STATIC AND DYNAMIC ANALYSIS OF ANCIENT IRANIAN STRUCTURAL SHELLS, ACCORDING TO CHANGING GEOMETRIC FACTORS

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1 STATIC AND DYNAMIC ANALYSIS OF ANCIENT IRANIAN STRUCTURAL SHELLS, ACCORDING TO CHANGING GEOMETRIC FACTORS Mohammad Shahriyari 1*, Ladan Shahzamani Sichani 1 and Amir Hossein Pouriaye Vali 2 1 Department of Architecture, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran 2 Department of Architecture, Natanz Branch, Islamic Azad University, Natanz, Iran *(corresponding Author: Mohammad Shahriyari 1 ) ABSTRACT Researchers have so far investigated the aesthetics or analysis of Iranian old architectural forms as a general form, but this paper shall consider the effects of geometric factors on the strength of old Iranianian structural shells, for the first time. By the use of limited element method and structural shell modeling in accordance with 3 geometric factors of the arc tools for the openings with greater depth (chafd) type, opening size and geometric sections, we are going to deal in this article with linear statistical analysis underweight loads and modal dynamic analysis under elcentro seismic loads. The conditions that we consider for studying the shells in this article are first according to to unilateral shell structure basis and secondly, we shall consider the materials of the shells to be of brick. Thus, we will consider our hypothesis on the basis: It seems that geometric factors have direct effects on the created static and dynamic stresses in the old structural shells in Iran. Regarding tensile strength, the best types of arch tools (chafd) are cylindrical (bastu), open 3-section arch (holochin), sharp arch (ozhiv), acute holochin, acute double fold (chamaneh), five-o-seven in orchidic form, sharp (goat horn), open 3-section arch, sarvak, acute horn-type, acute 3- sectional, acute clover-form and open clover-form, respectively. Regarding compressive strength, the best of the above arches include open holochin, chamaneh, ozhiv, acute holochin, open 3-sectional, bastu, five-o-seven, acute clover-form, open clover-form, acute 3-sectional, open horn-type, sarvak and acute horn-type. But, regarding shear strength, the best ones are open holochin, bastu, chamaneh, five-o-seven, open horn-type, acute horn-type, acute 3- sectional, acute holochin, ozhiv, open 3-sectional, acute clover-form, open clover-form and sarvak type, respectively. Regarding geometric factor, the opening size of the arches is 8m in the best status, and regarding the shell thickness, parabolic geometry is the best alternative. KEYWORDS: Ancient architecture in Iran, dynamic analysis, scientific and theoretical geometry, static analysis, unishell structures. INTRODUCTION AND CASE DESCRIPTION Practical usage of geometry in ancient Iranian architecture was quite common and played the optimizing role in engineering purposes to the extent that the formation and decorative requirements were provided and also the structural needs were considered. In the valuable book Al-Tafhim (Concept), Abu-Reihan Birouni defined geometry as the knowledge of dimensions and their comparisons, the knowledge of forms and shapes in a body and the science of numbers in general and in parts and estimates are becoming actual values by that. The Iranian architect used geometry, both practically and theoretically. For instance, geometry was used for different aspects for constructing domes, as a shell structure. An aspect was using chafd types (arch tools) for the physical form of the domes and transforming the created forces in that to compressive strength. Regarding the materials and the arch types, the dimensions of the dome opening considers the practical application of geometrical science. The section geometry or the shell thickness (known in Farsi as Tabareh ) is one of the most important geometric factors that the architect for structural shells considered for his design. As stated earlier, the static and dynamic analyses of the 3 above factors is the main aim of this article for investigating and identification of the geometrical effects on the strength of Iranian shell structures (domes). Using shell structures today has been considered important both due to their forms and also because of using appropriate compressed structures and construction of the shells in some constructions create proper architectural spaces without the presence of structural elements, such as columns and also 1 Architize@gmail.com. Tel: Volume-3 (Special Issue 3) DAMA International. All rights reserved. 11

