nu }_{{{\rm sim}}} }} {\equals}1$$

In addition, since Equation (26) can be obtained by substituting Equations (27) and (29) into Equation (23), there are no independent scaling laws derived from Equation (26).

Finally, by integrating Equations (27), (28) and (34), the scaling laws of the bending mode for the simplified cylindrical shell which satisfies the hypothesis of Timoshenko Beam are as follows:

(…35) $$\left\{ \matrix{ {\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} \hfill \cr {\rm \rlambda }_{R} {\rm {\equals}\rlambda }_{l} \hfill \cr {\rm \rlambda }_{k} {\equals}1 \hfill \cr {\rm \rlambda }_{{{\rm \nu }_{{{\rm sim}}} }} {\equals}1 \hfill \cr} \right.$$

3.3 Analysis of the equivalent similarity

To sum up, the process of equivalence is established in Fig. 2 and the equivalent criteria given by Equations (7), (10) and (21) are obtained. Through the deduce of scaling laws, we have completed the process of complete similarity in Fig. 2, and we derived the scaling laws given by Equation (35). Thus, the equivalent similarity, that is established by combining these equivalent criteria and scaling laws, is used to design scaled model for global bending mode of the stiffened cylindrical shells, and on the other hand, to predict the global bending mode of the prototype according to that of the scaled model.

Furthermore, this equivalent similarity is thoroughly analysed. Substituting Equations (4) and (16) into the expression of ν_sim given by Equation (31) yields

(…36) $${\rm \nu }_{{{\rm sim}}} {\equals}{1 \over 2}{{{{E_{1} I_{1} {\plus}E_{2} I_{2} } \over {2{\rm \rpi }R^{3} H}}} \mathord{\left/ {\vphantom {{{{E_{1} I_{1} {\plus}E_{2} I_{2} } \over {2{\rm \rpi }R^{3} H}}} {{{G_{1} A_{1} {\plus}G_{2} A_{2} } \over {2{\rm \rpi }RH}}}}} \right. \kern-\nulldelimiterspace} {{{G_{1} A_{1} {\plus}G_{2} A_{2} } \over {2{\rm \rpi }RH}}}}{\minus}1$$

Equation (36) can be written also using Equation (13) as

(…37) $${\rm \nu }_{{{\rm sim}}} {\equals}{1 \over 2}{{E_{1} } \over {G_{1} }}{{1{\plus}{{E_{2} A_{0} } \over {E_{1} A_{1} }}{{\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \over {R^{2} }}} \over {1{\plus}{{G_{2} A_{0} } \over {G_{1} A_{1} }}N}}{\minus}1$$

Then, according to similitude theory^{( 28 )}, by substituting Equation (37) into Equation (34), now Equation (38) is obtained

(…38) $$\left\{ \matrix{ {{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{G_{1} }} }}{\equals}1 \hfill \cr {{{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} } \over {{\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} }}{{\vskip--5pt \left. {{\rm \rlambda }_{x} } \right} {\vskip--3pt \big|}{{}_{{x{\equals}\mathop{\scale65%\sum}\limits_{j{\equals}1}^N {R_{j}^{2} }}}} \over {{\rm \rlambda }_{R}^{2} }}{\equals}1 \hfill \cr {{{\rm \rlambda }_{{G_{2} }} {\rm \rlambda }_{{A_{0} }} {\rm \rlambda }_{N} } \over {{\rm \rlambda }_{{G_{1} }} {\rm \rlambda }_{{A_{1} }} }}{\equals}1 \hfill \cr} \right.$$

or

(…39) $$\left\{ \matrix{ {\rm \rlambda }_{{{\rm \nu }_{{\rm 1}} }} {\equals}1 \hfill \cr {\rm \rlambda }_{{{\rm \nu }_{{\rm 2}} }} {\equals}{{{\rm \rlambda }_{N} {\rm \rlambda }_{R}^{2} } \over {\vskip--5pt \left. {{\rm \rlambda }_{x} } \right} {\vskip--3pt \scale 90%\big|}{{}_{{x{\equals}\mathop{\scale65%\sum}\limits_{j{\equals}1}^N {R_{j}^{2} }}}}}{\rm } \hfill \cr {\rm \rlambda }_{{G_{1} }} {\rm \rlambda }_{{A_{1} }} {\rm {\equals}\rlambda }_{{G_{2} }} {\rm \rlambda }_{{A_{0} }} {\rm \rlambda }_{N} \hfill \cr} \right.$$

