Structural Analysis of Historic Construction – D’Ayala & Fodde (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5

Continuous and discontinuous modeling of the structures inBayon temple, Angkor

T. Maeda & T. YamamotoDepartment of Architecture, Waseda University, Tokyo, Japan

T. HiraiObayashi Corporation, Tokyo, Japan

ABSTRACT: Bayon temple, Angkor, consists of dry-masonry structures made of sandstone. Strong wind inthe rainy season is suspected as one of the causes for its progressive deterioration. Since 2003, micro-tremor hasbeen measured at most of the tower-type structures and the continuum-equivalent elastic modulus was foundto be 1/9 to 1/27 of that of the sandstone specimen in laboratory tests. In 2006, micro-tremor was measured atlibraries in Bayon temple and the similar reduced equivalent modulus was also obtained for these frame-likesandstone-dry-masonry structures. This extremely low elastic modulus implies the inadequacy of continuummodeling for bearing capacity of dry-masonry structures. Then the Discontinuous Deformation Analysis wasapplied to safety evaluation of the library against wind load. It was an illustrative example; however, the evaluatedsafety margin for the wind velocity of 40 m/s is about two.

1 INTRODUCTION

Bayon temple, Angkor, built in late 12th century toearly 13th century consists of sandstone-dry-masonrystructures. It has been deteriorated possibly by rain,plant intrusion, settlements, and so on. Strong windof more than 20 m/s was observed in the rainy season(JSA 1995), which may vibrate the structures eitherdirectly or via ground motion and eventually causedamage on them.

The temple consists of a main tower, sub-towers,and two libraries as shown in Figure 1. The maintower is more than 30 m high, located at the top ofthe man-made mound of about 15 m high. Sub-towersstand from several different height levels of themound, which are interconnected via the inner andthe outer corridors. The libraries are located betweenthese corridors.

Vibration characteristics of the main tower andsub-towers in Bayon temple have been studied bymicro-tremor measurements since 2003, which is apreliminary study for constructing analytical modelsto evaluate the effects of the wind excitation. Thoughthe towers are merely a pile of stones, they definitelyhave vibration modes with predominant frequenciesof 2 Hz to 6 Hz according to their heights, with damp-ing factors of 2 to 3%, similar to modern ordinary

Figure 1. Plan and section of the Bayon temple.

reinforced concrete structures (Sugiura et al. 2004,Maeda et al. 2005, Maeda et al. 2007).

Equivalent continuum models of FEM for the tow-ers were constructed by simulating experimentally

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obtained base-fixed natural frequencies of horizontalvibration modes with adjusted elastic constants. Thesimulated equivalent Young’s modulus ranged from1/27 to 1/9 of the laboratory test. This extremelylow modulus may be attributable to spring effectsof some material existed between the interfaces ofsandstone blocks, or to reduced contact areas at theinterfaces.

In 2006, we had a chance to measure micro-tremor at the libraries. While the main tower andsub-towers are basically stonewalls, the libraries arerelatively small frame-like structures. Comparing theequivalent elastic modulus of these different types ofstructures, insight on the underlying common phys-ical phenomenon of sandstone-dry-masonry may befound.

Our final objectives are the evaluation of bearingcapacity for these dry-masonry structures. Since theextremely low continuum-equivalent elastic modulusmay reveal incapableness of continuum mechanics insimulating rupture processes of dry-masonry struc-tures, we have applied the Discontinuous DeformationAnalysis (DDA) (Shi 1993) to the evaluation of thesafety margin of the library against the wind load asan illustrative example.

2 MICRO-TREMOR MEASUREMENTS ANDFEM SIMULATION

2.1 Vibration characteristics of the libraries

InAugust 2006, micro-tremor measurements were car-ried out at the northern library and the southern libraryof Bayon temple (Maeda et al. 2007). They are about4.1 m high with area of 6.6 m times 14.4 m, stand-ing on a sandstone-covered manmade mound of 4.9 mhigh, which is made of sand, gravel, and laterite, aporous soft rock. Their appearances are similar to eachother as shown in Figure 2. The northern library wasfully restored in 2000 (JSA2000), by being disman-tled and reassembled mostly with original materialssupplemented by some new ones, while the southernlibrary is now under similar restoration.

We have arranged accelerometers to capture hor-izontal translation and rocking motion of the super-structure as shown in Figure 3. Accelerometers arecomposed of over damped oscillators, with frequencyrange of 0.1 Hz to 100 Hz and the sensitivity of 1 Gal/V.Acceleration is low-pass filtered at 30 Hz and sampledby 0.01 s. with 20 bit AD converter.

Figure 4 shows horizontal transfer functions of rooflevel to floor level for the northern library in NSand EW directions. Clear peaks and smooth transitionof phases are observed. Peak frequencies of the firstmodes are summarized in Table 1. In NS direction, thepeaks of the transfer functions for the first mode reveal

Figure 2. Appearance of the northern library (upper panel)and the southern library (lower panel).