2 as specific climatic characteristics, esprcially in warm regions. Since suitable geometry is emphasized in this article, we try to respond to 3 main questions as the research questions, as follows: 1-Which type of arch is suitable for the created loads in the shell and transforming them to compressive strength? 2-What effects do increasing the opening or reducing those have on static and dynamic stresses of old Iranian shell structures? 3-How does the change in thickness affect in stability of Iranian old shell structures? MATERIALS AND METHODS Research Methodology and Expressing The Hypothesis The main hypothesis of the article is as follows: It seems that geometric factors are effective in static and dynamic stresses created in Iranian old shell structures. Thus, the subordinate hypothesis could be achieved by considering the main one: There is a direct relation between Iranian ols unishells and their strength. Hence, by using documentary evidences and field studies, the geometric factors used in Iranian ancient architecture are considered, and then by using the descriptive research method, the contents of the data are analyzed. The method of analyzing the collected data is a quantitative analytical method. The methodology is to model the assumed shell structures as unishell structures by Auto CAD (V. 2013) software, and transferred to Sketch UP (V. 8) software through the file extension DXF, to be transformed to unishell dome-shaped surfaces. It is then transferred again by the extension IGES from Auto CAD to SAP2000 (V. 15). Regarding the very time consuming modeling of the shells by the parametric method in SAP2000 environment, the expressed methods causes to save time in the modeling process and more accuracy is used in that method as compared to the parametric method. It is to note that this modeling process is introduced for the first time by this article that provides modeling approaches for such structures in the software packages from CSI Co. Analyses will be done in SAP2000 by limited element method that is one of the strongest ways for analyzing structures. It seems that in case the resulted stresses from the analyses are relatively small as compared to the material strength and considering the geometric factors, a limited linear elastic analysis could be acceptable for the primary evaluation of the geometric behavior of such structures. The elements used in the modeling are solid elements and shell elements. (Fig. 2) The following hypotheses are considered in analyzing the dome-shaped shells: 1-Masonry materials have linear elastic behavior 2-Materials are isotropic 3-Replacement of all the supports is equal to zero 4-The material properties for dome-shaped shell structures are by brick and have the following characteristics: (table 1.) Table 1. The considered material properties for the analyses (Source: Hejazi) Modulus of elasticity E 7358 Poisson s ratio 0.1 Bulk density Allowable compressive stress 0.7 Allowable tensile stress 0.2 Allowable shear stress 0.1 ANALYZING GEOMETRIC FACTORS FOR IRANIAN ANCIENT SHELL STRUCTURES In old ages, when architecture was not yet based on academic knowledge, the building stability, decorations, dimensions and the appropriation between the building forms and beauty were based on the geometry that was tranferred from different generations. That geometry used simple and conceptual language and was quite accurate, with mathematical basis. (Mehdizadeh Seraj, Tehrani, Valibeig, 2011). Materials like ordinary brick and sun-dried brick are resistant against compressive strength, but are very weak against tensile strength. Thus, the first step for structural stability of mud-brick and ordinary-brick buildings is using the frames in which the force distribution is mainly Volume-3 (Special Issue 3) DAMA International. All rights reserved. 12

