Determination of Dynamic Characteristics of Historic Beyazıt Tower

Abstract

The Beyazıt Tower, located within the Beyazıt Campus of İstanbul University, was chosen as a reference structure to assess its dynamic characteristics due to its historical significance. The tower, which is a historic masonry structure, was analyzed using the finite element method with shell and solid 3D models. Ambient vibration tests were conducted to evaluate the accuracy of the analysis results. Based on the findings from the experimental study, the numerical models were updated, and the most suitable material properties were identified. The accuracy of the analysis results among the different models was also discussed. Additionally, a foundation was established for long-term monitoring studies by determining the dynamic characteristics through experimental methods. These studies will help identify any changes that may occur over time, allowing for the monitoring of potential damage and the detection of any impacts from natural disasters or human activities on this unique structure.

1. Introduction

Historical buildings and construction techniques represent a significant cultural and scientific heritage that has been passed down through the ages. Assessing the behavior, strength, and safety of these structures, as well as strengthening them when necessary, is a critical engineering challenge for the preservation of this heritage. The analysis models for historical masonry structures must be constructed realistically; however, many uncertainties arise due to a lack of knowledge about material properties. It is essential to determine whether the material properties defined in these analysis models accurately reflect actual behavior before proceeding with further analysis. Hence, this article presents experimental and numerical studies on Beyazıt Tower.

This section summarizes the historical and architectural features of Beyazıt Tower, discusses the material properties of masonry building elements found in the literature, and examines similar historical towers.

1.1. History of Beyazıt Tower

The Beyazıt Tower (41°0′46.0548″, 28°57′53.406″), built several times since the early 1700s with different forms and materials, was burnt or demolished many times over the years. The inscription on the tower’s base indicates that it was built in 1826, together with the Seraskerat Building (now Istanbul University Rectorate Building). It is known that it was built as masonry by architect Seyyid Abdülhalim Efendi. A lead-covered wooden cone existed on the Guard Floor in the first built structure. In 1849, three floors with an octagonal plan were added to the cone, and a steel flagpole was placed onto the tower. Until the 1950s, the tower was used as a fire watchtower to ensure communication between the districts of İstanbul in case of fire. Today, it is used to report the weather with the lighting placed on the top floors [1]. Figure 1 shows anonymous postcard images of the tower from the end of the 19th century.

1.2. Architectural Background

The masonry tower, standing at a height of 67 m (without flagpole), shown in Figure 2a, was constructed using rubble stone and limestone, and it rested on a truncated conical pedestal at the base. This base has a square plan, is 12 m wide at the bottom, and rises by narrowing to 10.30 m wide at the top. The pedestal height is 8.95 m. There is a section with pedestal corners cut and cornices at the upper part. The wall thickness varies between 3.40 and 4.30 m. The north facade features an entrance door that measures 97 cm in width and 205 cm in height. The upper part of the entrance door is arched and has a circular window above with a diameter of 40 cm (Figure 2b). A wooden staircase block with a 348 cm diameter is in the center of the tower. The stairs are helically shaped, and a 44 cm diameter wooden spar is in the middle. The tower, which has a cylindrical geometry on the base, rises by expanding and narrowing and has a fixed wall thickness between the elevations of +12.9 m and +38.40 m. Occasionally, there are ornaments at the expanding points. In the staircase, the windows were placed rotating on four sides in the plan with an elevation difference of 2.90 m.

The wall thickness between the elevations of 12.9 m and 38.4 m is approximately 1.80 m. Above 38.40 m, the tower expands with a double curvature form. At 43.30 m, the wall thickness increases to 4.60 m. Inside the tower, after 179 steps, the first floor is reached with an arched entrance featuring a circular plan at the elevation of +45.40 m, known as the Guard Floor. This floor has panoramic views with 12 windows. There are square and cylindrical tension rods positioned on and between these windows. The walls are finished with plaster, and the vaulted ceiling is adorned with landscape paintings (Figure 3). There are eaves on the Guard Floor, and the above floors’ eaves are surrounded by railings.

