With the bridge construction gradually from inland to coast and offshore, underwater structure of cross-sea bridge can encounter more complex hydrodynamic action. When elastic model is used to simulate the hydrodynamic response of underwater structure, the model scale and model material is the first problem must be solved. Based on principle of similitude, elastic model should meet the geometric similarity, kinematic similarity and mechanical similarity. Formula derivation gives the scale conditions for elastic model are as follows, geometric scale (Lr) equals elastic modulus scale (Er) and density scale (??r) equals 1. Then, how to find a material with same density and the elastic modulus is 1/ Lr of prototype is the key to make a model

Geng, B.L.

Liu, H.Y.

Zhang, H.Q.

Zheng, B.Y.

Hasanuddin University Repository

 470 
 
 
Proceedings of the 7th International Conference on Asian and Pacific Coasts  
(APAC 2013) Bali, Indonesia, September 24-26, 2013 
 
 
ELASTIC MODEL SCALE AND MATERIAL FOR UNDERWATER 
STRUCTURE OF CROSS-SEA BRIDGE 
 
B.L. Geng 
1
, B.Y. Zheng 
1
 , H.Q. Zhang 
1
 and H.Y. Liu 
1
 
 
 
ABSTRACT: With the bridge construction gradually from inland to coast and offshore, underwater structure of cross-
sea bridge can encounter more complex hydrodynamic action. When elastic model is used to simulate the 
hydrodynamic response of underwater structure, the model scale and model material is the first problem must be solved.  
Based on principle of similitude, elastic model should meet the geometric similarity, kinematic similarity and 
mechanical similarity. Formula derivation gives the scale conditions for elastic model are as follows, geometric scale 
(Lr) equals elastic modulus scale (Er) and density scale (ρr) equals 1. Then, how to find a material with same density and 
the elastic modulus is 1/ Lr of prototype is the key to make a model. 
 
Keywords: Elastic Model, scale, material, cross-sea bridge 
 
 
                                                 
1 Tianjin Research Institute for Water Transportation Engineering, National Engineering Laboratory for Port Hydraulic 
Construction Tecnology.  No. 2618, Xingang Erhao Road, Binhai New Area, Tianjin,  300456,  CHINA 
INTRODUCTION 
With the gradual development of bridge construction 
from the inland to the coast and offshore, extra large 
bridges are thus faced with harsher meteorological and 
hydrological conditions featuring by deep water, rapid 
flows and big waves. As the main mode of transportation 
across straits, extra large bridges have to withstand wave 
and current loads in the lower part and wind loads in the 
superstructure. In the coupling effect of wind, wave and 
current, bridge vibration or some displacement may 
occur in the construction and operation period, thus 
affecting the construction process or operational security.  
In the face of numerous natural factors, load field 
observation on extra large bridges in wind-wave-
circulation coupled environment and the load coupling 
characteristics are not clear. In addition, simulation 
technology and numerical analysis techniques remain in 
a blank state. The existing relevant design specification 
does not consider the coupling effect between multiple 
factors, so it is likely to overestimate the environmental 
design conditions or underestimate them. Therefore, in 
2011, "Study on the Theory and Method of 
Comprehensive Disaster Prevention and Mitigation of 
Oversize Bridge", a major project of the Ministry of 
Transport, will conduct physical model test and 
numerical simulation in this respect. This study mainly 
analyses the selection of scale method and model 
materials in the elastic substructure model test of cross-
sea bridges. 
 
 
 