3 compressive and hence, arches are used to cover such constructions and their stability depends on the applied geometry in the arches and the performing techniques (Mehrgan, Soleimanpour, 2013). Due to frequent use of geometrical knowledge in Iranian architecture, only 3 geometric factors are indicated and considered in this paper, which are applied in different shell structures. Primary Factor: Chafd (arch) The first geometric factor to be considered is chafd (arch). In case we want to have a definition similar to the past from this structural factor, we should consider the definition by the Muslim Iranian scientist, Qiasellid Jamshid Kashani, that according to him, an arch is a curved object for covering and its opening is greater than its depth. (Memariam, 1988) The main difference in a bend (Arabic: arc) with chafd is that arc is a part of an identified mathematical bend that is drawn with one or more mean points, but chafd in the geometric definition is a line or curve that is formed by 2, 4 or more arcs and architecturally, it is referred to the arch on the portal of a platform. (Fig. 1.). Arc balancing has a close relation with the base or the pillar and the torque imposed by the loads transferred to the arc shoulder place.the importance of arcs in architecture [Islamic Iranian] could be considered in two different aspects, first of which is the high rated application of this structure and its diversity in shapes that follow the static points in itself, as an important feature in architecture [Iranian]. The second aspect is about the building techniques of this structure that is advanced for its applications and time.the other building element that is similar to the arcs but different in static performance should not be mistaken with the real arcs. That type of arc, known as pseudo Volta, has been used in Mesopotamia, Egypt and other countries. (Memarian, 1988) Fig. 1. Types of chafd (arch) according to Asar 20 publication, stated by Pirnia (Drawn by: Authors) Since the studies for this paper are for showing the types of chafd, in the form of unishell structures, it is assumed that thin shells are the structures with stable frames with tensile and compressive and shear stresses imposed on their curved surface and they are not confronted with bending stresses due to being thin. The shell thickness is defined accirdind to bending disorders. But, since the shaells are thin, they are not affected by bending stress. Generally, these shells are called domes, in old architecture in Iran, and are identified as non-expandible surfaces, since they cannot be opened for a flat area, unless plenty of cuttings are done on them. Primary Factor: Opening size or Span length The size of opening, considered for each arch in constructing a dome-shape shell is an important geometric factor. Opening in Iranian traditional architecture is referred to the distance between two piers. Changing this geometric factor in construction of some openings used to be done through selecting the type of arch. In fact, the hypothesis to be expressed here is that selecting the arch type is related to the size of its opening. For example, Pirnia stated in Asar magazine that Sheikh Lotfollah dome in Esfahan is chamaneh-double folded, due to its base area (Pirnia, 1991). Volume-3 (Special Issue 3) DAMA International. All rights reserved. 13

4 Fig. 2. Geometric parts of an arch (A-B is the opening size) (Memarian, 1988) Studying of opening size in this article is according to chamaneh arch in opening sizes of 4m, 6m, 8m and finally 25m. (Fig. 3.) Fig. 3. Static analysis under load of the weight of chamaneh arch with different opening sizes Third Factor: Tabareh (Shell thickness) Iranian domes are made by two shells, the inner and the outer shells. The inner covering of the domes are facing the inner space and the outer parts are usually serrated (crinkled) that is constructed in step forms due to construction matters. (Pirnia, 1991). Thickness of chafds, arches or domes is referred to as tabareh that is different for different surfaces. The thickness considered for the dome pier is 1/16 of the opening. For example, if the opening id 16 gaz (a Persian unit), the thickness of the pier shall be 1gaz. The example is the Soltanieh dome with the opening of over 24 gaz and the thickness of 1.4 gaz, i.e. exactly 1/16 (Pirnia, 1991). Thus, it can be concluded that the opening size has direct mathematical relations with the type of arch and thickness has direct relations with the opening size. Using active frame geometry for the domes is a geometric principle, helping the stability of that. The active frame structure is one in which the tensile or compressive force is in conformity with the geometry of arcs and therefore no bending will occur in the structure (J.McDonalkd, 2004). A compressive active frame structure has parabolic shape and an active frame dome is elliptical (Foler Moor, 2011). Considering different domes in Iranian architecture in the past, using elliptical geometry is clearly evident and hence, the compressive stresses are transferred to the piers according to the geometry of the dome, to have high gravitational stability. ANALYTICAL STUDIES BASED ON ARCH TYPES Generally, the arches are drawn by two different methods: Circular method, by the use of compasses and elliptical method that is only applied for special types of chafds. It is presumed that drawing elliptical arches could have more loads; such as five-o-seven type. Also, a number of experts classify arches according to their gradient into 4 categories: sharp, open, acute and almost closed. Moreover, according to the experiences of Iranian architects and regarding the loads, the arches are divided into the 3 following groups: Volume-3 (Special Issue 3) DAMA International. All rights reserved. 14