For the commonly used metal materials in vehicle launchers, the difference between their Poisson’s ratios is very small, that is,

(…40) $${\rm \rlambda }_{{{\rm \nu }_{{\rm 2}} }} {\rm {\equals}}{{{\rm \rlambda }_{{E_{2} }} } \over {{\rm \rlambda }_{{G_{2} }} }}{\rm {\equals}}1$$

Then Equation (39) can be written also by using Equation (40) as

(…41) $$\left\{ \matrix{ {\rm \rlambda }_{{{\rm \nu }_{1} }} {\equals}{\rm \rlambda }_{{{\rm \nu }_{2} }} {\equals}1 \hfill \cr {\rm \rlambda }_{N} {\rm \rlambda }_{R}^{2} \vskip-0pt {\equals} {\vskip--0pt {{\rm \rlambda }_{x} } } {\vskip-2pt \scale 90%\big|}{\vskip---5pt {}_{{x{\equals}\mathop{\scale65%\sum}\limits_{j{\equals}1}^N {R_{j}^{2} }}}} \hfill \cr {\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} {\equals}{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} {\rm \rlambda }_{N} \hfill \cr} \right.$$

where $${\rm \rlambda }_{N} {\rm \rlambda }_{R}^{2} {\equals}\left. {{\rm \rlambda }_{x} } \right \vskip-2pt\big|_{\vskip-10pt {{x{\equals}\mathop{\sum}\nolimits_{j{\equals}1}^N {R_{j}^{2} } }} $$ represents the scaling law relevant to the distribution of the stiffeners.

The following is a division of Equation (41) in two specific equations used to solve different examples.

(1) If the number and distribution of stiffeners in the scaled model are exactly the same as those of the prototype, it means $$\left. {{\rm \rlambda }_{x} } \right|_{{x{\equals}\mathop{\sum}\nolimits_{j{\equals}1}^N {R_{j}^{2} } }} {\equals}\mathop{\sum}\nolimits_{j{\equals}1}^N {{\rm \rlambda }_{{R_{j} }}^{2} } ,\;{\rm \rlambda }_{N} {\equals}1$$ . Therefore, Equation (41) is simplified as

(…42) $$\left\{ \matrix{ {\rm \rlambda }_{{{\rm \nu }_{1} }} {\equals}{\rm \rlambda }_{{{\rm \nu }_{2} }} {\equals}1 \hfill \cr {\rm \rlambda }_{R}^{2} {\equals}\mathop{\sum}\limits_{j{\equals}1}^N {\rlambda _{{R_{j} }}^{2} } \hfill \cr {\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} {\equals}{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} \hfill \cr} \right.$$

Since $${\rm \rlambda }_{R}^{2} {\equals}\mathop{\sum}\nolimits_{j{\equals}1}^N {{\rm \rlambda }_{{R_{j} }}^{2} } $$ in Equation (42) is always satisfied in this case, it no longer needs to be listed. Accordingly, the scaling laws are obtained by using Equations (35) and (42) as

(…43) $$\left\{ \matrix{ {\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} \; \hfill \cr {\rm \rlambda }_{R} {\equals}{\rm \rlambda }_{l} \hfill \cr {\rm \rlambda }_{{{\rm \nu }_{{\rm 1}} }} {\equals}{\rm \rlambda }_{{{\rm \nu }_{{\rm 2}} }} {\equals}1 \hfill \cr {\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} {\equals}{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} \hfill \cr} \right.$$

(2) It is assumed that the prototype has 50 stiffeners and that only 20 stiffeners are designed in the scaled model, it means N _pro=50, N _m =20. Therefore, Equation (41) is simplified as

