Engineering Structures 32 (2010) 1466–1478 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Numerical models for the seismic assessment of an old masonry tower Fernando Peña a,∗ , Paulo B. Lourenço b , Nuno Mendes b , Daniel V. Oliveira b a Instituto de Ingeniería, Universidad Nacional Autónoma de México, Edificio 2 – 401, Circuito Escolar, Ciudad Universitaria, 04510 Mexico city, Mexico b University of Minho, Department of Civil Engineering, ISISE, Campus Azurem, 4800-058 Guimarães, Portugal article info Article history: Received 21 August 2008 Received in revised form 20 April 2009 Accepted 25 January 2010 Available online 19 February 2010 Keywords: Dynamics Non-linear analysis Pushover Tower Rigid elements abstract The present paper describes the seismic assessment of the Qutb Minar in Delhi, India. Three models with different levels of complexity and simplification were developed. The use of these models allows one to overcome the complexity of the study of the seismic behavior of ancient masonry structures; by combining the results of the different models it is possible to obtain a better and more comprehensive interpretation of the seismic behavior. The models were used for non-linear static (pushover) and nonlinear dynamic analyses. The static and dynamic analyses give different behaviors, indicating that the push-over analysis should be used carefully in the seismic assessment of masonry structures. For the static analysis, the base of the tower is the most vulnerable part; while according to the dynamic analysis, it is the upper part of the tower. This last behavior is according to the historical damage suffered by the tower due to earthquakes. The different behaviors can be explained by the influence of the higher modes of vibration. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Qutb Minar is the highest monument in India and one of the tallest stone masonry towers in the world. It is considered one of the most important monuments of Delhi. The minaret was built as a part of the Quwwatul-Islam Mosque to call the prayers and as a sign to glorify the victory of Islam against idolatry. Its construction started around 1200 and was finished by 1368 . The seismic history of the city is a testimony of the risk that historical constructions are subjected to. Delhi suffers near and far seismic activity. According to the Indian Seismic code , Delhi is in the Indian seismic zone IV, which is considered to have a severe seismic intensity. In general, earthquakes with magnitude of 5 to 6 are common, while a few earthquakes of magnitude 6 to 7 and occasionally of 7.5 to 8 have been also registered . Thus, restoration and conservation works have been carried out in the Qutb Minar since its construction. In the framework of the EU-India Economic Cross Cultural Programme ‘‘Improving the Seismic Resistance of Cultural Heritage Buildings’’, aimed at the preservation of ancient masonry structures with regard to the seismic risk, the seismic assessment of the Qutb Minar was carried out. The evaluation of the seismic behavior of ancient masonry structures requires specific procedures, since their response to ∗ Corresponding author. Tel.: +52 55 56233600x8404. E-mail addresses: [email protected] (F. Peña), [email protected].pt (P.B. Lourenço), [email protected] (N. Mendes), [email protected] (D.V. Oliveira). 0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.01.027 dynamical loads often differs substantially from those of ordinary buildings. In order to obtain a reliable estimate of the seismic risk, it is desirable to perform full dynamical analyses that describe the effective transmission and dissipation of the energy coming from the ground motion into the structure . However, sometimes the available analytical tools require a great amount of computational resources that are not commonly available. Therefore, procedures are necessary to overcome the complexity of the study of the seismic behavior of ancient masonry structures. In general, modeling the non-linear mechanical behavior of ancient masonry structures by means of three-dimensional models is not possible because it requires a great amount of computational resources. However, the use of simplified models in combination with very refined models allows one to overcome those restrictions. The results obtained from the complex models can be used as the basis for a better conception of the simplified models, which can resort to different analysis tools. Moreover, by combining the results of different models, it is usually possible to obtain a better and more comprehensive interpretation of the seismic behavior . The seismic assessment of the Qutb Minar has been performed by using a stepped strategy, conveniently divided into: • Background, which means gathering the previous information that it is possible to collect, as for example: the historical information (behavior, damages, repairs, etc); the materials and geometry description; and preliminary studies (survey of the structure, monitoring, field research and laboratory test, etc). • Numerical analysis tools, which is related to the selection of the different analysis tools. Here the following aspects F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 1467 should be considered : the relation between the analysis tool and the information sought; the cost and time requirements; and the use of an analysis tool that can be validated and assessed. • Process of tuning or the model’s calibration, where the data obtained in the first step, as well as the results of other models, can be used for the calibration. • Types of analysis, which depending on the analysis tools chosen and the computer resources available can be: elastic, non-linear, static, dynamic, etc. In general, the types of analysis can also be divided in an analysis for validation of the numerical tools and an analysis for the seismic assessment. • Parametric analyses, where, if the data collected or the tuning process are not enough to define all necessary variables, some values are proposed using as reference other similar structures or materials. Then, parametric analyses are required to overcome the