Finite element modeling of existing masonry towers : the Asinelli Tower

CONTENTS

1. INTRODUCTION

1.1. BACKGROUND

1.2. PURPOSE

1.3. METHOD

1.4. LIMITATIONS

2. CASE STUDY

2.1. GENERAL OVERVIEW

2.2. STRUCTURAL CHARACTERISTICS

2.3. HISTORY OF THE TOWER

2.4. INTERVENTIONS AND ANALYSES

2.4.1. Recent reasearch and restoration projects

3. THEORETICAL FRAMEWORK AND METHODOLOGY

3.1. FINITE ELEMENT METHOD

3.1.1. Masonry structures: The Asinelli tower

3.1.2.Solid models

3.1.3. Soil-structure interaction

3.1.4. Discretization elements and interpolation models

3.1.5. Comparisons

3.1.6. Material modeling

3.2. LINEAR STATIC ANALYSIS

3.2.1. Global formulation

3.3. MODAL ANALYSIS

3.3.1. Free vibration eigen value problem

3.4. NONLINEAR STATIC ANALYSIS (PUSHOVER)

3.4.1. Modal pushover analysis (MPA)

3.4.2. Capacity curve

3.4.3. Reduction factor due to ductility

3.4.4. Seismic demand

4. RESULTS AND COMPARISON

4.1. FIRST COMPARISON

4.1.1. Linear static analysis

4.1.2. Modal analysis

4.2. SECOND COMPARISON

4.2.1. Modal analysis

4.3. THIRD COMPARISON

4.3.1. Nonlinear static analysis (Pushover)

5. CONCLUSIONS AND FURTHER STUDIES.

BIBLIOGRAPHY

FIGURES LIST

Figure 1. Asinelli tower’s location in Bologna’s city center

Figure 2. Bologna landscape view from ‘San Michele in Bosco'

Figure 3. The two towers of Bologna: Asinelli (left) and Garisenda (right)

Figure 4. Tower structural South section and cross-sections (SCA 1:500 / Units: meters)

Figure 5. Towered Bologna in 1505, from Francesco Francia’s fresco at Palazzo Comunale

Figure 6. Internal space of the tower

Figure 7. Interventions and analyses in the tower (XII - XIX centuries)

Figure 8. Most recent interventions and analyses in the tower (XX and XXI centuries)

Figure 9. Asinelli tower average hourly FFT on the 2 horizontal orthogonal components

Figure 10. Beam finite element model by Riva et al. (1998)

Figure 11. Shell finite element model by Ceccoli (2011)

Figure 12. Three-dimensional finite element model by Carpinteri et al. (2013)

Figure 13. Terrestrial laser-scanning test performed to the Asinelli tower

Figure 14. Simple l, Simple2 and Complex l models geometry

Figure 15. Modeling processes example for Simple2 model

Figure 16. Simple1 model’s meshing (without and with embeded solids)

Figure 17. Structural Winkler model

Figure 18. Comparison between no shear transfer model between springs and opposite

Figure 19. Spring stiffness distribution for Winkler model with higher stiffness at edges

Figure 20. Types of three-dimensional finite elements

Figure 21. CTE30 element

Figure 22. Simple2 model foundation’s mide-side nodes

Figure 23. Bar particle in solid element

Figure 24. Steel cables position and section dimensions

Figure 25. CT36I element’s topology and displacements

Figure 26. Interface element in Complexl model with boundary constraints

Figure 27. Simplel, Simple2 and Complexl models mesh

Figure 28. Multi-directional fixed crack model for plain strain

Figure 29. Secant crack stiffness

Figure 30. Relation between traditional and secant crack parameters

Figure 31. Properties for Structural Linear Static Analysis in MeshEdit

Figure 32. Free vibration of a two-story frame system in its fundamental mode

Figure 33. Properties for Structural Eigenvalue Analysis in MeshEdit

Figure 34. Equivalent SDOF model and bilinear capacity curve

Figure 35. Graphical method to determine seismic demand

Figure 36. Top displacements due to self-weight [m]

Figure 37. Vertical stresses due to self-weight [N/m2]

Figure 38. Comparison of natural frequencies between models and its references

Figure 39. Simplel model modal shapes

Figure 40. Simple2 model modal shapes

Figure 41. Complexl model modal shapes

Figure 42. Frequencies for different damage parameters (Mode 1, 3, 5 and 6)

Figure 43. Pushover curves for Simplel, Simple2 and Complexl models

Figure 44. SDOF and bilinear capacity curves for Simplel, Simple2 and Complex1 models.

Figure 45. Simple1 nonlinear cracking pattern

Figure 46. Simple2 nonlinear cracking pattern

Figure 47. Complex1 nonlinear cracking pattern

TABLES LIST

Table 1. Mechanical characteristics of the structural materials (Palermo (2015))

Table 2. Mechanical characteristics of the soil surrounding the tower (Palermo (2015))

Table 3. Experimentally measured periods (ranges) (Palermo (2015))

Table 4. One and two-dimensional finite element models by Palermo et al. (2015)

Table 5. Simple1, Simple2 and Complex1 models material properties

Table 6. Elastic coefficients to be entered in FX+ to the Complex1 model

Table 7. Values of q for historical towers

Table 8. Natural frequencies (f) and periods (T) of vibration (First comparison)

Table 9. Modal mass participation and principal direction

Table 10. Natural frequencies (f) of vibration for different d [Hz]

Table 11. Natural periods (T) of vibration for different D [s]

Table 12. Participation factor r for Simple1, Simple2 and Complex1 models

Table 13. Reduction factors

EQUATIONS LIST

Equation 1. Total strain in smeared crack model

Equation 2. Local crack strain and relation with global strain

Equation 3. Crack stress-strain relation

Equation 4. Crack stress-strain relation with traditional parameters

Equation 5. Damage parameter d

Equation 6. Elastic coefficients for transversely isotropic materials

Equation 7. Stiffness Matrix modified by the damage parameter d

Equation 8. Conditions for verification

Equation 9. Global formulation of FEM

Equation 10. Virtual work

Equation 11. Approximate solution for the global formulation of FEM

Equation 12. Matrix eigenvalue problem

Equation 13. Modal pushover load distribution in DIANA

Equation 14. Equivalent SDOF system from MDOF system

Equation 15. Reduction factor calculation

Equation 16. Inelastic AD spectrum

Equation 17. Bilinear capacity curve transformation to AD format

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