1 INTRODUCTION

Ramses II statue was built 3200 years ago as a one piece of granite. It is 12 m high and weighs 83 tons. It was found in 1880 in six pieces in Meet Rahina, 40 kilometers to the south of Giza. It was moved to central Cairo in front of the Railway station in 1954 and the six pieces were joined together. This square was called Ramses Square, and was famous for many years as the centre of Cairo. The statue has been a symbol of Cairo, greeting the visitors who arrive by trains. The scenery of the square was very nice with a very beautiful fountain in front of the statue. Nowadays, the amount of traffic passing through the square has increased tremendously. Several bridges and new roads were built that they had to remove the fountain. A very big underground station was also built close to the statue’s foundation.

There were two main reasons to decide to relocate the statue. The first was that the square was so crowded that the scenery is not that nice any more. The second was the fear to harm the statue by either the air pollution from the exhaust gases of the traffic, or the vibrations caused by the ground and underground traffic. The project faced a lot of opposition from several individuals and organizations, but the debate was resolved after long discussions, deciding to move the statue. The new spot, close to the pyramids, will host the new Egyptian Museum, the biggest museum in the whole World, planned to be opened in 2011. This relocation project was registered in Guinness World Records (2008), the World Authority on Record Breaking, as the

Operational Modal Analysis of Ramses II Statue

Tamer Elnady ASU Sound & vibration Lab., Ain Shams University, Cairo, Egypt

ABSTRACT: Ramses II Statue is an invaluable monument that was created 32 centuries ago. The sculpture is made of an 83 tons, 12 m high granite rock. In fear of the harsh environment of the heavy traffic and smoke, the statue was moved in 2006 from downtown Cairo to the outskirts of the crowded city. The project was recorded in Guinness World Records, as the event holder of two records: the Farthest Building Relocation, and the Heaviest Building Moved on Wheels. There was a fear that the relocation project itself might ruin the cracked statue. The ASU Sound and Vibration Lab at Ain Shams University was appointed to monitor the vibrations of the statue during the project. The major concern was to avoid exciting the statue close to one of its natural frequencies. Therefore, the identification of the natural frequencies and mode shapes was crucial. There were many uncertainties regarding the material and structure of the statue which made the analytical simulation impossible and the numerical simulation less confident. Traditional experimental techniques were unacceptable for this huge structure. The main idea was to perform use the Operational Modal Analysis technique using the random excitation due to the construction work to determine the natural frequencies and mode shapes using. A full three-dimensional Finite Element Analysis also helped to assure that no single mode of vibrations was overlooked in the Operational Modal Analysis. This paper reports the analyses that were performed by the ASU Sound and Vibration Lab and which helped in providing the invaluable Ramses II statue a safe trip to its final destination.

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event holder of two records: the Farthest Building Relocation (35 km), and the Heaviest Building Moved on Wheels (83 tons).

The ASU Sound and Vibration Laboratory at Ain Shams University was appointed to monitor the vibration levels on the body of the statue during the relocation project. One major concern was to avoid exciting the statue close to one of its natural frequencies. Therefore, the identification of the natural frequencies and mode shapes was crucial. When the statue was found in 1886, it was found in 6 pieces that were joined together by the means of steel rods. There was a concern about the status of these steel rods, if they still do their job or not.

Traditional modal model identification methods and procedures are based on forced excitation laboratory tests during which Frequency Response Functions (FRFs) are measured. However, the real loading conditions to which a structure is subjected often differs considerably from those used in laboratory testing. Since all real-world systems are to a certain extent non-linear, the models obtained under real loading will be linearized for much more representative working points. Additionally, environmental influences on system behaviour (such as pre-stress of suspensions, load-induced stiffening and aero-elastic interaction) will be taken into account. On the other hand, forced excitation tests are sometimes very difficult, if not impossible, to conduct, at least with standard testing equipment. In such situations operational data are often the only ones available. Hence, extending classical operating data analysis procedures with modal parameter identification capabilities will allow a better exploitation of these data. Finally, the availability of in-operation established models opens the way for in situ model-based diagnosis and damage detection. Hence, a considerable interest exists in extracting valid models directly from operating data. This is a procedure that is called Operational Modal Analysis (OMA).