uncertainty associated with the use of these ‘‘estimated values’’. As a consequence of the above, the paper is based on a case study but also provides a very relevant discussion on an issue that has received insufficient attention: the seismic analysis of masonry structures without a box behavior. The seismic analysis of masonry structures with a box behavior received much attention in the past and is relatively well consolidated in the research environment, e.g. [6–9]. On the contrary, the seismic analysis of masonry structures without a box behavior remains complex, with some apparent consensus in the use of a limit analysis of block mechanisms, e.g. [10–12]. The use of a limit analysis is not trivial for special structures, which constitute most monumental structures. Thus, this paper also presents a significantly wider contribution to the field, by demonstrating the possible inadequacy of push-over analysis to tackle the seismic assessment of masonry structures without a box behavior. 2. Background 2.1. Historical information The Qutb Minar is one of the tallest stone masonry towers in the world. The tower has five balconies (Fig. 1). The construction began during the reign of Qutb-ud-din around 1202, but the erection stopped at the first storey. The next ruler, Iltutmish, added the next three storeys. The tower was damaged by lightning in 1326 and again in 1368. In 1503 Sikandar Lodi carried out some restoration and enlargement of the upper storeys . The minaret has five storeys. The topmost of the original four storeys was replaced by two storeys in 1368. Balconies are placed at the end of each storey. The minaret had originally a cupola, which fell during an earthquake in 1803 and was replaced by a new cupola in late Mughal style in 1829. However, it was removed in 1848. In 1920 some facing stones needed some repairs and major structural work was carried out in 1944. The damaged stones were replaced using lime mortar and stainless-steel dowels. The foundation was strengthened by grouting in 1971, and further work was carried out in 1989 to replace damaged facing blocks and to strengthen the inner core . 2.2. Geometrical and structural description The Minar is a typical example of the classical Indo Islamic architecture. It directly rests on a 1.7 m deep square ashlar masonry platform with sides of approximately 16.5 m, which in turn overlies a 7.6 m deep lime mortar rubble masonry layer, also square, with sides of approximately 18.6 m. The bedrock is located around 50–65 m below the ground level. The Minar cross-section is Fig. 1. Qutb Minar in Delhi, India. circular/polilobed, the base diameter being equal to 14.07 m and tapering off to a diameter of 3.13 m at the top, over a height of 72.45 m (Fig. 2(a),b). The tower is composed of an external shell corresponding to a three leaf masonry wall and a cylindrical central core . The core and the external shell are connected by a helicoidal staircase and by 27 ‘‘bracings’’ composed of stone lintels with an average cross section of 0.40 × 0.40 m2 . The staircase is spiral, disposed around the central masonry shaft, and it is made of Delhi quartzite stone. Each storey has a balcony and the uppermost storey finishes with a platform. Each storey has a different pattern in construction and ornamentation (Fig. 2(b)): (a) the first storey has angular flanges alternating with rounded flutes; (b) the second storey has circular projections; (c) the third storey has star shaped projections; and (d) the fourth and fifth storey projections are round. In plan the minaret can be considered as approximately circular, with a base of 14.07 m of diameter and tapering to a diameter of 3 m at the top, with a total height of 72.45 m. The Minar is also provided with diffuse ventilation openings that can be divided in some smaller openings on three levels and larger openings as windows and doors (Fig. 2(c),(d)). In correspondence with the second and third levels of the smaller openings the cross section of the tower decreases almost to 50% of the total . 2.3. Materials The Minar outer shell is built of three leaf masonry (Fig. 2(a)). In the first three storeys the external veneering is of ashlars of red and buff colored sandstone, whereas the internal veneering is of ashlars in Delhi quartzite stones. In the two upper stories the external veneer is made of white marble stones and the internal veneering is of red sandstone. The infill is composed of rubble stone masonry, mainly with stones taken from the temples destroyed during the Islamic dominion. These three layers are held up with iron dowels incorporated frequently in between the masonry layers. The central shaft is of rubble masonry with a Quartzite stone facing. The mortar used is of lime with brick powder as an aggregate. The thickness of the outer shell tapers from 3 m at the base to 0.6 m at the top [1,13]. 1468 F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 41.71 m c a b d Fig. 2. Geometrical survey: (a) vertical section of the minaret; (b) cross sections at different levels; (c) small size ventilation openings; (d) tapered windows. Fig. 3. Mode shapes and the associated frequencies in Hz for the first 10 modes . 2.4. Survey of the structure Ambient vibration tests were performed by the University of Minho with the aim of defining the modal parameters (natural frequencies, mode shapes and damping coefficients) of the Minar, as well as to evaluate the degree of connection between the central shaft of the tower and the external shell. For details on the experimental testing program, please refer to . These tests were carried out considering several test positions, at different heights, in order to proceed with the modal identification of the tower. Several natural frequencies and the corresponding modes were defined. The Unweigth Principal Component (UPC) technique was used to define the dynamic parameters. It operates in the time domain and is based on the Stochastic Subspace Identification (SSI) method, which allows a modal parameters estimation with high frequency resolution . Fig. 3 depicts the modal shape and the associated frequency for the 10 measured modes. Eight bending, one torsional and one axial mode shapes were estimated. It is stressed that the two first bending modes were not clearly defined at the top, especially at the fourth balcony. The bending modes directions are almost perpendicular for the closely spaced pairs of frequencies. This is due to the axisymmetric cross section of the tower. 