Three octagonal planned floors were built by removing the cylinder cone on the Guard Floor in 1849. The three levels are named as follows: the first level is called the Sign Level, the second is the Basket Floor, and the third is the Starboard Floor. Each of these floors features wooden flooring supported by steel IPN profiles in both directions. One end of the I-profiles is connected to the walls, and the other end to the other I-profiles or the middle wooden spar. All three floors are gradually reduced in the plan. Each floor has circular metal windows on four sidewalls. The balconies on the floors have stone balustrades. While rubble stone was used throughout the tower, limestone was used to construct these floors with steel clamps between them in a steel carcass. Figure 4a,b shows these steel clamps and carcasses. Above the last floor of the tower, there is a steel flagpole with a height of approximately 12 m. The bottom diameter is about 38 cm, and the tapered top diameter is about 20 cm, with a thickness considered to be 5 mm.

1.3. Literature Preview of Historical Masonry Towers

Many historical masonry tall buildings were built on the Earth centuries ago and survived or suffered minor or severe damage during a natural disaster. In addition to the historical importance of these masonry structures, due to the characteristics of the period in which they were built and material differences, examining their structural behavior is also very important and exciting in engineering. These towers were built using stone, brick, or a combination of both and with different mortar materials. Towers with different geometry, wall thickness, and heights have been the subject of many studies. These studies in the literature include subjects such as numerical models and simulated studies [2,3,4,5,6,7,8,9,10], studies on the effects of environmental conditions on dynamic characteristics [11,12,13,14,15,16], comparison of experimental and numerical studies, and updating the analysis models [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

In these studies, the dynamic characteristics of historical buildings were determined using both experimental and numerical methods. Factors such as building height, geometry, wall thickness, and natural stone or brick building materials are of great importance in determining the frequencies of the buildings. Table 1 was created to compare these differences and establish relationships between them. Dynamic characteristics of most of the towers given in Table 1 were also evaluated by Diaferio et al. [32]. Based on this table, it is seen that similar structures with approximately the same height and wall thickness have close frequencies.

1.4. Scope of the Study

This study aims to determine the dynamic characteristics of a tall historical building with different numerical models and experimental methods by taking the Beyazıt Tower as a case study due to its historical significance, which has been undamaged from several earthquakes and has been in service for approximately 150 years. Numerical models are considered in two cases: (i) a simplified shell model and (ii) a detailed 3D solid model. The comparison of these models with the same material properties and with the experimental data was investigated to consider which numerical model provides more realistic results regarding the actual behavior of the tower. Modal frequencies were obtained through experimental studies, and a comparative analysis with numerical results was conducted. Additionally, uncertainty arising from the wide range of elastic modulus values complicates the determination of the actual behavior of masonry structures. The global modulus of elasticity of the structure was also examined using a model updating technique. Based on the findings from the studies, this work aims to provide essential insights into the analysis and behavior of tall masonry structures in the literature, particularly regarding Beyazıt Tower [33].

Table 1. Dynamic characteristics of historical masonry towers determined by experimental studies.

Masonry Building	Location	Year Built	Height [m]	Wall Thickness [m]	Plan Geometry *	Material **	Natural Frequency [Hz]			Ref.
							1st Mode	2nd Mode	3rd Mode
Gabbia Tower	Italy	1227	54	2.4	Rec	B	0.92	0.99	3.89	[12]
Santa Maria del Carrobiolo bell-tower	Italy	1339	33.7	-	Rec	-	1.92	2.01	2.37	[13]
Benedictine Abbey of San Pietro bell tower	Italy	13th c.	61.45	-	Hex	S	1.45	1.52	4.34	[14]
San Vittore bell tower	Italy	15th c.	36.72	1.30–1.35	Rec	S	1.22	1.28	3.60	[15]
The Basilica of San Pietro (Bell Tower)	Italy	996	61.4	-	Hex	B + S	1.48	1.55	4.29	[16]
Matilde Tower	Italy	-	35	-	Rec	B	1.20	1.91	3.22	[17]
Annunziata Bell Tower	Greece	1394	20	1.00–0.55	Rec	S	2.62	2.83	5.51	[18]
Sultan Ahmet Mosk minaret	Türkiye	1616	50.91	1.5	Cir	S	0.87	3.22	4.85	[19]
Torre Grossa	Italy	1240	61	2.6–1.6	Rec	B + S	1.31	1.33	3.41	[20]
Masonry Chimney	Spain	20th c.	38.3	1.71–0.5	Cir	B	0.76	8.41	23.05	[21]
Bell Tower of Capua	Italy	861	41	-	Rec	S	1.26	1.29	3.1	[22]
Bell Tower of Averse	Italy	1080	45.5	-	Rec	S	1.05	1.37	4.81	[22]
Ghirlandina Tower	Italy	1319	88.82	-	Rec	-	0.74	0.85	2.77	[23]
Bell tower of Santa Maria del Carmine	Italy	14th c.	68	3.54–0.84	Rec	S	0.69	0.76	2.28	[24]
Cathedral of Colima	Mexico	1889	31	1.5	Rec	B	1.40	1.62	-	[25]
Qutb Minar	India	1368	72.45	-	Cir	S	0.79	0.81	1.95	[26]
Santas Justa and Rufina Church bell tower	Spain	15th c.	35.5	1.5	Rec	B + S	2.15	2.24	5.95	[27]
Becci tower	Italy	-	39.4	1.5	Rec	S	1.37	1.67	-	[30]
Bell Tower of St. Sisto’s Church	Italy	1754	40	-	Rec	S	1.36	1.47	4.25	[34]
Minaret of Little Hagia Sophia	Türkiye	1762	35.39	0.8–0.35	Cir	S	1.449	1.495	6.302	[35]
Hagia Sofia Tower	Türkiye	1250–1260	23	0.9–1.5	Rec	-	2.59	2.71	6.37	[36]