5 Bearing chafds: The arches that have great role in transferring main loads of the building are the examples for this category, such as clover-form, sarvak, chamaneh, 3&4 expression, sharp arch or bastu, sharp holochin, acute arch or acute holochin, five-o-seven (elliptical) and overlap (pa-to-pa) arches. Bearing and decorative arches: Clover-form, five-o-seven (sharp, acute, almost closed), and overlap (pa-to-pa) Non-bearing arches (Amodi): These arches have no main roles in transfer of main building loads and are mainly decorative, such as Kalil (Azari, Partian, Kameshi), horn-type (sharp, acute), Paniz (keel), 3-sectional (sharp and acute) and ozhiv arches. 13 types of the above arches that are mainly used for bearing loads are described in this paper. A) STATIC ANALYSIS (Table 2.) Table 2. Table of stresses and displacements Shear(N/mm) Compress(N/mm) Tensile(N/mm) Displacement(mm) Analysis weight load acute 3-section Mode weight load open 3-section Mode weight load Five-o-seven Mode weight load bastu Mode weight load chamaneh Mode weight load Acute holochin Mode weight load open holochin Mode weight load ozhiv Mode weight load sarvak Mode weight load acute cloverform Mode weight load Open cloverform Mode weight load acute horn-type Mode weight load open horn-type Mode 1 DATA ANALYSIS: Max. Tensile stress in 3-sectional arches is 1.517Nmm, with regards to the blue part in the graph. This rate is more than the permissible tensile stress of 0.2N/mm of the materials and the dome may encounter cracks or breakage in those areas. Max. Permissible compressive stress for the domes with bricks is 0.7N/mm. Max. Compressive stress in the Volume-3 (Special Issue 3) DAMA International. All rights reserved. 15

6 debris part of domes is 0.606N/mm. it is less than the permissible range and hence the domes are resistant against compressive stresses. S12 stress is a shear stress and the max. shear stress in brick materials should not exceed 0.1N/mm. the rate of created shear stress in the domes is Thus, these domes will not have 45 cracks. In sharp 3-sectional arches, the max. rate of tensile stress will be 1.319N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress at debris part of the dome is 0.396N/mm that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. the created shear stress is 0.11N/mm and hence, there would be the possibility of cracks. In sharp five-o-seven arches with elliptical shape the max. rate of tensile stress will be 1.005N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress at debris part of the dome is N/mm that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. the created shear stress is 0.068N/mm and hence, the dome would not face 45 cracks. In bastu arches the max. rate of tensile stress will be 0.609N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is N/mm at the sharp end that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. the created shear stress is 0.055N/mm and hence, the dome would not face 45 cracks. In chamaneh arches the max. rate of tensile stress will be 0.857N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress at debris part of the dome is N/mm at the sharp end that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.063N/mm and hence, the dome would not face 45 cracks. Max. tensile stress in acute holochin arches is 0.782N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is N/mm at the sharp end that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.098N/mm and hence, the dome would not face 45 cracks. Max. tensile stress in sharp holochin arches is 0.616N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is N/mm at the sharp end that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.050N/mm and hence, the dome would not face 45 cracks. Max. tensile stress in ozhiv arches is 0.746N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.394N/mm at the sharp end and the debris part that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.105N/mm and hence, the dome may face 45 shear cracks. Max. tensile stress in sarvak arches is 1.465N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. Volume-3 (Special Issue 3) DAMA International. All rights reserved. 16