3. Numerical analysis tools Three different numerical models were considered to evaluate the structural behavior of the minaret. Two models use the well known Finite Element Method, both are three-dimensional models but one uses 3-D solid elements (Solid model) while the other one was performed with 3-D composite beams (Beam model). The third model uses 2D in-plane elements based on the Rigid Element Method (Rigid model). 3.1. Three-dimensional Solid model The Solid model was implemented by using the Finite Element (FE) software DIANA . The external shell was modeled as a three layered wall. The central shaft was modeled using a single type of masonry. The foundation, the doors and windows of the minaret were also modeled (Fig. 4). The three levels of openings situated below the first, second and fifth balconies were represented in the FE model with different materials, with the mechanical properties defined as a percentage of the materials in the surrounding areas, namely the outer layer of the outer shell. In order to include the openings, the material’s properties of the first, second and third openings were thus considered as 70%, 50% and 60% of the material in the outer layer near each opening. It should be noted that these layers cross all three vertical layers. The helicoidal staircase was modeled using flat shell elements, forming horizontal slabs, where its thickness and elastic modulus were optimized, in order to minimize the differences between the experimental results and the FE model, as explained below in Section 4. The stairs have an important role, because they make the connection between the inner shaft and the outer wall. The full model involves 65,912 elements with 57,350 nodes, resulting in about 172,000 degrees of freedom (DOF). The base of the foundation was considered as fully restrained, given the existence of the foundation block. F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 a 1469 c b y z x Fig. 4. Solid model: (a) general view; (b) staircases modeled as horizontal slabs; (c) detail of the FE model. a b Fig. 5. Material constitutive axial laws: (a) Solid and Beam element ; (b) Rigid element . 3.2. Three-dimensional Beam model The Beam model was performed using the DIANA code  based on the Finite Element Method. Three dimensional tapered beam elements of three nodes based on the Mindlin–Reissner  theory were used. Composite beams were adopted in order to take into account the different layers of materials, as well as the center core of the minaret, in a cross section. Each composite cross section was defined with four different materials according to the layer. The balconies were considered as additional localized masses. The model has 41 nodes (120 DOF) and 20 elements. It is noted that, in this model, it is not possible to model the influence of the staircase, thus a perfect connection between the shaft and the core is considered. In addition, the openings were neglected. These simplifications were made after the analysis of the results obtained by the Solid model, which shown that they have minor influence in the global behavior of the structure. 3.3. In-plane Rigid model A simplified in-plane model of the minaret based on the Rigid Element Method was developed. The Rigid Element Method idealizes the masonry structure as a mechanism made of rigid elements and springs. For further details about the method please refer to . The rigid elements used can be defined only with a rectangular cross section. For this reason an equivalent square cross section and equivalent isotropic material were considered (Table 1). In total, the numerical model has 39 elements and 117 DOF. 3.4. Material model Isotropic materials have been considered for all models because the masonry is not made by blocks and layers of mortar, where an orthotropic behavior can be expected . The masonry found is instead a rubble material made with irregular stones embedded in thick lime mortar, where no weak planes can be detected. Therefore, the mechanical behavior of this composite can be approached as an isotropic material, similar to concrete . Orthotropic behavior can be expected for the external veneer since it is made of stone blocks and layers of mortar. However, the global behavior of the minaret seems to be not influenced very much by the external veneer, as the rigid element model shows, and bending along the vertical direction will be the predominant behavior. Therefore, this layer can also be considered as a homogenized isotropic material in order to reduce the computational effort. The models consider that the materials of levels 1 to 3 are different from the materials of levels 4 and 5. This hypothesis was adopted because the last two levels were constructed in a later period, the adopted stone is different and the experimental modal shapes show that the deformation is concentrated on the last two levels, see Fig. 3. Finally, the elastic properties of the material were obtained by calibration of the models with the frequencies measured in the tower (see next Section). The strength properties were taken from the literature [1,18–20] and parametric analyses, varying these parameters, were performed in order to evaluate the influence of these properties in the global behavior of the Minaret. The constitutive model used to simulate the non-linear properties of the Solid and Beam model was a traditional smeared cracking model, with an exponential tension softening and a constant shear retention of 0.01 . A parabolic compressive behavior was considered for all materials (Fig. 5(a)), see  for details. The compressive strength (fc) was different for each material, while for the tensile strength (ft) a very low value was assigned (50 kPa). The tensile fracture energy (Gf ) was set equal to 20 Nm/m2 . The objective is to simulate a very weak material in tension. 