There are several studies in the literature which involve either Finite Element Analysis or Operational Modal Analysis on different types of structures in order to assess the dynamic behaviour of these structures for different reasons. Casciati and Borja (2004) constructed a full three-dimensional dynamic soil–foundation–structure interaction (SFSI) analysis of a famous landmark in Luxor, Egypt, the South Memnon Colossus, is performed to investigate the response of this historical monument to seismic excitation. The analysis is carried out using the finite element (FE) method in time domain. The statue is modelled using 3D brick finite elements constructed from a photogrammetric representation that captures important details of the surface and allows the identification of probable zones of stress concentration. The modelling also takes into account the presence of a surface of discontinuity between the upper part of the statue and its fractured base. These studies are useful for future conservation efforts of this historical landmark, and more specifically for designing possible retrofit measures for the fractured base to prevent potential collapse of the monument from overturning during an earthquake.

Magalhães et al. (2006) described the application of three output-only modal identification techniques to the data collected during an ambient vibration test performed at the Braga Sports Stadium suspended roof. The identified natural frequencies and modes shapes are compared with the ones predicted by a finite element model that took into account the geometrical non-linear behaviour and the construction process. On the other hand, the estimated modal damping coefficients are compared with the ones obtained in previously developed forced/free vibration tests. These comparisons allowed evaluating the accuracy of the results provided by the experimental and numerical tools used. It is shown that the available techniques can provide very accurate estimates of natural frequencies and mode shapes, the obtained estimates being very coherent and well correlated with the results provided by the developed numerical model.

The Operational Modal Analysis Techniques have been tested against Forced Modal Analysis and have proven to give good results. OMA is also used for damage detection, a feature which will be used later in this paper. Ramos et al. (2005) performed a dynamic identification analysis of a masonry construction to verify if the operational modal analysis is able to assess the damage in an earlier stage in the structure. The masonry model was built with limestone units and lime mortar joints with polymeric grid reinforcement placed on the horizontal joins. The dynamic identification analysis was divided in several tasks. For the calculation of the expected dynamic parameters a preliminary FEM analysis was carried out. Two types of operational modal analysis were used, the EFDD and SSI methods. One of the analysis was to compare the classical input-output experimental modal analysis with the

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ambient based modal analysis. During the several tests it was possible to observe the effect of introducing a damage by decreasing of all resonant frequencies. The corresponding mode shapes also suffers significant changes, especially for the higher modes. In particular, a new mode shape associated with localized damage was detected, either by ambient and forced vibration identification tests.

Brincker and Andersen (2002) performed modal identification of the Swiss highway bridge Z24. A series of 15 progressive damage tests were performed on the bridge before it was demolished in autumn 1998, and the ambient response of the bridge was recorded for each damage case. Modal properties are identified from the ambient responses by frequency domain decomposition (FDD). 6 modes were identified for all 15 damage cases. The identification was carried out for the full 3D data case i.e. including all measurements, a total of 291 channels, a reduced data case in 2D including 153 channels, and finally, a 1D case including 20 channels. The modal properties for the different damage cases are compared with the modal properties of the undamaged bridge. Deviations for frequencies and damping ratios are used as monitoring variables. From these results it can be concluded, that frequencies and model shapes for the structure changed significantly during damage. Further, it can be concluded, that the spatial information gained by a large number of channels does not seem to result in significant better estimates of frequency and mode shape deviations.