Figure 3. Sensor arrangements for the library.

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Frequency (Hz)

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Figure 4. Transfer functions of the northern library in NSand EW directions.

Table 1. The first mode frequencies of the libraries.

Northern library Southern libraryHz Hz

Soil-coupled NS 5.6 4.7EW 8.3 9.0

Base-fixed NS 5.8 4.8EW 9.4 9.2

non-uniform motion at the roof level. For the south-ern library, similar mode shapes are identified, and thefrequencies of its first modes in horizontal directionsare also summarized in Table 1.

2.2 FEM modeling of libraries

The FEM model for the superstructure of the north-ern library was constructed as shown in Figure 5. Ithas 4,891 elements, 7,968 nodes, and 206 ton of totalmass. Continuum-equivalentYoung’s modulus is eval-uated by simulating base-fixed frequencies of the firsthorizontal modes summarized in Table 1. The Young’smodulus is evaluated as 850 N/mm2 for both of NS andEW directions, which is about 1/15 of 13,000 N/mm2,a value obtained by the laboratory test.

Figure 6 shows first horizontal translation modesobtained by the finite element model for the north-ern library in NS and EW directions. The NS mode

Figure 5. FEM model for the superstructure of the northernlibrary.

Figure 6. First horizontal translation mode for the northernlibrary in NS and EW directions.

has larger amplitude along the north and the southedges compared to the east and the west edges, whichis in accordance with the variation of peaks in transferfunctions shown in Figure 4.

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Figure 7. First horizontal translation mode for the southernlibrary in NS and EW directions.

Table 2. The first mode base-fixed frequencies of thelibraries by measurements and analyses.

Northern library Southern libraryHz Hz

Measurement NS 5.8 4.8EW 9.4 9.2

Analysis NS 5.8 4.7EW 10.3 8.2

The equivalent modulus for the southern library isevaluated as 480 N/mm2, which can explain first fre-quencies in NS and EW directions and is about halfof that of the northern library. The first horizontaltranslation modes obtained by FEM for the south-ern library are shown in Figure 7; the mode in NSdirection exhibits similar in-plane deformation to thatof the northern library and in accordance with theobservation. Comparison of the observed base-fixedfrequencies and analytical ones are summarized inTable 2.

2.3 Summary of equivalent elastic modulus of thestructures in the temple

We have two FEM models for the main tower, onewas made in 2004 with reference to literatures andphotos, the other was in 2006 based on the three dimen-sional topography survey (Ikeuchi et al. 2004). Eithermodel was based on micro-tremor measurements in

Table 3. Equivalent Young’s modulus of the structures.

Young’s modulus Fraction of theN/mm2 laboratory test

Main tower 2004 1,200 1/11Main tower 2006 1,500 1/9Sub tower 950 1/14Northern library 850 1/15Southern library 480 1/27Laboratory test 13,000 1/1

2004 for equivalent elastic modulus. One of sub-towerswas also modeled in 2004. They all exhibit extremelylow elastic moduli compared to the one in the labo-ratory test (Sugiura et al. 2004, Maeda et al. 2005).Continuum equivalent Young’s modulus ranges from480 N/mm2 to 1,500 N/mm2; fraction of the laboratoryvalue ranges 1/27 to 1/9 as summarized in Table 3.

These extremely low moduli are obtained regard-less of structure sizes and structure types and may beattributable to spring effects of some material exist-ing between the interfaces of sandstone blocks, or toreduced contact areas at the interfaces. Unless we canfind the reason for that, we cannot use the continuummechanics for evaluation of bearing capacity of thestructures in Bayon temple.

3 AN EXAMPLE OF SAFETY EVALUATIONOF THE LIBRARY

3.1 DDA analysis

Most of studies dealing with the bearing capacity ofmasonry structures are based on stresses computedby continuum mechanics. This scheme may be takengranted for a bonded masonry, since it implies that gen-eration of a crack in a block or at an interface limitsthe bearing capacity. On the other hand, dry-masonryhas its initial cracked state, so that the application ofthis scheme is questionable.

We have been studying the applicability of the Dis-continuous Deformation Analysis (DDA) (Shi 1993)to dry-masonry structures. DDA is formulated asincremental dynamic equilibrium for blocks based onthe theory of the least total potential energy with itera-tion scheme for preventing intrusion and tensile stressat interfaces of blocks. DDA can guarantee existenceand uniqueness of the solution and can be practical onusual PC platforms.