1470 F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 Table 1 Equivalent square cross section and the mesh considered in the Rigid model. Area (m2 ) Inertia (m4 ) Level Equivalent thickness (m) Real Equivalent Real Equivalent Error(%) 0 11.67 127.36 136.25 6.98 1762.80 1547.10 −12.24 1 7.75 54.69 60.08 9.86 360.07 300.85 −16.45 2 5.91 31.62 34.97 10.58 123.43 101.91 −17.43 3 4.45 18.08 19.81 9.54 38.94 32.71 −16.00 4 3.45 10.80 11.91 10.26 14.25 11.82 −17.00 5 2.41 5.18 5.85 12.94 3.58 2.85 −20.51 Error (%) In the case of the Rigid model, the constitutive law for axial springs is parabolic in compression and bi-linear in tension with softening (Fig. 5(b)). A Mohr–Coulomb law is considered for the shear springs in order to relate the shear stresses with the axial stresses . For this model only compressive (f 0 c), tensile (ft) and shear (fs) strength are necessary, as well as the friction angle (ϕ ). The compressive strength was different for each material, while the tensile and shear strength as well as the friction angle were considered equal for both materials, as was done in the smeared cracking model. The damping is considered as a viscous damping C , depending on the mass M- and stiffness K by means of the Rayleigh formulation C = aM + bK , where a is the mass proportional damping constant and b is the stiffness proportional damping constant. The value of these two parameters was optimized in order to obtain similar values as the experimental damping for each frequency derived from the environment vibration tests, being: a = 0.2256, b = 0.00057. 4. Process of tuning the model Model updating was performed in order to match the natural frequencies arising from the experimental investigation. Each model was updated independently and the elastic modulus was the parameter considered for the calibration of the different sets of materials, ranging between reasonable values for the different types of masonry. Considering that the models present an axial symmetry, the numerical pairs of corresponding bending mode shapes and frequencies are identical (and orthogonal). Thus the averaged experimental values were considered for the model calibration. The modal updating was performed by using the methodology proposed by Douglas and Reid , in which the frequencies of the structure can be estimated by means of: fiD (X1 , X2 , . . . , XN ) = Ci + N X Aik Xk + Bik XkN (1) i=1 where f D is the frequency estimated, Xk are the variables to calibrate, A, B and C are constants, and N is the number of frequencies used in the calibration. Thus, the variables are obtained by an optimization of the function J: J = m X wi εi2 (2) i=1 εi = fiEMA − fiD (X1 , X2 , . . . , XN ) (3) where f is the experimental value of the frequencies and w is a weight factor. The optimization process took place in the GAMS software  that determines the optimal values by a comparison of the EMA optimization variables in limit cases with the experimental values, and then determines the values that minimize the error between the numerical and experimental values. Matching the frequencies does not guarantee that the mode shapes are also correctly matched. It is also straightforward to incorrectly match one frequency but it is unexpected to incorrectly match several frequencies, as done here. The mode shapes obtained with the analyses will be only qualitatively compared with the measured modal shapes in order to corroborate the results. This is due to the low quality of some of the mode shapes and the low number of stations used in the dynamic identification procedure . As the Solid model has a very refined description of the geometry of the structure, ten variables were optimized. They were the elastic modulus of the different layers of the shaft and core, as well as the thickness and the elastic modulus of the staircase (Table 2). Five variables were considered in the Beam model that corresponds with the different materials of the minaret (Table 3). Finally, due to the simplifications made in the Rigid Element model, only two isotropic homogeneous materials were considered, one for levels 1 to 3 and the other for levels 4 and 5 (Table 4). 5. Types of analyses The different types of analysis performed can be divided into two main groups: (a) validation of the analysis tools; and (b) evaluation of the seismic behavior of the tower. In the first group we have the eigenvalue and self-weight analyses. The calculus of the frequencies and modal shapes allow one to validate the models in the elastic field; while the self-weight analysis allow one to verify the correct geometrical description of the structure and the properties of the materials. In this context, the 3D Solid model was used to validate the hypothesis in which the other two models are based, mainly: the stairs are sufficiently stiff to consider that the inner shaft works together with the external shell; the openings can be neglected, since they do not influence the global behavior of the minaret. In the second group, the non-linear static (pushover) and dynamic analyses are considered. It is worth noting that the Solid model was not used for a dynamic analysis because the great amount of computational resources and time required. The dynamic analyses were carried out using five synthetic accelerograms compatible with the design spectrum of the Indian Seismic code for Delhi considering the experimental damping for the first mode of 2.5%. Fig. 6 shows the design spectra for the different values of damping. The vertical lines correspond to the periods of the structure. For the minaret the design spectrum corresponds to a damping of 2.5%, which is the damping emerged from the environment tests. For this spectrum, the maximum ground acceleration corresponds to 0.25 g, while the constant branch of the spectrum has a spectral acceleration of 0.63 g. It can be seen that the first period is in the lower part of the spectrum F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 1471 Table 2 Optimized values of the materials used in the Solid model. Material Elastic modulus (GPa) Specific mass (Kg/m3 ) Poisson’s coefficient f 0 c (kPa) ft (kPa) Shaft 