 
HYDROELASTIC MODEL SCALE 
 
Hydroelastic model 
The flow-induced vibration of hydraulic structures is 
caused by the structural vibration of hydrodynamic loads, 
where there exists a dynamic interaction between the 
structure and the fluid. According to its excitation 
mechanism, the flow-induced vibration can be divided 
into forced vibration, object-vibration-controlled 
excitation, self-excitation and parametric excitation. In 
fact, the flow-induced vibration of hydraulic structures is 
ubiquitous. In most cases, the vibration is within the 
permissible range. But in some cases, the vibration can 
be harmful and even cause an accident. 
The flow-induced vibration is a kind of fluid-
structure interaction, so it is closely related to hydraulic 
conditions and structural dynamic characteristics. To 
study its vibration state and the mitigation measures, it is 
necessary to use a physical model of fluid-structure 
interaction, namely, hydroelastic model experiments. 
The hydroelastic model is a development of the 
hydraulic model, which has similarities in both hydraulic 
conditions and structural dynamic characteristics. 
The concept of hydroelasticity appeared in the field 
of hydraulics before 1958. The flow-induced vibration of 
gates was generally studied in indirect ways in the world 
before 1958. The hydraulics experiment and the 
structural dynamics experiment were carried out 
respectively. In 1964, Heller pointed out the concept of 
hydroelasticity is concerned with the interaction between 
inertia, hydrodynamic, and elastic forces. In 1965, 
Toebes defined the concept of fluid-elastic. When 
studying fluidelastic features of flow, he considered the 
 
Elastic Model Scale And Material For Underwater Structure Of Cross-Sea Bridge 
 
471 
 
interdependence between hydrodynamic force, inertial 
force generated by the structure, and elastic force. For 
hydroelastic modeling on the flow-induced vibration of 
hydraulic structures, the vibration phenomena caused by 
the interaction between hydrodynamic force, structural 
inertial force, and elastic force are studied, without 
consideration of the compressibility and elasticity of 
water. 
 
The principles of similitude 
To ensure that the main phenomena and 
performances of the prototype can be reflected from the 
model and that the quantitative predictions of the 
prototype can be made, the fluid motion and the 
structural vibration of the model and those of the 
prototype have to maintain a certain scale between 
corresponding physical quantities on corresponding 
points. To be specific, the model and the prototype must 
be similar in geometry, dynamics, and mechanics. 
For dynamic similarity, several similitude criteria are 
to be satisfied. For two flow-induced vibration systems 
which are completely similar, the scales of Froude 
number, Cauchy number, Euler number and Reynolds 
number should all be 1. It should be noted that the 
Reynolds number and Froude number cannot be 
simultaneously satisfied, but as long as the Reynolds 
number is large enough, the similitude criteria can be 
automatically satisfied. They are not only the similitude 
criteria for two flow-induced vibration systems, but also 
the criteria for model design. 
In addition, the friction of structure has various types, 
and different types of friction have different simulation 
methods. The frictions of structure are important because 
they can exert damping effect on vibration. Simulating 
coefficients of friction is not impossible but difficult as it 
requires high processing skills. An easier way is to not to 
simulate but to eliminate all frictions of the model, 
because by eliminating the frictions, the experimental 
results can be reliable for assessing the harmfulness of 
flow-induced vibration. 
 
Selection of elastic model scales 
The selection of elastic scale mainly involves inertia 
force, gravity, and elastic force. The scale factors can be 
derived as follows: 
(1) Inertia force 
 
inertF ma                                                               (1) 
 
here, m is mass; a is acceleration. The scale of inertia 
force is given below: 
 
3 2 4 2 2 2
inert,
p p p p p
r r r r r r r r r r r
m m m m m
m a V a
F L L T L T L v
m a V a

  

 

    

 
(2) 
 
here, V is volume; ρ is density; the corner mark p 
represents the prototype and the corner mark m 
represents the model; ρr, Tr, vr are the density scale, time 
ratio, and speed scale respectively. 
      (2) Gravity 
 
gravF mg                                                                 (3) 
 
here, g is gravity acceleration. The gravity scale is given 
below: 
 
3
grav,
p p p p p
r r r r
m m m m m
m g V g
F L g
m g V g



                     (4) 
 
here, gr is the scale of gravity acceleration. In the 
gravitational field, gr =1. 
(2) Elastic force 
 
elast
L
F EA
L

                                                          (5) 
 
here, E is elasticity modulus; A is the cross sectional area 
under the elastic force; ΔL is the dependant variable of L; 
ΔL/L=ε, ε is the strain value; and according to geometric 
similarity, εp/εm=1. The scale of elasticity force is given 
below: 
 
2
,elast r r r r rF E A E L                                             (6) 
 
According to the criterion of Froude number, the 
scale of inertia force is equal to that of gravity, namely, 
 
2 2 3
r r r r r rL v L g                                                       (7) 
 
considering gr=1, the simplified equation is as follows 
 
2
r rv L                                                                       (8) 
 