* Rec: rectangular; Cir: circular; Hex: hexagonal. ** B: brick; S: stone.

Table 1. Dynamic characteristics of historical masonry towers determined by experimental studies.

Masonry Building	Location	Year Built	Height [m]	Wall Thickness [m]	Plan Geometry *	Material **	Natural Frequency [Hz]			Ref.
							1st Mode	2nd Mode	3rd Mode
Gabbia Tower	Italy	1227	54	2.4	Rec	B	0.92	0.99	3.89	[12]
Santa Maria del Carrobiolo bell-tower	Italy	1339	33.7	-	Rec	-	1.92	2.01	2.37	[13]
Benedictine Abbey of San Pietro bell tower	Italy	13th c.	61.45	-	Hex	S	1.45	1.52	4.34	[14]
San Vittore bell tower	Italy	15th c.	36.72	1.30–1.35	Rec	S	1.22	1.28	3.60	[15]
The Basilica of San Pietro (Bell Tower)	Italy	996	61.4	-	Hex	B + S	1.48	1.55	4.29	[16]
Matilde Tower	Italy	-	35	-	Rec	B	1.20	1.91	3.22	[17]
Annunziata Bell Tower	Greece	1394	20	1.00–0.55	Rec	S	2.62	2.83	5.51	[18]
Sultan Ahmet Mosk minaret	Türkiye	1616	50.91	1.5	Cir	S	0.87	3.22	4.85	[19]
Torre Grossa	Italy	1240	61	2.6–1.6	Rec	B + S	1.31	1.33	3.41	[20]
Masonry Chimney	Spain	20th c.	38.3	1.71–0.5	Cir	B	0.76	8.41	23.05	[21]
Bell Tower of Capua	Italy	861	41	-	Rec	S	1.26	1.29	3.1	[22]
Bell Tower of Averse	Italy	1080	45.5	-	Rec	S	1.05	1.37	4.81	[22]
Ghirlandina Tower	Italy	1319	88.82	-	Rec	-	0.74	0.85	2.77	[23]
Bell tower of Santa Maria del Carmine	Italy	14th c.	68	3.54–0.84	Rec	S	0.69	0.76	2.28	[24]
Cathedral of Colima	Mexico	1889	31	1.5	Rec	B	1.40	1.62	-	[25]
Qutb Minar	India	1368	72.45	-	Cir	S	0.79	0.81	1.95	[26]
Santas Justa and Rufina Church bell tower	Spain	15th c.	35.5	1.5	Rec	B + S	2.15	2.24	5.95	[27]
Becci tower	Italy	-	39.4	1.5	Rec	S	1.37	1.67	-	[30]
Bell Tower of St. Sisto’s Church	Italy	1754	40	-	Rec	S	1.36	1.47	4.25	[34]
Minaret of Little Hagia Sophia	Türkiye	1762	35.39	0.8–0.35	Cir	S	1.449	1.495	6.302	[35]
Hagia Sofia Tower	Türkiye	1250–1260	23	0.9–1.5	Rec	-	2.59	2.71	6.37	[36]

* Rec: rectangular; Cir: circular; Hex: hexagonal. ** B: brick; S: stone.