1–3 Shaft 4–5 External shell inner layer 1–3 External shell external layer 1–3 External shell core layer 1–3 External shell core layer 4–5 External shell inner layer 4–5 External shell external layer 4–5 Stairs 1.545 0.300 6.171 2.000 0.785 0.300 6.602 2.000 3.689 1800 1800 2600 2300 1800 1800 2600 2600 2000 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1000 600 5200 2500 1000 600 3000 5200 elastic 50 50 50 50 50 50 50 50 Table 3 Optimized values of the materials used in the Beam model. Material Elastic modulus (GPa) Specific mass (Kg/m3 ) Poisson’s coefficient f 0 c (kPa) ft (kPa) Masonry infill 1–3 Masonry infill 4–5 Sandstone Quartzite Marble 1.00 0.60 2.50 5.21 3.00 1800 1800 2300 2600 2600 0.2 0.2 0.2 0.2 0.2 1000 600 2500 5200 3000 50 50 50 50 50 Table 4 Optimized values of the materials used in the Rigid model. Material Elastic modulus (GPa) Specific mass (Kg/m3 ) Poisson’s coefficient f 0 c (kPa) ft (kPa) fs (kPa) ϕ (o ) Masonry 1–3 Masonry 4–5 3.18 0.57 1900 1900 0.2 0.2 3000 1500 35 35 45 45 15 15 6. Analyses for validation model presents the higher errors for the frequency associated to the torsion (Table 5). This model calculated the fifth mode as torsional, while the ambient vibration presented this mode as the seventh. The reason for this difference is possibly the staircase. The Solid model calculated the ‘‘correct sequence’’ of the modes, with an error associated to this torsional mode of about 8%. On the other hand, the model that presents in general the smaller errors in the calculated frequencies (less than 5%) is the simplified Rigid Element model; except for the vertical mode that has an error up to 10%. Here, it is noted that the vertical mode was difficult to calibrate. Solid and Beam models consider this mode as the 8th, while in the experimental test this is the 10th mode. Note that the true structure and the Solid model are not axisymmetric, meaning that a clear orientation of the modes in two essentially orthogonal directions x and z have been found. On the contrary, the Beam model is fully axisymmetric, meaning that the directions x and z lose their meaning. As any set of orthogonal directions would provide two equal eigenvectors, the frequencies related to the consecutive bending modes in two orthogonal directions have been lumped to a single value in Table 5. Finally, the Rigid model is two-dimensional, meaning that only one frequency is found for each pair of consecutive bending modes and a torsional mode cannot be found. The Solid model considers the different openings (windows, doors and the small ventilation openings) of the minaret. These openings should give some kind of asymmetry to the structure. However, the differences between the pairs of bending modes obtained are very low and showing that the differences in the pairs of bending modes obtained experimentally are not only related with the openings. These differences can be related to other aspects of the structure as differences in the quality and properties of the materials, differences in the geometry of the minaret, some cracks, etc. 6.1. Eigenvalue analysis 6.2. Self-weight analysis In general, all the models have a good match between the experimental and calculated frequency values (Fig. 7). The Beam A self-weight analysis was performed in order to evaluate the stress pattern. All three models give similar results. The Fig. 6. Design spectrum of the Indian Seismic Code for Delhi for different values of damping. (0.20 g), while the third to tenth periods are located in the constant branch of the spectrum (0.63 g). It is noted that the response spectrum can have some influence on the result and several codes define a near- and far-fault earthquake, which must be associated with a macro-zonation map. It is believed that the use of historical earthquake records, particularly if scaled to a different PGA, raise also a lot of doubts on representativeness. For the purpose of discussing the safety of the particular structure and discussing the difficulty of using a push-over analysis for the monumental structures without a box behavior, it is believed that the choice of using the code response spectrum is adequate. The procedure adopted is aligned with the proposal of recent codes, e.g. Eurocode 8 , which clearly states that the code’s elastic response spectra should be used for a nonlinear time history analysis. 1472 F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 Fig. 7. Modal shapes for the models, Solid (left), Beam (center) and Rigid (right). Table 5 Comparison among experimental frequencies and the frequencies obtained with the three different models. Mode shape 1 2 3 4 5 6 7 8 9 10 Comment Frequencies (Hz) 1st Bending, z 1st Bending, x 2nd Bending, x 2nd Bending, z 3rd Bending, x 3rd Bending, z 1st Torsion 4th Bending, x 4th Bending, z 1st Vertical Experimental Solid Error (%) Beam Error (%) Rigid Error (%) 0.789 0.814 1.954 2.010 3.741 3.862 4.442 5.986 6.109 6.282 0.71 0.71 2.07 2.09 3.55 3.59 4.80 5.98 6.02 5.35 −10.01 −12.77 0.734 −8.42 0.778 −3.17 2.257 13.87 1.886 −4.84 4.129 8.62 3.582 −5.77 — — 5.93 3.98 −5.10 −7.04 8.06 −0.10 −1.45 −14.83 3.656 −17.69 6.665 10.21 6.419 5.65 6.098 −2.93 7.061 12.41 Table 6 Compressive stresses at the bottom of each material layer (middle plane) of the Beam model and the map of the vertical stresses in the Rigid model. Material Maximum axial stress σmax (KPa) Compressive strength f 0 c (kPa) σmax /f 0 c (%) Masonry infill 1–3 494 2000 24.70 Masonry infill 4–5 102 600 17.00 Sandstone 617 2500 24.68 Quartzite 1290 5208 24.77 127 3000 4.23 Marble total weight of the minaret is around 75,000 kN and the average compressive stress at the base is 0.6 MPa. The maximum compressive stress is 1.29 MPa at the bottom of the external layer of the shell (the reference adopted in the middle plane of each layer). The materials remain in the elastic range and the maximum stresses at the base of the minaret are around 24% of the compressive strength (Table 6). 