Cunha et al. (2004) reanalysed of the ambient vibration data of Vasco da Gama cable-stayed bridge with the purpose of testing the efficiency and accuracy of two recent and promising identification methods in a large application: the Frequency Domain Decomposition (FDD) and the Stochastic Subspace Identification (SSI) methods. The modal estimates obtained using these alternative approaches are compared, taking also into account the estimates previously obtained with the conventional Peak Picking technique from the free vibration test of the bridge, performed at the end of construction. The analysis shows that both the Frequency Domain Decomposition (FDD) and the Stochastic Subspace Identification methods are powerful methods that allow an objective identification procedure, providing sufficiently accurate estimates of natural frequencies and mode shapes of large bridges. With regard to the identification of modal damping ratios, which can play a very important role in terms of aerodynamic stability of this type of structures, the SSI method seems to enable rather satisfactory estimates, in comparison with accurately measured values in a free vibration test performed at the end of construction.

Ventura et al. performed a model updating study conducted on a 15-storey reinforced concrete shear core Heritage Court building. The output-only modal identification results obtained from ambient vibration measurements of the building were used to update a finite element model of the structure. The starting model of the structure was developed from the information provided in the design documentation of the building. Different parameters of the model were then modified using an automated procedure to improve the correlation between measured and calculated modal parameters. Careful attention was placed to the selection of the parameters to be modified by the updating software in order to ensure that the necessary changes to the model were realistic and physically realizable and meaningful. But at the end of a model updating exercise it is up to the analyst to accept the changes suggested by the modal updating program and to justify how realistic are the changes to be done.

In the present work, the dynamic analysis of Ramses II Statue is presented. It was decided to perform an Operational Modal Analysis and a Finite Element Analysis on the statue for two-fold objectives. The first is to identify the natural frequencies and mode shapes of the structure in order to avoid damaging it during the construction phase of the relocation transporter around the statue. The second is to assess the integrity of the steel rods joining the six parts together by comparing the results of the OMA to that of the FEM model of the healthy one-piece statue.

2 OPERATIONAL MODAL ANALYSIS

The purpose of this procedure is to extract modal frequencies, damping and mode shapes from data taken under operating conditions. This means that under the influence of its natural excitation. Theoretically, one could consider the case where the input forces are measured in such conditions which means that conventional FRF processing and modal analysis techniques

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could be used. However the Test.Lab Operational Modal Analysis module is aimed specifically at applications where the inputs cannot be measured and works when only responses such as accelerations signals are available. The ideal situation is when the input has a flat spectrum. In our measurements, this condition was satisfied because of the nature of the drilling process in the concrete base. Although the drilling speed was constant, the feed was not constant. The drill hits a variety of rock types and sizes generating excitation at several frequencies.

The simplest method to estimate the modal parameters of a structure subjected to ambient loading is the so−called peak−picking method. An interesting, more advanced alternative for peak picking is the stochastic subspace identification method. Originally intended for application to frequency response functions (FRFs), the frequency domain maximum likelihood method was extended to use output spectra as primary data. Maximum likelihood identification is an optimization based method that estimates the parameters of a model by minimising an error norm. After having pre-processed output data into output spectra, it is now the task to identify a modal model. This technique is called PolyMAX, which has proven better performance in discarding the incorrect mode in the stabilization diagram.

3 FINITE ELEMENT MODEL

The Ramses II statue is treated as a solid structure built of isotropic material (granite) whose modulus of elasticity E is 60 GPa, Poission’s ratio υ is 0.25, and density ρ is 2600 kg/m3. The displacement vector u at any point of this structure is divided into three components

,,vu and w in the ,, yx and z directions, respectively. The strain vector � has six components { }zxyzxyzzyyxx εεεεεε which are related to the displacement components by the strain-displacement relationship (Reddy 1993)

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(1)

The stress vector � also has six components {�xx �yy �zz �xy �yz �zx} which are related to the strain vector by the stress-strain relationship

�C� ⋅= (2)

where C denotes the matrix of elastic stiffness which is defined for isotropic elastic materials as (Zienkiewicz and Taylor 2000)

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A 4-node tetrahedral element is considered. Each node has three degrees of freedom. Within this element the displacement is approximated by

N.Uu ≈ (4)

where U is the vector of degrees of freedom of the element (12 degrees of freedom per element) while N is a 123× matrix including Lagrange interpolation functions. The variation in the potential energy of the element due to a virtual displacement uδ is