7 Permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.617N/mm at the sharp end and the debris part that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.131N/mm and hence, the dome may face 45 shear cracks. Max. tensile stress in acute clover-type arches is 1.65N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.415N/mm at the debris part that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.118N/mm and hence, the dome may face 45 shear cracks. Max. tensile stress in sharp clover-type arches is 1.682N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.424N/mm at the sharp end and the debris part that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.12N/mm and hence, the dome may face 45 shear cracks. Max. tensile stress in acute horn-type arches is 1.504N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.730N/mm at the debris part (place of debris) that is less than the permissible range and thus, we may face cracks due to breaking or pressing. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.088N/mm and hence, the dome would not face 45 cracks. Max. tensile stress in sharp horn-type arches is 1.275N/mm according to the blue region in the pier. This rate is more than the permissible tensile stress of 0.2N/mm and it is possible for the dome to have cracks or breakage in that area. Max. permissible compressive stress for these domes with brick material is 0.7N/mm. Max. compressive stress of the dome is 0.611N/mm at the debris part that is less than the permissible range and thus, the dome is resistant against compressive stresses. S12 is the shear stress that should not exceed 0.1Nm. The created shear stress is 0.077N/mm and hence, the dome would not face 45 cracks. B) TIME-HISTORY ANALYSIS The seismic analysis is done according to El-Centro earthquake, with 300 acceleration-time pairs, and the stresses and displacements are analyzed for different periods. The graphical analysis was done for El-Centro earthquake time that is about 5 seconds. (Table 3.) Table 3: History of displacements in a dome and max. and min. stresses creted in the most critical times Step Time(Sec) Dmax Smin Smax S E-6-2 E-9 11 E E E-6-24 E E-9 35 E E-6-12 E-9 66 E-9 18 E E-6-60 E E-9 84 E E E E E-9 Acute E E E E-9 sectional E E E E E-6-75 E E-9 70 E E-6-66 E E-9 78 E E-6-50 E E-9 77 E E-6-56 E E-9 77 E-9 Volume-3 (Special Issue 3) DAMA International. All rights reserved. 17

8 E-9 55 E E E E E E E E E E E E E E E E E-9 Open E E E E-6 sectional E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-5 Five -o E E E E-5 seven E E E E E E E E E E E E E E E E E E E E E bastu E-6-32 E E E E E E E E E E E E E E E E E E E E E-6 14 E E-6 chamaneh E E E E E E E E E E E E E E E E E E E E E E E Acute holochin Volume-3 (Special Issue 3) DAMA International. All rights reserved. 18

9 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-6-60 E E-9 51 E E E E E E E E E E E E E E E E E E E E-5 7 E E E E E E E E E E E E E E E E E E E E E E E E-9 60 E E E E E-9 Open holochin ozhive sarvak Acute cloverform Volume-3 (Special Issue 3) DAMA International. All rights reserved. 19 Open

10 E E E E-9 cloverform E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-6-70 E E E E-6-48 E E E E E E E E E E E-9 Acute E E E E-9 horn-type E-6-80 E E E E-6-70 E E E E E E E E E E E E E E E E E E E E E-9 85 E E E-6-15 E-9 50 E-9 12 E E-6-42 E E-9 56 E E E E E E E E E E-6-21 E-9 70 E E E-6-33 E E E E-6-60 E E E E-6-50 E E-9 91 E E-6-66 E E-9 84 E-9 DATA ANALYSES Open horn-type Most rate of displacement due to earthquake conditions is observed in 2.5sec. for 3-sectional arches. Smin stress is a max. pressure. The compressive stress of -225E-9 is according to the table and the min. rate of compressive stress is - 2E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 850E-9 and the min. rate of tensile stress is 11E-9. S12 is a shear stress and max. shear stress is equal to 270E-9 and its min. rate is 2.8E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 0.5sec. for sharp 3-sectional arches. Smin stress is a max. pressure. The compressive stress of -135E-9 is according to the table and the min. rate of compressive stress is -0.48E-6. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 6.163E-6 and the min. rate of tensile stress is 55E-9. S12 is a shear stress and max. shear stress is equal to 2.105E-6 and its min. rate is 2.8E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 3sec. for sharp five-o-seven arches with elliptical shape. Smin stress is a max. pressure. The compressive stress of E-5 is according to the table and the min. rate of compressive stress is -225E-9. Thus, the rate of compressive stress in all the structure according to the table and in the Volume-3 (Special Issue 3) DAMA International. All rights reserved. 20