6.3. Performance of the numerical models The objective in making different models was to combine the results of different models in order to obtain a better and more comprehensive interpretation of the seismic behavior. Thus the 3D Solid model was, mainly, used to validate the hypothesis on which the other two models are based. In this context, the Rigid model (Pa) computational efficiency of the simplified models (Beam and Rigid models) compared with the 3D Solid model was exploited in order to perform a full non-linear dynamic analysis. This type of analysis is precluded in the Solid model due to the level of refinement of the mesh and the high computational cost (CPU time and space in hard disk), see Table 7. The Solid model takes 24 more times to run than the Beam model and 12,000 more times to run than the Rigid model, making the non-linear time integration analysis unwieldy. It is worth noting that the Beam model requires 480 more times to run than the Rigid model; although both models have almost the same number of degrees of freedom. This is mainly due to the number of integration points of each model. The Rigid model has 195 integration points, while the Beam model has 14,000. The Rigid model is able to perform fast non-linear dynamic analysis with few computational resources. For example, the Beam F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 1473 Table 7 Comparison between the degrees of freedom (DOF) and the computational cost for a non-linear static analysis using the three different models. Model Beam (3D) Rigid (2D) Solid (3D) 1 2 DOF 120 117 172,000 DOF ratio CPU time (min)1 CPU ratio2 Space in hard disk (MBytes) Space ratio2 1 1 1400 120 0.25 2880 1.0 0.002 24 92.3 0.7 6958 1.0 0.008 75.4 Using a one-core Pentium IV Processor (3 Ghz), 1 Gb in RAM and for non-linear static analysis. Using the Beam model as reference. a b c Fig. 8. Pushover analysis, capacity curves. model (in the scope of general purpose software) requires 23 h to perform a non-linear dynamic analysis, while the Rigid model (in the scope of specific, stand alone software) needs only 20 min for the same analysis. 7. Analyses for the evaluation of the seismic behavior 7.1. Non-linear static analysis A non-linear static pushover analysis was carried out considering a uniform acceleration distribution. The load was applied with an increasing acceleration in the horizontal direction and a control point at the top of the tower was considered. Fig. 8 shows the capacity curve lateral displacement–load factor (shear base/selfweight). Rather similar behaviors were found for the models. It can be seen that the average load factor is 0.21 for the Beam and Rigid models, while the load factor for the Solid model is 0.18. It is worth noting that the materials do not fail by compressive stresses and the tower collapses by overturning at the base. 7.2. Non-linear dynamic analysis A non-linear dynamic analysis was performed with the Rigid and Beam model, considering a Rayleigh model for damping with the damping values emerged from the investigation campaign a Fig. 10. Results of the Rigid model for record 1 at time 18.3 s: (a) deformed shape; (b) axial stresses [Pa]; (c) axial strains. (approximately equal to 2.5% for the first frequency). The dynamic analyses were carried out using five synthetic accelerograms compatible with the design spectrum of the Seismic Indian code. Fig. 9 shows the load factor (shear force/own weight) and the absolute maximum displacements for each level obtained with the Rigid model. It is worth noting that these values are the maximum and do not necessary occur at the same time. The average load factor at the base is 0.20 and remains almost constant for the first level, while for the second balcony the average load factor is 0.25. The average load factors of the third and fourth balconies are of 0.75 and 1.10 respectively. This means that the amplification of the seismic loads is concentrated at the last two levels. Displacements of levels 1, 2 and 3 increase practically in a linear way, while the displacements of level 5 are in general almost a double of the displacements of level 4. Fig. 10 shows the deformed shape and the map of axial stresses and strains at the instant 18.3 s of record 1 of the Rigid model. As can be seen, only levels 4 and 5 present a lateral deformation (Fig. 10(a)). The axial stresses in levels 1 and 2 are practically uniform (Fig. 10(b)), since the bending moment is not present. Axial compressive stresses on levels 4 and 5 are concentrated on one side due to the bending moment and therefore the axial strains are concentrated on the fourth balcony (Fig. 10(c)). b Fig. 9. Maximum absolute results along the height of the minaret for dynamic analyses with the Rigid model: (a) load factor; and (b) lateral displacements. 1474 F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 a b Fig. 11. Maximum absolute results along the height of the minaret for dynamic analyses with the Beam model: (a) load factor; and (b) lateral displacements. a b Fig. 12. Comparison between the Rigid and Beam models for average results along the height of the minaret for dynamic analyses: (a) load factor; and (b) lateral displacements. These results show that levels 4 and 5 are the most vulnerable. Especially level 5 presents the highest drift (3.0%). Analyses show that the higher modes of vibration have a great influence on the seismic response of the minaret. Fig. 11 shows the maximum load factor and displacements for each level with the Beam model. The average load factor at the base is 0.16 and increases to 0.18 for the first level. The second balcony has an average load factor of 0.28, while the third and fourth balconies have an average load factor of 0.47 and 0.9 respectively. The maximum displacements at the top vary from 0.35 to 0.65 m, having a maximum drift of 6%. The results show that for all the five records levels 4 and 5 are the most vulnerable too. On the other hand, Fig. 12 shows the comparison between the Rigid and the Beam models for the dynamic analysis. The compared results are the average of the maximum load factors and displacements for each storey. Both models have similar behavior: the two top storeys are the most vulnerable part of the minaret. The load factors calculated by the Rigid model are, in average, 20% greater than the Beam model; while the displacements are 30% lower. This means that Rigid model is slightly more rigid than the Beam model. This is mainly due to the simplifications considered in the Rigid model (see Section 3.3). 