(…44) $$\left\{ \matrix{ {\rm \rlambda }_{{{\rm \nu }_{{\rm 1}} }} {\rm {\equals}\rlambda }_{{{\rm \nu }_{{\rm 2}} }} {\rm {\equals}1} \hfill \cr {\vskip4pt{{20} \over {50}}{\rm \rlambda }_{R}^{2} {\equals}\left. {{\rm \rlambda }_{x} } \right\Big|}_{{x{\equals}\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } }} \hfill \cr {\rm \rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} {\equals}{{20} \over {50}}{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} \hfill \cr} \right.$$

where $${{20} \over {50}}{\rm \rlambda }_{R}^{2} {\equals}\left. {{\rm \rlambda }_{x} } \right\Big|_{{x{\equals}\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } }} $$ can be rewritten as

(…45) $${{20} \over {50}}{{R_{m}^{2} } \over {R_{{{\rm pro}}}^{2} }}{\equals}{{\left. {\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right|_{m} } \over {\left. {\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right|_{{{\rm pro}}} }}$$

Similar to the simplification of Equation (14), now Equation (46) is obtained

(…46) $$\left. {\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right|_{{{\rm pro}}} {\equals}25R_{{{\rm pro}}}^{2} $$

By substituting Equation (46) into Equation (45), next Equation (47) is obtained

(…47) $$\left. {\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right|_{{m}} {\equals}10R_{{m}}^{2} $$

Finally, by integrating Equations (35), (44) and (47), the scaling laws in this special case are obtained as

(…48) $$\left\{ \matrix{ {\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} \hfill \cr {\rm \rlambda }_{R} {\equals}{\rm \rlambda }_{l} \hfill \cr {\rm \rlambda }_{{{\rm \nu }_{{\rm 1}} }} {\rm {\equals}\rlambda }_{{{\rm \nu }_{{\rm 2}} }} {\equals}1 \hfill \cr \left. {\mathop{\sum}\limits_{j{\equals}1}^N {R_{j}^{2} } } \right|_{m} {\equals}10R_{m}^{2} \hfill \cr {\rm 2}{\rm .5\rlambda }_{{E_{1} }} {\rm \rlambda }_{{A_{1} }} {\equals}{\rm \rlambda }_{{E_{2} }} {\rm \rlambda }_{{A_{0} }} \hfill \cr} \right.$$

In summary, the above analysis shows that the scaling laws relevant to the thickness of the cylindrical shell and the section size of the stiffener are reduced in Equation (35). Only the scaling laws between the cross-sectional area, the amount and the distribution of the stiffener are retained. Compared to traditional similarity method, the stiffeners in scaled model are no longer required to maintain the same amount and distribution as the prototype. Moreover, since Equation (35) does not limit the sectional shape of the stiffeners, then the stiffeners of scaled model can be designed with different cross-sectional shapes with that of the prototype. As a consequence, Equation (35) can make the design of scaled model more freely.

4.1 Design of scaled models

As the finite element model shown in Fig. 3, a stiffened cylindrical shell is set as the prototype with mid-radius of the shell R=2 m, axial length l=40 m and thickness δ=80 mm. Fifty axial stiffeners are distributed equidistantly in the circumference. The geometrical parameters of T-shaped cross section of stiffeners are shown in Fig. 4(a). The shell and the stiffeners are both made of aluminium alloy with elastic modulus 70 GPa, density 2700 kg/m³ and Poisson’s ratio 0.3.

Figure 3 (Colour online) Finite element model of the prototype. (a) Overall view; (b) Partial view.

Figure 4 (Colour online) Cross sections of the stiffeners in the prototype and the scaled models. (a) Prototype; (b) Model-1; (c) Model-2; (d) Model-3.

According to the scaling laws (Equation (35)), three models with different scales are established, which are Models 1, 2 and 3, respectively. In order to verify the analysis of equivalent similarity in Section 3.3, the Model 1 is designed with the same thickness as the prototype, the quantity of stiffeners of the Model 2 is different from that of prototypes and the cylindrical shell and the stiffener in the Model 3 are composed of different materials. Moreover, the cross-sectional shape of the stiffeners of the three models is different from that of the prototype, and the simpler rectangle shape is adopted (as shown in Fig. 4). The related parameters and scaling factors of each model are shown in Tables 1 and 2, respectively. The relevant design is presented in the next paragraph.