11 first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 7.846E-5 and the min. rate of tensile stress is 750E-9. S12 is a shear stress and max. shear stress is equal to 2.318E-5 and its min. rate is 280E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 3.5sec. for bastu arches. Smin stress is a max. pressure. The compressive stress of is according to the table and the min. rate of compressive stress is E- 4. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is and the min. rate of tensile stress is S12 is a shear stress and max. shear stress is equal to and its min. rate is that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 5sec. for chamaneh arches. Smin stress is a max. pressure. The compressive stress of E-6 is according to the table and the min. rate of compressive stress is -2E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 2.591E-5 and the min. rate of tensile stress is -32E-9. S12 is a shear stress and max. shear stress is equal to 6.389E-6 and its min. rate is 38.5E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 3sec. for acute holochin arches. Smin stress is a max. pressure. The compressive stress of is according to the table and the min. rate of compressive stress is E-4. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is and the min. rate of tensile stress is S12 is a shear stress and max. shear stress is equal to and its min. rate is that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2.5sec. for sharp holochin arches. Smin stress is a max. pressure. The compressive stress of is according to the table and the min. rate of compressive stress is Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is and the min. rate of tensile stress is S12 is a shear stress and max. shear stress is equal to and its min. rate is that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 0.1sec. for ozhiv arches. Smin stress is a max. pressure. The compressive stress of 2.354E-6 is according to the table and the min. rate of compressive stress is 11E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 8.303E-6 and the min. rate of tensile stress is 60.5E-9. S12 is a shear stress and max. shear stress is equal to 2.106E-6 and its min. rate is 13.3E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2sec. for sarvak arches. Smin stress is a max. pressure. The compressive stress of 3.823E-4 is according to the table and the min. rate of compressive stress is Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is and the min. rate of tensile stress is S12 is a shear stress and max. shear stress is equal to and its min. rate is that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 4.5sec. for acute cover-type arches. Smin stress is a max. pressure. The compressive stress of 9.784E-4 is according to the table and the min. rate of compressive stress is 60E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 3.139E-5 and the min. rate of tensile stress is 200E-9. S12 is a shear stress and max. shear stress is equal to 9.1E-6 and its min. rate is 51E-9 that is less than the permissible range of 0.1N/mm. Volume-3 (Special Issue 3) DAMA International. All rights reserved. 21

12 Most rate of displacement due to earthquake conditions is observed in 3sec. for sharp clover-type arches. Smin stress is a max. pressure. The compressive stress of 15-E-9 is according to the table and the min. rate of compressive stress is 0.84E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 800E-9 and the min. rate of tensile stress is 2.8E-9. S12 is a shear stress and max. shear stress is equal to 660E-9 and its min. rate is 1.753E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2.5sec. for acute horn-type arches. Smin stress is a max. pressure. The compressive stress of -240E-9 is according to the table and the min. rate of compressive stress is 0.2E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 800E-9 and the min. rate of tensile stress is 1.056E-9. S12 is a shear stress and max. shear stress is equal to 840E-9 and its min. rate is 7.8E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2.5sec. for sharpe horn-type arches. Smin stress is a max. pressure. The compressive stress of -225E-9 is according to the table and the min. rate of compressive stress is 1.95E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 850E-9 and the min. rate of tensile stress is 6.5E-9. S12 is a shear stress and max. shear stress is equal to 245E-9 and its min. rate is 2.45E-9 that is less than the permissible range of 0.1N/mm. ANALYTICAL STUDIES BASED ON THE OPENING SIZE (CASE STUDY: CHAMANEH) Three openings that are mostly used are considered for the studies in this section and most of the dome shells are constructed in Iran with the required dimensions as 4m, 6m and 8m openings. But, since all the samples in this section are modeled according to Soltanieh dome and tested, the 25m opening is also considered as the max. opening rate, that is presumably similar to the dome opening part of Soltanieh dome. A) STATIC ANALYSIS (Table 4.) Table 4: Table of stresses and displacements Shear(N/mm) compressive(n/mm) Tensil(N/mm) Displacement(mm) Weight load 4m Mode Weight load 6m Mode Weight load 8m Mode Weight load 25m 5.074E E Mode 1 DATA ANALYSIS Max. tensile stress with regards to the blue section of the graph for the 4m opening is 0.857N/mm. It is more than the permissible range of 0.2N/mm and it is possible for the dome to have cracks in that area. Max. compressive stress for 4m opening of brick domes is 0.7N/mm. Max. compressive stress is N/mm at debris part that is less than the persmissible range and hence the dome is resistive against compressive stresses. S12 for 4m opening is the shear stress that its max. should not exceed 0.1N/mm. the created shearing stress for the dome is and hence this type of dome would have no 45 cracks. Volume-3 (Special Issue 3) DAMA International. All rights reserved. 22