2. Materials and Methods

2.1. Numerical Modeling: Finite Element Models

Two structural finite element (FE) models developed using shell and 3D solid elements were subjected to numerical analyses to investigate the differences and accuracy in their overall behavior. The numerical analysis models are based on the dimensions given in the restoration project prepared by Haznedar et al. [37], which is provided courtesy of the Construction and Technical Affairs Department of Istanbul University.

The ANSYS v.19.0 [38] program was used for solid model analysis. In the solid model, the window openings with truncated conical geometry on four sides of the structure, arched doors and arched windows on the Guard Floor, and the vaulted ceiling on the Guard Floor are modeled in detail. The three floors added later to the structure are modeled with different material properties from the primary material forming the structure, and the connection between the body of the structure and these three floors is defined. The helical staircase in the tower’s center is modeled as a 3D solid, and the properties of the wooden material are defined. Stairs and their connection with the main wall are defined as bonded in the analysis model. Square and circular cross-section tension rods that exist on the Guard Floor and steel floor beams of the Sign and Basket floors are also modeled in detail, and the connection between them is defined as bonded. Tension rods likely to be found in thick walls were not included in the analysis due to their unknown status and location. Steel clamps on the Sign, Basket, and Starboard floors are also not included in the analysis. The entire 3D solid model consists of 126,928 pieces of ten-node tetrahedral elements and 356,758 joints. Figure 5a,b shows general views of the solid model and details of the staircase and tension rods.

The shell model was analyzed using the SAP2000 v.15.0.0 [39] FE program. In the shell model, entrance doors, facade windows, arched windows on floors, and door openings are also taken into consideration. This model is more straightforward and does not include the staircase. Also, the tension rods and steel floor beams are not included. Shell elements representing the masonry walls are modeled by the axis through the centers of the wall thicknesses. Continuity is provided with rigid horizontal shell elements to comply with the geometry in cases where the cross-section is enlarged and narrowed. These rigid shell elements are considered non-weight elements with 1000 times the normal shell element stiffness and modulus of elasticity. The shell model has 3471 joints and 3412 thick shell elements.

In the shell and solid models, the support condition was defined as pinned for each node on the ground level. An overview of the shell model is given in Figure 5c.

2.2. Modal Analysis Using Finite Element Models

The tower was constructed using natural stone for the main walls and paving stone for the outer facades, which can be observed in the little window openings of the staircase, yet its material properties and origin remain unknown. Since there is a wide range of the modulus of elasticity, information found in the literature and regulations for these materials was investigated, and the average values for the material’s mechanical properties were considered for the preliminary analysis.

Table 2. The density and Young’s modulus of the rubble stone and limestone as reported in the literature.

	Density [kg/m3]	Ref.	Young Modulus [MPa]	Ref.
Rubble stone	2700	[40]	20,000	[40]
	1600	[41]	1100	[41]
	2579	[42]	1124	[43]
	2050–2120	[44]	240	[45]
	2400	[46]	1050–9000	[47]
Limestone	1870–2700	[48]	-	-
	1700	[49]	10,000	[49]
	1920–1970	[35]	11,500–13,000	[35]
	2190	[50]	24,000	[50]
	2390	[51]	5840	[51]

presents the density of rubble stone and limestone, as well as the modulus of elasticity values found in the literature. Since a destructive test by sampling in the Beyazıt Tower was not allowed, the actual mechanical properties could not be determined. Therefore, an average value was determined by considering the literature review and codes for the unit weight and modulus of elasticity used in the analysis models. For this reason, 1900 kg/m³ and 2400 kg/m³ values were chosen for the unit volume weight of limestone in the additional floors and rubble stone wall in the original part, respectively. The initial value of the modulus of elasticity was taken as 5000 MPa for the overall structure in both models.

The modal analysis results of the shell and solid models for the first and second modes revealed the same mode shapes and directions. The mode shapes for the shell and solid models are shown in Figure 6 and Figure 7, respectively, and the natural frequencies obtained in each analysis are given in

Table 3. Modal frequencies obtained from shell and solid FEM analyses.