� ⋅=V

dVPE ��δδ (5)

while the virtual change in the kinetic energy due to the virtual displacement is

� ⋅=V

dVKE uu ��ρδδ (6)

In the absence external forces, the structure can undergo free oscillations at a natural circular frequency nω satisfying the system of equations

( ) 0UMK 2 =− nn� (7)

where K and M are the global stiffness and mass matrices which can be calculated using the Finite Element approximation in 4 and the stress-strain-displacement relations in Eqs. (1) and (2). The vector Un is the global vector of degrees of freedom. Eq. (7) represents an eigenvalue problem which can be solved for the natural frequencies �n and the corresponding mode shapes Un.

3.1 Importing the geometry into COMSOL Multiphysics

The available statue geometry was in the form of a 100 MB stereolithography CAD file (.stl), Fig. 1a. STL files describe only the surface geometry of a three dimensional object without any representation of color, texture or other common CAD model attributes. This file was created by scanning the full-scale statue. By definition, this geometry is hollow and cannot be used for structural analysis. Therefore, it was required to form a solid geometry out of this hollow shape. Several ways were tried using different CAD software.

(a) (b) (c) (d) (e)

Figure 1: The steps of importing the statue geometry.

The procedure which was successful is the following: Import the STL file into a software

called Rhino where we could take cross-sections at equal distances across the main axis of the

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statue (Fig. 1b), then delete the hollow shell. This results in a set of solid 2D planes on top of each other. The new geometry is then imported into Autodesk Inventor where all planes are selected, and the loft command was applied to them (Fig. 1c). The loft command fills the gap between two 2D geometries in different planes to form a 3D solid. The lateral profile is determined as an interpolation between the perimeters of the two geometries. The geometry can now be saved as a solid 3D IGES file, that can be imported directly into COMSOL Multiphysics (Fig. 1d). An automatic unstructured mesh is used resulting in 122.5 Thousand Degrees of Freedom (Fig. 1e) in approximately 24.5 Thousand tetrahedral elements.

The process of finding the correct way to convert the hollow geometry into a solid one consumed a lot of time. We wanted to test if it was really necessary to use the exact 3D geometry or it was enough with a simplified one. Several simplified geometries consisting of Rectangular Cuboids and cylinders were tested. None of these simplified geometries succeeded to capture the modal behaviour of the original 3D statue geometry.

3.2 Results

The commercial software COMSOL Multiphysics was used to numerically solve the system of Eq. (7). The solution time for the number of degrees of freedom discussed earlier is a few minutes. The first five mode shapes are shown in Fig. 2, and the corresponding natural frequency is shown below each mode. The first two modes are the first bending modes in two horizontal directions. The third mode is the twisting mode around the vertical axis. The fourth and fifth modes can be seen as the second bending modes in the two horizontal directions. It is noted that most of the mode motion is in the top crown of the statue.

5.3 Hz 12.84 Hz 35.60 Hz 48.48 Hz 57.16 Hz

Figure 2: The first five mode shapes calculated using FEM.

4 MEASUREMENTS AND OMA ANALYSIS

The operational time data was collected at 9 measurement points in the two horizontal directions. It was proven earlier that a small number of measurement points can still capture the dynamic behaviour of the structure (Brincker and Andersen 2002). The locations and directions of these points are shown in Fig. 5. The time data was collected using an LMS Pimento System (Fig. 4). The accelerometers used were B&K 4507 fixed to its plastic base which was glued to the concrete body of the statue using special glue that cannot be affected by the heat of the sun and can be removed easily using boiled water. Before every measurement, a standard calibration of the microphones was done using the hand held B&K vibration calibrator. This calibration was very useful to detect any problems with the route of the long cables. The

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working conditions at the site were very difficult as several other construction activities were taking place at the same time.