We are interested in the response of masonry struc-tures to wind load exerted horizontally on the struc-tures. Equilibrium under the gravitational load wasfirst achieved, and then horizontal load was appliedgradually to let sliding occur between blocks. Figure 8

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Figure 8. Response of series of blocks to horizontal loadssimulated by DDA.

shows the responses of a series of square blocks linedin vertical, which is subjected to horizontal load eitherat upper portion (upper panel) or at all of the blocks(lower panel). The different collapse modes can besimulated by DDA, i.e. sliding at the interface rightbelow the loaded portion for partial loading and upliftof whole blocks for whole loading.

3.2 Response of the northern library tohorizontal load by DDA

The north-south section of the northern library includ-ing columns and walls is simplified by plane strainmodel with consideration on symmetry shown inFigures 9–10. Friction angle is set to 30 degrees, massdensity 2.52 g/cm3, Poisson’s ratio 0.11; no cohesionand no dynamic friction are considered. Uniformlydistributed horizontal load along the wall height isreplaced by nodal forces exerted gradually at the rate of7,937 N/m2/s after the settlement under gravitationalload. Figure 11 shows development of deformation andvariation of principal stress with increase of horizon-tal load. The principal axes for the initial gravitationalload in vertical (a) are firstly inclined by combina-tion of axial force and shear force due to friction atinterfaces (b), and back to vertical again due to theoccurrence of slide (c). The first sliding occurs at20,117 N/m2 of uniform load, which can be viewedas a bearing capacity of the model.

Figure 9. Modeled portion of the northern library by DDA.

Figure 10. DDA model for the northern library.

3.3 Bearing capacity of the library to the wind load

The wind load density can be evaluated by (1).

where D: horizontal force [N], S: area [m2], C: resis-tance factor, ρ: density [kg/m3], V : wind velocity[m/s]. For wind velocity of 40 m/s, (1) gives about10,000 N/m2, which is less than a half of the evaluatedbearing capacity of the first slide at 20,117 N/m2.Thenthe safety margin for the wind load corresponding towind velocity of 40 m/s is more than twice.

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Figure 11. Deformation and principal stress in the northernlibrary DDA model for horizontal load.

4 DISCUSSION

The continuum-equivalent Young’s modulus forsandstone-dry-masonry structures in Bayon templeranges from 1/27 to 1/9 of the laboratory value

regardless of sizes and types of structures, based onmicro-tremor measurements. Since the extremely lowmodulus is not yet clearly explained, we cannot relyon a continuum model to evaluate bearing capacity ofdry-masonry structures. DDA was selected to replaceFEM and tested for fundamental features by simpleproblems. Finally DDA was applied to the northernlibrary with plane strain model. The bearing capacityfor the first slide occurrence is evaluated as about twicelarger than the wind load corresponding to 40 m/s.

ACKNOWLEDGEMENT

This paper is based on the 2006 Bachelor’s thesisof Takeya Yamamoto and Yusuke Wako and 2006Master’s thesis of Tomonari Hirai at the Departmentof Architecture, Waseda University. We would like toexpress our gratitude to the Japan government team forsafeguarding Angkor, lead by Prof. Takeshi Nakagawaof Waseda University to provide us a chance to studyon Bayon temple. We are indebted to Prof. NaoyukiKoshiishi of Waseda University for laboratory test dataof sandstone.

REFERENCES

Ikeuchi, K., Hasegawa, K., Nakazawa A., Takamatsu, J.,Oishi, T. & Masuda, T., Bayon Digital Archival Project,10th International Conference on Virtual Systems andMultimedia (VSMM2004), November 2004.

JSA (Japanese Government Team for Safeguarding Angkor),1995. T. Nakagawa (ed.), Annual Report on the TechnicalSurvey of Angkor Monument 1995.

JSA, 2000. T. Nakagawa (ed.), Report on the Conservationand Restoration Work of the Northern Library of Bayon,Angkor Thom, Kingdom of Cambodia, 2000.

Maeda,T., Sugiura,Y. & Hirai,T. 2005.Vibration characteris-tics of the Bayon temple main tower, Angkor, Cambodia,In C. A. Brebbia & A. Torpiano (eds), Structural Stud-ies, Repairs and Maintenance of Heritage Architecture 9,Malta 2005: 255–264, Southarnpton: WIT PRESS.

Maeda, T., Yamamoto, T., Wako, Y. & Hirai, T. 2007. Vibra-tion characteristics and equivalentYoung’s modulus of theNorthern Library and the main tower, Bayon, Cambodia,In C. A. Brebbia (ed.), Structural Studies, Repairs andMaintenance of Heritage Architecture 10, Prague 2007:493–502, Southarnpton: WIT PRESS.

Sugiura, Y., Fukumoto, Y. & Maeda, T. 2004. Vibration char-acteristics of the main tower, the Bayon temple, 21st Inter-national congress of theoretical and applied mechanics,2004.

Shi, G-H. 1993. Block system modeling by discontinuousdeformation analysis, In C. A. Brebbia & J. J. Connor(eds), Topics in Engineering Volume 11, Southampton:Computational Mechanics Publications.

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