Mode	Mode Shape	Modal Frequencies [Hz]
		Shell	Solid
1	Bending (N–S)	0.5761	0.5891
2	Bending (E–W)	0.5767	0.5896
3	Bending (E–W)	2.0842	2.2890
4	Bending (N–S)	2.0847	2.2907
5	Bending (E–W)	2.7997	3.0490
6	Bending (N–S)	2.8440	3.0568
7	Torsional	3.7354	3.5452

. Since the near-symmetry of the tower, the natural frequencies of the bending modes in two directions were close; in the third mode, frequency differences were increased, according to the differences in the details such as the steel floor beams in the additional floors and the tension rods in the Guard Floor.

2.3. Experimental Study

In historic masonry structures, destructive tests are needed to determine material properties such as modulus of elasticity, Poisson ratio, unit weight, and compressive strength, and this is often not permitted due to the historical importance of such structures. Determining some of these uncertain parameters for numerical analysis is essential for the accuracy of the analysis results. In this sense, destructive experiments were not allowed in Beyazıt Tower due to its historical importance. Therefore, modal analysis experiments, frequently applied to such structures today, are more suitable for determining dynamic characteristics and global elastic modulus. Operational Modal Analysis (OMA) through ambient vibration tests, one of the modal analysis methods, was preferred in terms of its applicability, considering the tower’s height. This section gives the implementation of modal analysis.

Ambient Vibration Tests

Vibration amplitudes will be very low as the wall thickness of the tower is high, and its flexibility is low. The appropriate test equipment was selected considering the frequency range obtained by the modal analysis carried out by FE analysis. For this reason, IEPE-type piezoelectric accelerometers, which are classified as seismic accelerometers, are used for precise measurements of low-amplitude vibrations. Two Dytran brand, 3191A1 model piezoelectric accelerometers with a sensitivity of 10 V/g in the range of ±0.5 g and a National Instruments brand NI 9234 model data acquisition unit with four analog input channels, including 24-bit resolution delta-sigma ADC, were used for recording the vibrations.

Since the piezoelectric accelerometers are uniaxial, the sensors were positioned perpendicularly to measure vibrations in two directions. No holes were drilled in the walls or floors for installation, considering the tower’s historical nature. For this reason, a rigid connection element was used for the orthogonal placement of the sensors, and this connection element was bonded to the floor with 3M brand transparent double-sided mounting tape. Figure 8a shows the 10 × 10 cm concrete connecting element, and Figure 8b shows a photograph taken during experiments in place.

Measurements were completed in 2 days, and 24 vibration records were obtained with piezoelectric sensors. The mode shapes obtained from the FE analysis were considered, and attention was paid to determining the measurement points regarding the modal amplitudes in each elevation.

Vibration recording started on 11 February 2018. Tests were carried out on the Guard (+45.40) and Sign floors (+55.50) at points 1 and 2. Among the sensors located close to the eastern façade of the building, A1 was placed parallel to the north–south direction, and A2 was placed radially in the east–west direction for orientation a. The measurements on the Sign Floor were made by positioning the sensors in the window opening. The position and direction of the sensors are drawn on the plans obtained from the restoration project prepared by Haznedar et al. [37] and are given in Figure 9 and Figure 10. The sampling frequency was taken as 1650 Hz, the default setting of NI 9234, in all records.

On 17 February 2018, the last day of the tests, measurements were carried out at four points. The first ambient vibration test was carried out at point 3 (see Figure 9a) on the Guard Floor, inside the North Facing Wall, close to the north–south direction of the structure. However, due to the unevenness of the floor ground on the radial direction representing the northern axis of the structure, the measuring point was placed with a slight offset. Measurement point 4 was chosen to identify the torsional mode because the modal amplitude of torsion was expected to be too low at the center of the tower (see Figure 9a).

The fifth and sixth tests were carried out in the staircase, on window openings, at elevations of +30.04 and +27.17. The sensor directions at these points are given in Figure 10a and Figure 10b, respectively.

The front facade of the tower presented in Figure 11 shows elevations for all measurements.

Table 4. Ambient vibration test points and their locations.