13 B) TIME-HISTORY ANALYSIS (Table 5.) ISSN: (p); (e) Table 5: History of displacements in a dome and max. and min. stresses creted in the most critical times Step Time(Sec) Dmax Smin Smax S12 HISTORY ANALYSIS E-6-32 E E E-9 4m E E E E E E E E E E E E E E E E E E-6 14 E E E E E E E E E E E E E E E E E E E E E E E E-9 7 E E-9 6m E-6-24 E-9 80 E E E E-9 65 E-9 21 E E-6-84 E E-9 84 E E E E E E E E E E-6-84 E E-9 84 E E-6-90 E E-9 91 E E E E E E-6-39 E E-9 42 E E-6-70 E E E E E E E-9 8m E-6-70 E E-9 84 E E-6-28 E E E E E E E E E E E E E E E E E E-9 91 E E-6-90 E E-9 98 E E E E E E E E E E E E E E-9 24 E E-9 25m E-6-80 E E E E-6-44 E E-9 63 E E-6-80 E E E E E E E E E E E E-6-60 E E-9 66 E E E E E E E E E E E E E E E E E-9 Volume-3 (Special Issue 3) DAMA International. All rights reserved. 23

14 Max. tensile stress with regards to the blue section of the graph for the 6m opening is 1.287N/mm. It is more than the permissible range of 0.2N/mm and it is possible for the dome to have cracks in that area. Max. compressive stress for 6m opening of brick domes is 0.7N/mm. Max. compressive stress is 0.548N/mm at debris part that is less than the permissible range and hence the dome is resistive against compressive stresses. S12 for 4m opening is the shear stress that its max. should not exceed 0.1N/mm. the created shearing stress for the dome is and hence this type of dome would have no 45 cracks. Max. tensile stress with regards to the blue section of the graph for the 8m opening is 1.716N/mm. It is more than the permissible range of 0.2N/mm and it is possible for the dome to have cracks in that area. Max. compressive stress for 8m opening of brick domes is 0.7N/mm. Max. compressive stress is 0.730N/mm at debris part that is less than the permissible range and hence the dome would face compressions and breaking due to to the imposed loads. S12 for 4m opening is the shear stress that its max. should not exceed 0.1N/mm. the created shearing stress for the dome is and hence this type of dome would have 45 cracks. Max. tensile stress with regards to the blue section of the graph for the 25mopening is 5.052N/mm. It is more than the permissible range of 0.2N/mm and it is possible for the dome to have cracks in that area. Max. compressive stress for 25m opening of brick domes is 0.7N/mm. Max. compressive stress is N/mm at debris part that is less than the permissible range and hence the dome would face compressions and breaking due to to the imposed loads. S12 for 4m opening is the shear stress that its max. should not exceed 0.1N/mm. the created shearing stress for the dome is and hence this type of dome would have 45 cracks. DATA ANALYSES Most rate of displacement due to earthquake conditions is observed in 5sec. for 4m openings. Smin stress is a max. pressure. The compressive stress of E-6 is according to the table and the min. rate of compressive stress is -32E- 9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 2.951E-5 and the min. rate of tensile stress is 176E-9. S12 is a shear stress and max. shear stress is equal to 6.389E-6 and its min. rate is 38.5E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2.5sec. for 6m openings. Smin stress is a max. pressure. The compressive stress of -255E-9 is according to the table and the min. rate of compressive stress is -2.1E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 850E-9 and the min. rate of tensile stress is 7E-9. S12 is a shear stress and max. shear stress is equal to 280E-9 and its min. rate is 1.5E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 2.5sec. for 8m openings. Smin stress is a max. pressure. The compressive stress of o.48e-6 is according to the table and the min. rate of compressive stress is -4.8E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 2.438E-6 and the min. rate of tensile stress is 26.4E-9. S12 is a shear stress and max. shear stress is equal to 630E-9 and its min. rate is 6.3E-9 that is less than the permissible range of 0.1N/mm. Most rate of displacement due to earthquake conditions is observed in 0.1sec. for 25m openings. Smin stress is a max. pressure. The compressive stress of -0.72E-6 is according to the table and the min. rate of compressive stress is -7.2E-9. Thus, the rate of compressive stress in all the structure according to the table and in the first 5sec. is less than the permissible compressive stress. The max. tensile strength for tensile Smin is 3.329E-6 and the min. rate of tensile stress is S12 is a shear stress and max. shear stress is equal to 720E-9 and its min. rate is 11.4E-9 that is less than the permissible range of 0.1N/mm. Volume-3 (Special Issue 3) DAMA International. All rights reserved. 24