1. (i) According to Equation (43), a 1/5 scaled model named Model 1 is designed. Its mid-radius is 0.4 m, the axial length is 8 m and the thickness is 80 mm the same as the prototype. Fifty axial stiffeners are distributed equidistantly in the circumference. Their cross sections are rectangular with a width of 32 mm and a height of 10 mm (as shown in Fig. 4(b)). The shell and the stiffeners are both made of aluminium alloy. Accordingly, the scaling factors of Model 1 are obtained and shown in Table 2. By substituting the relevant scaling factors into $${\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} $$ given from Equation (43), λ _f = 5 is obtained, that is, the scale factor of frequency similarity is 5.

2. (ii) According to Equation (48), a 1/10 scaled model named Model 2 is designed. Its mid-radius is 0.2 m, the axial length is 4 m and the thickness is 40 mm. Twenty axial stiffeners are distributed equidistantly in the circumference. Their cross sections are rectangular with a width of 20 mm and a height of 10 mm (as shown in Fig. 4(c)). The shell and the stiffeners are both made of aluminium alloy. Accordingly, the scaling factors of Model 2 are obtained and shown in Table 2. By substituting the relevant scaling factors into $${\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} $$ given from Equation (48), λ _f = 10 is obtained, that is, the scale factor of frequency similarity is 10.

3. (iii) Directly according to Equation (35), a 1/20 scaled model named Model 3 is designed. Its mid-radius is 0.1 m, the axial length is 2 m and the thickness is 20 mm. Twenty axial stiffeners are distributed equidistantly in the circumference. Their cross sections are rectangular with a width of 15 mm and a height of 10 mm (as shown in Fig. 4(d)). The stiffeners are made of aluminium alloy, the same material as the prototype, while the material of the shell is switched to alloy steel with an elastic modulus of 210 GPa, a density of 7,800 kg/m³ and Poisson’s ratio of 0.3. Accordingly, the scaling factors of Model 3 are obtained and shown in Table 2. By substituting the relevant scaling factors into $${\rm \rlambda }_{f} {\equals}{1 \over {{\rm \rlambda }_{l} }}\sqrt {{{{\rm \rlambda }_{{E_{1} }} } \over {{\rm \rlambda }_{{{\rm \rrho }_{{{\rm sim}}} }} }}} $$ given from Equation (35), λ _f = 20.35 is obtained, that is, the scale factor of frequency similarity is 20.35.

Table 1 Parameters of the prototype and the two scaled models

Table 2 Scale factors of the three-scaled models

4.2 Validation of the scaling laws

The modal analysis of the prototype and the three-scaled models are carried out by use of the finite element software MSC. NASTRAN. The results of the first four global bending modes and their corresponding natural frequencies are shown in Fig. 5 and Table 3, respectively. Among them, because the mode shapes of the prototype and the scaled models are the same, the mode shape of the prototype is shown only.

Figure 5 (Colour online) First four global bending mode shapes of the prototype. (a) 1st mode shape; (b) 2nd mode shape; (c) 3rd mode shape; (d) 4th mode shape.

Table 3 Natural frequency of prototype and the three-scaled models

As shown in Table 4, the similar deviations of each natural frequency of the three-scaled models are calculated according to the results shown in Table 3. The similar deviations are calculated as follows: the natural frequency of the scaled model is multiplied by the scale factor of frequency to obtain the predicted frequency of the scaled model for the prototype, then the predicted frequency and the frequency of the prototype are compared to calculate the error of similarity, that is,

(…49) $$f_{{{\rm pre}}} {\equals}f_{{m}} \cdot {\rm \rlambda }_{f} ,{\rm {\varepsilon}{\equals}}\left| {{{f_{{{\rm pre}}} {\minus}f_{{{\rm pro}}} } \over {f_{{{\rm pro}}} }}} \right|{\times}100\%\,,$$

where the subscript ‘pre’ represents the predicted value of the scaled model for the prototype, and ε represents the error of similarity.

Table 4 The error of similarity for each natural frequency

From Table 4, it can be observed that the similar deviations of each natural frequency of the three-scaled models are not more than 1.03%. The results show that the deduced scaling laws in Equation (35) have a high degree of similarity, which can verify the equivalent similar method proposed in this paper. On the other hand, according to the special design of the three models, it is proved that the equivalent similar method can reach higher freedom in the design of the parameters such as thickness, number of stiffeners, cross-sectional shape and size of stiffeners, materials and so on.