No.	Elevation [m]	Location	Orientation	No.	Elevation [m]	Location	Orientation
1	+45.40	1	a	13	+45.40	3	a
2	+45.40	1	a	14	+45.40	3	a
3	+45.40	1	a	15	+45.40	3	b
4	+45.40	1	a	16	+45.40	3	b
5	+45.40	1	a	17	+45.40	4	a
6	+55.50	2	a	18	+45.40	4	b
7	+55.50	2	a	19	+30.04	5	a
8	+55.50	2	a	20	+30.04	5	a
9	+55.50	2	b	21	+30.04	5	b
10	+55.50	2	b	22	+27.14	6	a
11	+45.40	3	a	23	+27.14	6	a
12	+45.40	3	a	24	+27.14	6	b

summarizes the information for the tests performed, including test numbers, elevations, locations, and sensor orientations defined as a and b.

3. Results

3.1. Modal Analysis Results

Vibration recordings taken in two directions at six different points of Beyazıt Tower were evaluated by the Frequency Domain Decomposition (FDD) method using Power Spectral Density (PSD), First Singular Value Decomposition (SVD) through Cross Power Spectral Density (CPSD), and stabilization diagram (SD) plots using the least-squares complex exponential (LSCE) algorithm of all records with the MATLAB v.2017b [52] program [53,54,55]. Figure 12, Figure 13, Figure 14 and Figure 15 show the PSD, first SVD, and SD results of the sensors A1 and A2 in the a and b orientation.

SD plots in Figure 12c, Figure 13c, Figure 14c and Figure 15c show that there is a good coherence between modal frequencies obtained by SVD for the determination of the tower’s modal frequencies. However, frequencies obtained at 4.738 and 5.285 Hz in the SVD diagrams needed to be clarified for the torsional mode. For this purpose, fast Fourier transform (FFT) was applied to the ambient vibration records taken at accelerometer locations 1 and 4 to determine the frequency associated with the torsional mode. Frequencies such as 4.738 or 5.285 Hz, observed at accelerometer locations 1 and 3 farther from the center of the structure, which could be associated with the torsional mode, will not be visible or will be visible with low amplitudes in the frequency decomposition of the vibration records at point 4. This phenomenon is illustrated in Figure 16 and Figure 17, which show the results of the FFT applied to the single time history records taken at accelerometer locations 1 and 4 for both directions.

Considering the results, the first and second modes are close to each other in both directions. The first bending modes were assumed to be 0.823 and 0.832 Hz, the second bending mode considering the third and fourth modes (flagpole-induced mode) was 2.15 and 2.193 Hz, the third bending mode considering the fifth and sixth modes was 2.86 Hz, and the torsional mode was 5.28 Hz.

3.2. Comparison of Solid and Shell Models

When comparing the numerical analysis results obtained from the 3D solid model analyzed by ANSYS [38] and the shell model analyzed by SAP2000 [39] under the initial conditions, significant differences were observed between the natural frequencies and the experimental results (see

Table 5. Comparison of experimental and non-updated numerical analysis results for shell and 3D solid models.

Mode	Experimental	Shell Model		Solid Model
	Frequency [Hz]	Frequency [Hz]	Difference [%]	Frequency [Hz]	Difference [%]
1	0.823	0.576	30.60	0.589	29.04
2	0.832	0.576	30.60	0.589	29.04
3	2.150	2.084	3.07	2.289	−6.47
4	2.193	2.084	3.07	2.290	−6.51
5	2.860	2.803	1.99	3.049	−6.61
6	2.860	2.848	0.42	3.056	−6.85
7	5.280	3.738	29.20	3.545	32.86

). One of the most essential facts for the differences is the inability to determine the modulus of elasticity correctly. In addition, the following issues could also be considered as the reasons why actual natural frequencies cannot be detected precisely in numerical models:

• Due to the difficulties in the modeling technique of the shell model, the inability to achieve actual geometry in the regions where the body thickness of the tower changes.
• Uncertainties about the presence of tension bars remaining in the walls, which cannot be included in the 3D solid and shell models.
• Inability to model the actual support conditions of architectural elements such as stairs.
• Neglecting the soil–structure interaction [56,57,58,59].

In this study, since the differences are considerable, the actual natural frequencies of the tower are examined with experimental results, and the numerical models are evaluated using the model update technique through the global modulus of elasticity for the main structural parts of the tower.