7.3. Remarks about the analyses performed The different models present a similar behavior under different types of load. However, the results obtained from the non-linear static and dynamic analyses indicate a quite different response of the structure to earthquakes. The non-linear static analysis shows that the lowest part of the structure exhibited a diffuse cracking and a base overturning mechanism could be detected. On the other hand, the non-linear dynamic analyses carried out indicated that the part of the Qutb Minar more susceptible to seismic damage coincides with the two upper levels, where the highest accelerations and drifts were found. The differences in the results between the static and dynamic analyses can be explained by the high influence of the higher modes in the seismic behavior of the tower. In fact, the standard non-linear static analyses do not take into account the participation of the different modes. The results of the non-linear dynamic analyses are more representative of the real seismic behavior of the tower, since the damage by previous earthquakes has been concentrated in the last two levels. In this context, it is possible to conclude that the most vulnerable part of the Qutb Minar is the two top storeys. 8. Parametric analyses For the study of the seismic assessment of ancient masonry structures, parametric studies should be made when the real properties of the material or the loads that the structure could suffer are not well known or defined. In this particular case, as the strength properties of the materials were taken from the literature and differences found in the behavior between the static and dynamic analyses, parametric analyses were carried out in order to evaluate their influence. It is worth noting that the parametric analyses were only performed with the Beam model in order to have a better description of the geometry and the material’s distribution (cross section and different material layers), as well as to take into account the influence of the torsional mode. F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 Fig. 15. Damping curves used in the parametric analyses. Fig. 13. Pushover analyses, capacity curves with different distribution of lateral loads. 8.1. Non-linear static analysis In order to study the influence of the distribution of the lateral force into the pushover analysis, a second non-linear static analysis was performed. Three different configurations of lateral loads were considered: the forces proportional to the mass (uniform acceleration); the linear distribution of the displacement along the height as proposed by the Seismic Indian Code ; and the forces proportional to the first modal shape. Fig. 13 shows the three computed capacity curves. The maximum load factor that the structure can resist depends very much on the distribution of the forces: 0.205, 0.108 and 0.072 for the proportional mass, the linear displacement and the first mode’s load distribution, respectively. The load factor proportional to the first mode is only 35% of the load factor proportional to the mass, while the load factor proportional to the linear distribution is 53%. It is worth noting that the collapse section changes too. In the case of the analysis considering the forces proportional to the mass, the section of the collapse is located at the base, while for the other two analyses the Minaret collapses at the first balcony. A modal pushover analysis (MPA) was performed in order to take into account the higher modes . The first mode has an effective mass equal to 37.9% of the total mass. Thus, in order to take into account at least the 90% of the mass, it is required to consider seven modes for each direction, see Table 8. Fig. 14 shows the displacement and drift of each storey obtained with the MPA and they are compared with the average results obtained with the dynamic analyses. As it can be observed, the obtained behavior is similar for the first three levels, where the displacements and drifts are similar to the non-linear dynamic analyses. However, the behavior of levels four and five are quite different. The MPA cannot reproduce satisfactory the failure mechanism of the last two levels. a 1475 This difference is due to the change of the dynamic properties of the tower during the damage process (see Table 8). The change of the dynamic properties means that the hypotheses of the modal pushover analysis cannot be fulfilled, namely: (a) the influence (participation factors and effective mass) of each mode cannot be considered constant during the analysis; (b) the peak acceleration for each mode will be different since the frequency changes; (c) the number of the required modes to have at least the 90% of the effective mass also changes, in this particular case changing from 7 to 19; (d) the lateral force distribution for each mode will change; (e) the modal damping will vary as the frequencies change; for the first frequencies it will be greater, while for the higher modes it will decrease (see Fig. 15). The change of the dynamic properties can be possibly taken into account considering an adaptative modal pushover analysis. However, this type of analysis becomes impractical. In this case, it is possibly better to perform a non-linear dynamic analysis, in which the full dynamic behavior is directly taken into account. 8.2. Non-linear dynamic analysis In order to evaluate the influence of the most relevant parameters in the seismic behavior of the Minar, non-linear dynamic analyses were performed varying the tensile strength, the tensile fracture energy and the damping. In total 75 analyses were carried out, since three different values for each variable and non-linear dynamic analyses using the previous five compatible accelerograms with the Indian Seismic code were considered. The values of the tensile strength were considered as the 2.5, 5 and 10% of the compressive strength of the masonry, while the values of the fracture energy were considered proportional to the values of the tensile strength, according to [18–20]. Table 9 shows the values taken into account in these analyses. b Fig. 14. Comparison between the modal pushover analysis and the non-linear dynamic analyses; (a) displacements; and (b) drifts. 