Structural Dynamic Characteristics of an Ancient Egyptian Obelisk
Abstract
Ancient Egyptian obelisks have existed for several millennia and have withstood numerous natural disasters among which is the Hatshepsut Obelisk. This study investigates the structural dynamic characteristics of the Obelisk using a twostaged approach. The first, involved a series of tests to study the mechanical properties of the Red Aswan granite of which the obelisk was made while the second incorporates the test results in modeling the Obelisk pyramid using finite element software. An Eigenvalue modal analysis was performed in order to generate the natural periods and corresponding modes of vibration of the structure. These natural periods were compared to the dominant periods of earthquakes in order to analyze the obelisk’s response to loading and its degree of resonance. Following, a timehistory analysis was performed in which the structure was subjected to an earthquake signal. The results show that the Hatshepsut obelisk could safely withstand seismic loads.
Keywords
Structural dynamics Earthquake engineering Obelisk Modal analysis Dynamic analysis Granite1 Introduction
The ancient Egyptians used numerous materials in building different structures for religious and sacral purposes such as pyramids, temples, obelisks and various forms of tomb structures. While buildings used for the dwelling of royalty and nobility were built using bricks made from the mud available by the annual flooding of the Nile, funeral and religious structures were built using more resistant material to last for eternity (Klemm and Klemm 2001).
Obelisks are regarded as one of the key attractions of the ancient Egyptian civilization that are still present today. There are several obelisks today decorating major international cities like Paris, London and New York, after being moved from Egypt during the last two centuries, in addition to the Hatshepsut obelisk in the Karnak temple in Luxor, Egypt. According to (Engelbach 1923), the number of obelisks built throughout various dynasties of ancient Egypt must have been numerous. He reported that there must have been more than 50 obelisks with a length exceeding 10 m. He attributed the decrease in number of obelisks present today to earthquakes and soilsubsidence problems.
Most obelisks studied by historians and archeologists are reported to be made of a red granite from quarries located to the south of Aswan in Southern Egypt. The discovery of an “unfinished obelisk” on site in that area was a clear evidence that ancient Egyptian obelisks were extracted from that quarrying area (Engelbach 1923; Klemm and Klemm 2008; Kelany et al. 2009). This type of granite is generally named “Red Aswan Granite”. This type of granite consists of large reddish feldspar crystals in a fabric that also contains quartz, plagioclase and biotite (Klemm and Klemm 2008; Serra et al. 2010). The practical reasons for selecting this type of granite to build the obelisks is because its natural joints are far enough to enable the extraction of large structures like obelisks as a single piece of stone mass with no fissures or cracks (Engelbach 1923).
While extensive research was conducted on the red Aswan granite from the archeological, geological and chemical perspective (Kelany et al. 2009; Klemm and Klemm 2001; Serra et al. 2010), little information is available on some of the mechanical properties of this material such as its modulus of elasticity. Moreover, there is a scarcity in the information regarding the seismic behavior of monumental structures built using this material such as obelisks and tombs. Hence, the objective of this work is to conduct experimental testing to determine the mechanical properties of red Aswan granite such as the unit weight, ultimate compressive strength and modulus of elasticity. Consequently, these parameters are used as input parameters to study the structural dynamic characteristics of the Hatshepsut obelisk using a finite element model. Finally, a timehistory analysis is conducted in which a real earthquake signal is applied on the structure to check its ability to withstand the earthquake vibrations.
2 Material Testing
2.1 Mechanical Tests on Red Aswan Granite
2.2 Results of Mechanical Tests
Mechanical properties of red Aswan granite.
Specimen #  Unit Weight (Kg/m^{3})  Ultimate compressive strength (MPa)  Elastic modulus (MPa) 

1  2580.39  143.15  5687 
2  2525.49  121.14  5202 
3  2517.64  146.04  5608 
4  2556.86  153.02  5508 
5  2529.79  139.1  5073 
6  2256.86  143.76  5137 
7  2517.64  142.68  5679 
8  2544  135.62  5509 
Average  2541.09  140.56  5425.38 
Standard dev  22.43  9.34  249.91 
3 Numerical Model
3.1 Finite Element Model
The modeled obelisk had a total height of 29.57 m, a squared crosssection of 2.41 m × 2.41 m and tapered until having a 1.77 m × 1.77 m crosssection at the base of the pyramidal located at the top of the obelisk at a point 26.61 m above the obelisk base (Engelbach 1923).
3.2 Modal Analysis
The natural periods and the modal participation factors.
Mode #  Period (s)  Modal participation factors  

Translation in X  Translation in Y  Translation in Z  
1  1.166  0%  57%  0% 
2  1.166  57%  0%  0% 
3  0.230  1%  21%  0% 
4  0.230  21%  1%  0% 
5  0.115  0%  0%  0% 
6  0.089  9%  0%  0% 
7  0.089  0%  9%  0% 
8  0.066  0%  0%  80% 
9  0.048  1%  4%  0% 
10  0.048  4%  1%  0% 
11  0.047  0%  0%  0% 
12  0.030  2%  1%  0% 
As shown in Table 2, the period for the first two modes of vibration was 1.166 s which is within the range of the dominant periods of typical earthquakes and typical boundary winds which exceed 0.5 s (Tedesco et al. 1999). That implies that the Hatshepsut obelisk is expected to resonantly respond due to earthquakes and winds and will mainly behave in a dynamic manner when subjected to such dynamic loads. Hence, it is necessary to perform a dynamic analysis when structurally analyzing this obelisk as a static analysis is not expected to produce accurate results.
3.3 TimeHistory Analysis
Following on the modal analysis, a timehistory analysis was performed in order to determine the timevarying response of the structure while subjected to a real earthquake using the Newmark direct integration method. The earthquake chosen was the 1940 El Centro earthquake that happened in California, USA with perceived intensity of X on the Mercalli intensity scale (Wikimedia Foundation, Inc. 2016). This earthquake is significantly stronger than any recorded earthquake that has ever been recorded in Egypt in which the obelisk is located. Hence, the signal of the earthquake event was scaled down to match the peak ground acceleration of Luxor, Egypt (where the Hatshepsut obelisk is located in the Karnak temple), which is 0.125 g according to the Egyptian loading code (Housing and Building National Research Center 2008).
The Newmark direct integration method used applies the concept of proportional damping in which the coefficients α and β are multiplied by the stiffness and mass matrices. These two coefficients were calculated based on a conservatively assumed damping of 1%. This percentage of damping is based on conservatively assuming that the granite has a similar performance as concrete which is typically considered to have a damping even larger than this percentage (Tedesco et al. 1999). Another factor taken into account is the time step size as according to (Bathe 2006) the solution will not converge if the time step exceeds T/π and it could only produce accurate results if it is less than or equal to T/10 where T is the smaller of the natural period of the highest mode of interest and the dominant period of loading.
3.4 Results of TimeHistory Analysis
4 Conclusions

The natural periods for the twelve modes of vibrations of the studied obelisk ranged between 1.166 s and 0.03 s.

The principal natural periods of vibration for the various modes were within the range of the dominant periods of the earthquake ground motions implying that the resonant component of vibration is expected to be significant.

According to the results produced by the timehistory analysis, the maximum spectral displacement occurred at a period equal to the principal natural period of the structure confirming the occurrence of resonance due to the earthquake signal.

The maximum stresses are significantly less than the ultimate strength of the red Aswan granite proving that the structure could safely withstand the earthquake load.

According to the results produced by the timehistory analysis, the horizontal drift at the top of the obelisk was 1/300 of the obelisk height which is within the safe limits of vibrations explaining how that obelisk existed for thousands of years.
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