3.3. Comparison of Updated Numerical and Experimental Results

Experimental studies did not determine the experimental mode shapes due to single point measurements, so the model update was performed using only the natural frequencies obtained experimentally. Model update was performed iteratively by a calibration coefficient (C) given in Equation (1) to modify the modulus of elasticity (E) of the tower parts [60]. f_exp and f_num represent experimentally and numerically obtained model frequencies to be considered. The modulus of elasticity is modified as given in Equation (2) for sequent iteration steps. In Equations (1) and (2), i denotes the iteration steps.

$$C_i={\left({\displaystyle \frac{f_{exp}}{f_{num}^i}}\right)}^2$$

(1)

$$E_{i+1}=E_i{C}_i$$

(2)

Table 6. Update steps for 3D solid model according to modulus of elasticities.

Mode	Experimental	1st Iteration		2nd Iteration		3rd Iteration
	Frequency [Hz]	Frequency [Hz]	Difference [%]	Frequency [Hz]	Difference [%]	Frequency [Hz]	Difference [%]
1	0.823	0.828	−0.61	0.824	−0.12	0.823	0.00
2	0.832	0.829	0.36	0.825	0.84	0.824	0.96
3	2.150	2.226	−3.53	1.972	8.28	2.061	4.14
4	2.193	2.226	−1.50	2.034	7.25	2.116	3.51
5	2.860	4.25	−48.60	2.936	−2.66	2.957	−3.39
6	2.860	4.26	−48.95	2.964	−3.64	2.988	−4.48
7	5.280	4.983	5.63	4.937	6.50	4.937	6.50
		Em1 = 9869 MPa		Em1 = 9869 MPa		Em1 = 9869 MPa
		Efp = 185,270 MPa		Efp = 185,270 MPa		Efp = 201,617 MPa
				Em2 = 700 MPa		Em2 = 700 MPa

gives the update sequence for the solid model and the values of the modulus of elasticity in each update step. After the first iteration, the fifth and sixth modes could not be approached by proportional updating using a single modulus of elasticity for the masonry body. Therefore, a separate modulus of elasticity was defined for the additional tower stories where the highest modal deformations occur. In Table 6, E_m1 represents the modulus of elasticity of the original structure, E_m2 represents the additional floors, and E_fp represents the flagpole. Similar operations were performed for the shell element model, and the results are given in

Table 7. Update steps for shell model according to modulus of elasticities.

Mode	Experimental	1st Iteration		2nd Iteration
	Frequency [Hz]	Frequency [Hz]	Difference [%]	Frequency [Hz]	Difference [%]
1	0.823	0.828	−0.61	0.823	0.00
2	0.832	0.829	0.36	0.824	0.96
3	2.150	2.188	−1.77	2.144	0.28
4	2.193	2.188	0.23	2.145	2.19
5	2.860	4.021	−40.59	2.657	7.10
6	2.860	4.087	−42.90	2.676	6.43
7	5.280	5.374	−1.78	5.31	−0.57
		Em1 = 10,208 MPa		Em1 = 10,208 MPa
		Efp = 210,000 MPa		Efp = 210,000 MPa
				Em2 = 700 MPa

. For both models, the modulus of elasticity obtained after the update process was considered to be around 10,000 MPa for the original structure and 700 MPa for the additional stories.

The updated modulus of elasticity is compared with the experimental results as a percentage, as well as a graphical comparison suggested by Rainieri and Fabbrocino [54]. Figure 18 shows the relationship between the experimental and updated numerical analysis results for each analysis model. The results of the modal frequencies obtained for the shell and solid models are homogenous and distributed evenly around the 45° reference line. The coefficient of determination, the R² values, indicate that there is a good fit of the updated solid and shell model frequencies with the actual behavior of the tower. These results showed that the determined global modulus of elasticity is reliable for investigating the actual behavior of the tower.

4. Discussion

Two different analysis models were developed to determine the dynamic characteristics of the Beyazıt Tower. The preliminary results obtained from the models were evaluated before the experimental study; the frequency range was determined approximately, the modal shapes were examined, and points of low modal amplitudes were avoided in determining the elevations of the measurements.

In the shell model, wall thicknesses, windows, and door spaces are given in accordance with the original structure, but the geometry of the masonry body is not fully reflected. The 3-D solid model is the most detailed model in which the cross-section geometry and continuity of the masonry body are provided, and the tensions in the Guard Floor and the steel beams in the additional floors are added. However, due to the difficulty of modeling the steel carcass elements in the additional floors and the unknown situation of probable tensioners in the walls, it was not included in the tower model.