A good excitation is important to get good results. If the excitation is strong enough, this will enhance the correlation of the measurement data to support better analysis. We collected time domain data of the acceleration at different points on the statue during the drilling of holes in the concrete base which provided a strong excitation signal (Fig. 5). This time data was fed to LMS Test.Lab together with the STL Geometry for post processing and extraction of the modes. The PolyMAX algorithm was further implemented to the data.

Table 1 shows a comparison of the calculated natural frequencies of the first four modes by Finite Element Analysis, and those extracted by the Operational Modal Analysis. There is a good agreement (less than 10% error) between the results obtained by the two analyses. This gives credibility to the obtained natural frequencies. Moreover, this also proves that the assumption used during the Finite Element Analysis, that the statue is considered as one object, is a valid and realistic assumption. This implies that the steel rods joining different parts of the statue are in good condition. This is useful information that we can get of OMA, to detect structural damages. On the other hand, it was proven that the statue is safe from hitting its fundamental resonance by the excitation forces during the relocation project. The frequencies of construction equipment are much higher than the critical natural frequency, whereas the frequencies of transporter movement are much lower than the critical natural frequency because the speed is very low (5 km/hr).

Figure 3: The location of the measurement points on the statue.

Table 1 Comparison between the natural frequencies calculated by FEM and OMA

FEM OMA 5.5 6.8

12.84 11.86 35.6 30.5

48.48 44.29

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Figure 4: The process of drilling the holes in the concrete

base of the statue. Figure 5: The LMS Pimento system during

the collection of the data for OMA.

5 SUMMARY AND CONCLUSIONS

Ramses II Statue was relocated from central Cairo for a distance of 35 km close to the new Egyptian Museum. During the preparation of the statue and the site for the relocation, several construction activities were taking place around the statue and there was a fear that these activities would affect the statue structure or the steel rods joining its six parts. The ASU Sound and Vibration Laboratory at Ain Shams University was responsible for the vibration monitoring and dynamic analysis during the project. In order to estimate the natural frequencies and mode shapes of the statue, both Finite Element Analysis and Operational Modal Analysis were performed. FEM and OMA results were in good agreement. This gives credibility to the obtained results, and proves that the assumption of considering the statue as one rigid body is a reasonable one. This implies that the steel rods joining different parts of the statue are in good condition. Moreover, it is proven that the statue is safe from hitting its fundamental resonance by the excitation forces during the relocation project. On the other hand, it is very important to use the original and actual geometry during the FE analysis in order to obtain accurate results.

REFERENCES

Brincker R. and Andersen P. 2002. Identification of the Swiss Z24 Highway Bridge by Frequency Domain Decomposition. Proceedings of The 20th International Modal Analysis Conference (IMAC), Los Angeles, California, February 4-7.

Casciati S. and Borja R. 2004. Dynamic FE analysis of South Memnon Colossus including 3D soil–foundation–structure interaction. Journal of Computers and Structures, 82, p. 1719–1736.

Cunha A., Caetano E., Brincker R. and Andersen P. 2004. Identification from the Natural Response of Vasco Da Gama Bridge. Proceedings of The 22nd International Modal Analysis Conference (IMAC), Dearborn, Michigan, January 26-29.

Glenday C. 2008. Guinness World Records. Magalhães F., Caetano E. and Cunha A. 2006. Operational Modal Analysis of the Braga Sports Stadium

Suspended Roof. Proceedings of The 24th International Modal Analysis Conference (IMAC), St. Louis, Missouri.

Ramos L., Costa A. and Lourenço P. 2005. Operational Modal Analysis for Damage Detection of a Masonry Construction. 1st International Operational Modal Analysis Conference, Copenhagen, Denmark, April 26-27.

Reddy J. 1993. An Introduction to the Finite Element Method, second ed., McGraw-Hill, New York. Ventura C., Brincker R., Dascotte E. and Andersen P. 2001. FEM Updating of the Heritage Court

Building Structure. Proceedings of The 19th International Modal Analysis Conference (IMAC), Orlando, Florida, February 5-8.

Zienkiewicz O. and Taylor R. 2000. The finite element method: solid mechanics 4th ed. Butterworth-Heinemann.