According to the criterion of Cauchy number, the 
scale of inertia force is equal to that of elastic force, 
namely, 
 
2 2 2
r r r r rL v E L                                                           (9) 
 
by simplifying the equation, the following formula can 
be obtained: 
 
 B.L. Geng, et al. 
472 
 
2
1r r
r
v
E

                                                                   (10) 
 
substituting the equation (8) into the above, it is 
concluded that when ρr =1, 
 
r rE L                                                                     (11) 
 
It can be concluded that the requirements of 
hydroelastic model to be met by the model material are 
Er=Lr, ρr =1. Therefore, the key to establishing the model 
is to find a kind of material that has the same density 
with the prototype and the elasticity modulus is 1/Lr of 
the prototype. 
 
HYDROELASTIC MODEL MATERIAL 
No materials satisfying the above similitude criteria 
can be found currently in the market. According to 
relevant documents, the hydroelastic materials used in 
model experiments of water resources & hydropower 
engineering were specially developed. They were 
manufactured by relevant manufactures with given 
parameters. 
The bridge substructure is mostly concrete, and the 
model material of the concrete structure should meet the 
above similitude criteria When Changjiang Scientific 
Research Institute was preparing "Hydroelastic Model 
Test Research on Releasing Flood Vibration of Arch 
Dam for Goupitan Project", they conducted a 
monographic study on the hydroelastic model materials 
of the dam body and the dam base for their physic-
mechanical properties. Heavier rubber materials were 
employed to meet the elasticity criterion, while barite 
powder was used as filler to meet the density criterion. 
For example, a hydroelastic model with a scale of 150 : 1 
was used in the test. The physic-mechanical parameters    
are listed in Table 1 below:
 
Table 1 physic-mechanical parameters of the hydroelastic model with 150:1 scale 
Dynamic elastic modulus 
(MPa) 
Density  
(t / m3) 
Damping ratio 
(%) 
Poisson ratio 
prototype model prototype model prototype model prototype model 
2.7×104 
~3.15×104 
198~220 2.4 2.43 2 5 0.167 0.2~0.25 
 
It is noted that the elastic modulus of such material 
is largely affected by temperature. When the temperature 
is high, the elastic modulus will decrease significantly. 
As a result, when doing the experiment, the ambient 
temperature should be consistent with the temperature 
when elastic modulus was tested. Figure 1 is the waste 
model material and Figure 2 is the dam model after the 
test. 
 
 
Fig. 1 The waste model material 
 
Fig. 2 Dam model with hydroelastic material 
 
CONCLUSION 
This study explores how to select the elastic model 
scale of cross-sea bridge substructure. It points out that 
the key to establishing the elastic model is to find a type 
of material that has the same density with the prototype 
and that the elasticity modulus is 1/Lr of the prototype. 
On the basis of field visits, this study describes the 
features and methods of concrete elastic model material, 
and provides a reference for relevant elastic model tests. 
 
Elastic Model Scale And Material For Underwater Structure Of Cross-Sea Bridge 
 
473 
 
ACKNOWLEDGEMENTS 
The authors gratefully acknowledge the financial 
support provided by Science and technology project of 
the Ministry of Transport (Grant No. 2011318223170 & 
2011318494150) and Central Commonweal Research 
Institute Basic R&D Special Foundation of TIWTE 
(Grant No. TKS110201) 
 
REFERENCES 
Cao, X. L., Zhang, L. R. (2006) Hydroelastic model test 
research on releasing flood vibration of arch dam for  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Goupitan project. Changjiang Science and research 
institute, Wuhan. 
Kolkman P. A. (1979) Analysis of vibration and 
damping measurements on a reversed tainter valve. 
IAHR Symposium, Stockholm. 
Wu, J. F., Cao, X. L. (1996) Hydroelastic model material 
development report. Changjiang Science and 
research institute, Wuhan. 
Wu, J. F., Zhang, L. R. (2008) Hydroelastic model 
research and application for Flow-induced vibration 
of  hydraulic structure. Changjiang Press, Wuhan. 
 
 


Elastic model scale and material for underwater structure of cross-sea bridge

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