Due to the limited number of sensors, mode shapes could not be determined experimentally since only a biaxial vibration measurement was performed at a single point. Therefore, it was used to determine the measurement points in the tower by considering the mode shapes and natural frequencies obtained from the FE models. Measurements were conducted at various levels and different points within the structure to identify which modal frequency corresponds to the frequency peaks obtained experimentally through PSD and SVD; i.e., for determining the torsional mode at the center of the Seizure Floor and the determination of the second bending mode, vibration records were taken at the elevations of +30.05, +27.04, and +60.50. As a result of these measurements, although the frequency of the torsional mode was observed at points 1, 2, and 3, which are distant from the center of the Guard Floor, it was not observed at point 4 close to the center of the floor plan. In the vibration records carried out at different elevations, it was observed that the FFT amplitudes of the second and third bending modes were higher than the other modes.

As a result of ambient vibration tests conducted in Beyazıt Tower, the modal frequencies of the tower were determined as follows: first bending modes, 0.823 Hz and 0.832 Hz; second bending modes (induced by the flagpole), 2.150 Hz and 2.193 Hz; third bending modes, 2.860 Hz in both directions; and torsional mode, 5.28 Hz.

Based on the material properties identified in the literature, considerable discrepancies were detected in the natural frequency values of the initial analysis results, even when using the most detailed model. Therefore, updating the analysis models with appropriate material properties is necessary. Since the structure is close to symmetry and the mode shapes in both directions and natural frequencies are very close to each other, only the modal update methods considering natural frequencies are taken into account, and the tower analyzed by two different FE models was calibrated following the results obtained by the ambient vibration test. When the modulus of elasticity obtained for each updated model is compared, it is determined that the 3D solid model best matches the experimental results. The reason for this is that the geometry and details of the structure are reflected most realistically. In the model consisting of shell elements, although the details of the tower could not be given accurately, the wall thicknesses were provided by the dimensions given in the architectural plans. Therefore, the results obtained from the shell model coincide with the experimental results, and it is seen that the material properties are acceptable. As a result of the update processes, the overall modulus of elasticity for the material is found to be around 10,000 MPa in the original main body of the masonry tower and 700 MPa in the additional three stories.

It should be noted that in the experimental results, the natural frequencies observed at 2.150 Hz and 2.193 Hz are associated with the flagpole. In the preliminary modeling, this flagpole was not considered in either model. Even though it has a relatively lightweight mass compared to the mass of the structure, several difficulties were encountered in associating the numerical modal frequencies of the tower during the interpretation of the OMA results. If considered as an architectural element and not included in the model, these frequencies and mode shapes will not appear in numerical analyses and result in misleading interpretations. Therefore, it should be emphasized that the dynamic effects of a non-structural element, which is assumed to have no impact on structural behavior, cannot be ignored and may lead to misinterpretation of modal parameters unless reflected in the structural model.

Recent studies [58,59] reveal that soil–structure interaction has an influence on determining the damping and modal frequencies of structures through numerical models, especially for higher modes in bending rather than the first bending modes. This phenomenon could also be included in the analysis models for precise model updating processes after fieldwork is carried out for soil parameters.

The displacements obtained from the analysis with an improperly determined modulus of elasticity do not reflect the actual structure behavior. Within the scope of this study, the global modulus of elasticity of the main parts of the tower was determined by using ambient vibration tests, and thus, the displacements to be obtained by the structural analysis were provided to be closer to the actual behavior.

5. Conclusions

Using two different numerical analysis models, solid and shell, the modal frequencies and mode shapes of the historic Beyazıt Tower were determined. Additionally, the impact of model sensitivity on the results was evaluated through a model update process conducted via ambient vibration tests. Given the historical significance of the tower, the obtained modal frequencies contribute valuable information about the structure’s overall behavior to the literature. Due to the wide range of modulus of elasticity values for masonry structures, the global modulus of elasticity for Beyazıt Tower was examined through the model update process, leading to the determination of a specific value of about 10 GPa. Consequently, through various vibration measurements to be conducted on the tower over time, a foundation was established for monitoring the building’s dynamic characteristics and any changes that may occur due to natural disasters or human-induced effects on this unique structure.