1476 F. Peña et al. / Engineering Structures 32 (2010) 1466–1478 Table 8 Periods and effective mass of the modes considered during the modal pushover analysis (original) and calculated after the non-linear dynamic analysis for the seismic record 1 (damaged). Mode 2 3 5 8 14 17 21 Period (s) Effective mass (%) Original Damaged Difference (%) Original Damaged Difference 1.36 0.44 0.24 0.15 0.10 0.08 0.06 1.67 0.85 0.51 0.36 0.26 0.22 0.18 22 92 112 138 149 183 183 37.9 20.3 11.9 8.8 5.6 3.3 2.5 25.8 17.3 10.1 4.5 11.5 4.0 1.5 −32 −15 −15 −49 90.3 74.7 −17 Total 105 20 −39 Table 9 Variables considered in the parametric analyses. Parameter Reference value Mid value Maximum value Tensile Strength (kPa) Tensile Fracture Energy (Nm/m2 ) Damping for the first frequency (%) Mass proportional damping a Stiffness proportional damping b 50 20 2.5 0.2256 0.00057 100 50 5.0 0.4361 0.00119 200 100 8.0 0.7194 0.0017 The initial analyses were performed considering the damping values emerged from the investigation campaign. However, these damping ratios were obtained through low amplitude vibrations (ambient vibration), which are sometimes not representative of the damping that the structure can present during a seismic event. Historical masonry constructions subjected to seismic events can present damping ratios around 8%–10% . Therefore, three different damping ratios were used: the damping obtained from the ambient vibration and a proportional damping to ambient vibration but considering 5% (a = 0.4361, b = 0.00119) and 8% (a = 0.7194, b = 0.00220) for the first frequency (Fig. 15). Fig. 16 shows the average results for the five synthetic records with damping of 2.5% (experimental) varying the tensile strength (left) and the fracture energy (right). It can see that the fracture energy has practically no influence on the average behavior of the minaret in terms of the forces and displacements. The tensile strength has more influence in the global behavior. This influence can be noticed in the top storey, in which the lateral displacement and the drifts have differences around 10% between extreme values. The shear forces and load factors increase with this parameter. On the other hand, the parameter that has more influence on the global behavior is damping. Fig. 17 shows the average maximum displacements and drifts for different values of damping. As can be seen, the global behavior of the tower is similar to previous analyses, being independent of the damping ratio considered. The last two storeys of the minaret are the most vulnerable part of the tower. However, considerable differences in the values of the displacements and the drift were found. In general, the first 2 storeys have a similar behavior independently of the damping and strength characteristics. But in the last three storeys the drifts and displacements increase as the damping decreases. 9. Conclusions A simple strategy of analysis for the seismic assessment of the Qutb Minar in Delhi, India was presented. Three different models with different levels of complexity and simplifications were developed. The use of these three models allows one to overcome the complexity on the study of the seismic behavior of ancient masonry structures; since combining the results of the different models it is possible to obtain a better and more comprehensive interpretation of the seismic behavior. It is clear that this strategy can be used for other similar structures. The models present a similar behavior under the same loads and types of analysis. However, the results obtained from the non-linear static and dynamic analyses indicate quite a different response of the structure to earthquakes. The non-linear static analysis shows that the lowest part of the structure exhibited a diffuse cracking and a base overturning mechanism could be detected. On the other hand, the nonlinear dynamic analyses carried out indicated that the part of the Qutb Minar more susceptible to seismic damage coincides with the two upper levels, where the highest accelerations and drifts were found. The differences in the results between the static and dynamic analyses are due to the high influence of the higher modes in the seismic behavior of the tower. In fact, the non-linear static analyses do not take into account the participation of the different modes. The modal pushover analysis, which considers the influence of the higher modes, cannot reproduce the appendix like behavior of the last two levels satisfactorily. This is due to the change of the dynamic properties of the tower during the damage process. The results of the non-linear dynamic analyses can be considered more representative of the real seismic behavior of the tower, since the historical damage by earthquakes has been concentrated in the last levels. In this context, it is possible to conclude that the most vulnerable part of the Qutb Minar is the two top storeys. The results from the different approaches allow one to conclude that: (a) the distribution of the lateral forces has a large influence on the pushover analyses. The section at the minaret fails and the load factor depends significantly on the lateral force distribution; (b) it is not recommended to perform pushover analyses to study the seismic behavior of masonry towers, since it is not possible to neglect the influence of the higher modes; (c) in this particular case, the modal pushover analysis cannot reproduce the results obtained with the non-linear dynamic analysis; (d) damping has a large influence on the dynamic time integration results in terms of displacements, whereas the effect of the tensile strength and the fracture energy is only minor; (e) it is recommended to perform a full non-linear dynamic analysis instead of a static non-linear analysis for the seismic assessment of the tower. 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