15 ANALYTICAL STUDIES BASED ON THE SHELL THICKNESS (NABAREH) (CASE STUDYOF THE ARCH: SHARP KEEL) (Fig. 4) A) STATIC ANALYSIS (Table 6.) Table 6:Table of stresses and displacements Fig. 4: Shell thickness Shear(N/mm) Compressive(N/mm) Tensil(N/mm) Displacement(mm) Weight load Parabulic E Mode 1 section Weight load Pitch E Mode 1 section DATA ANALYSIS Max. tensile stress for parabolic geometry is equal to 0.223N/mm. This rate is more than the permissible range of 0.2N/mm and it is possible that the dome encounters cracks and breaks in that area. Max. compressive stress for brick domes is 0.7N/mm. Max. compressive stress in the dome is 0.249N/mm. It is lower than the permissible range and hence the dome would be resistive against compressive stresses. S12 is shear stress that should not exceed 0.iN/mm in brick materials. The created shear in the dome is and thus, 45 cracks appear in the dome. Max. tensile stress for pitch geometry is equal to 1.461N/mm. This rate is more than the permissible range of 0.2N/mm and it is possible that the dome encounters cracks and breaks in that area. Max. compressive stress for brick domes is 0.7N/mm. Max. compressive stress in the dome is 0.365N/mm at its sharp edge. It is lower than the permissible range and hence the dome would be resistive against compressive stresses. S12 is shear stress that should not exceed 0.1N/mm in brick materials. The created shear in the dome is and thus, 45 cracks appear in the dome. B) ANALYSIS OF FREE VIBRATION (MODAL) AND DETERMINATION OF THE DOMES VIBRATING MODES (Table 7.) (Fig.5) Table 7: 12 Primary modes of the structures under free vibration StepNum Period(Sec) Frequency(Cyc/sec) CircFreq(rad/sec) Eigenvalue(rad2/sec2) Parabulic section Volume-3 (Special Issue 3) DAMA International. All rights reserved. 25

16 Pitch section Fig. 5. Two types of considered thicknesses in parabolic form and step (pitch) form (modeling by opened arch) C) TIME-HISTORY ANALYSIS (Table 8.) Table 8: History of displacements in a dome, and max. and min. stresses creted in the most critical times Step Time(Sec) Smin Smax S E E-4 Parabulic section E Pitch Volume-3 (Special Issue 3) DAMA International. All rights reserved. 26