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Seismic Behavior of Masonry Minarets: Nonlinear Static and Dynamic Assessment in DIANA

Comparative evaluation of pushover and nonlinear time history methods in DIANA for slender heritage structures.

Masonry minarets are particularly vulnerable under seismic loading due to their high aspect ratio, geometric discontinuities, and limited tensile capacity. Post-earthquake field surveys consistently report damage concentrated near the transition segment between the polygonal base and cylindrical shaft; a region characterized by abrupt stiffness change and stress concentration.

The referenced study evaluates the seismic response of a representative masonry minaret modeled in DIANA using two analysis strategies: nonlinear static (pushover) analysis and nonlinear dynamic time-history analysis. The objective is to quantify how each procedure represents displacement demand, cracking patterns, and collapse progression under increasing seismic intensity.

Technical Background

Previous analytical and field investigations of masonry towers have identified recurring damage mechanisms, primarily horizontal cracking above the transition segment and tensile stress concentrations at geometric discontinuities.

Pushover analysis is widely adopted in masonry assessment due to computational efficiency and alignment with Eurocode 8 procedures. However, static load patterns are approximations. They do not explicitly incorporate inertia effects or frequency-dependent amplification inherent in recorded ground motion. By directly comparing pushover results with nonlinear dynamic time-history analysis, the study evaluates how these methodological differences influence predicted structural response.

Geometry and Structural Configuration

The minaret examined in this research is a 15 m tall, brick-masonry structure representing a common typology across the Middle East, Central Asia, and South Asia. Its geometry is architecturally representative: an octagonal Pulpit base (outer diameter 2.40 m), a cylindrical shaft (outer diameter 2.20 m, wall thickness 0.40 m), a spiral internal staircase, a central column, and a timber Belfry (3.60 m tall) at the crown.​

The transition segment, where the octagonal Pulpit steps up into the cylindrical shaft, is the structural linchpin of the whole system. Every experienced engineer working with minarets knows it as the zone where stiffness changes abruptly and stress concentrations accumulate. This study set out to quantify exactly that, and to test how different numerical methods capture it.

The three-dimensional geometry shown in Figure 1 was generated in AutoCAD and imported into DIANA.

Figure 1. Geometry of the investigated minaret: (a) 3D model, (b) sectional view, (c) Section A–A, (d) octagonal pulpit geometry (dimensions in meters).

Finite Element Modeling in DIANA: Discretization and Numerical Rigor

Modeling Strategy

A continuum macro-modeling strategy was adopted to represent masonry as a homogenized material, capturing global structural behavior with high precision. Micro-modeling of individual bricks and joints was not employed due to calibration complexity and computational cost relative to the study’s global-response objective.

Discretization

  • Element type: HX24L solid brick element
  • Eight-node isoparametric formulation
  • Linear interpolation and Gauss integration
  • Element size: 0.10 to 0.20 m
  • Minimum three elements through thickness in critical regions
  • Six integration points through thickness

Boundary condition:

  • Base fully fixed in translation
Figure 2. Numerical model of the minaret: (a) HX24L, an eight-node isoparametric solid element used in the model, (b) general view and details of the FEM.

Constitutive Modeling

Material nonlinearity was represented using the Total Strain Rotating Crack Model (TSRCM). This strain-based formulation:

  • Simulates tensile cracking with softening behavior
  • Accounts for shear retention after cracking
  • Allows crack orientation to rotate with principal strain directions

The rotating crack assumption enables realistic simulation of crack reorientation under cyclic or multidirectional loading.

Material parameters were defined according to values reported in the study, including compressive strength, tensile strength, fracture energy, and elastic modulus consistent with masonry properties documented in the table 1 below.

Table 1. Linear elastic and nonlinear properties of the materials.

Eigenvalue (Modal) Analysis

Modal analysis was performed prior to nonlinear seismic simulation using the Implicitly Restarted Arnoldi method.

Six vibration modes were extracted, achieving approximately 93% cumulative effective modal mass participation in both horizontal directions, satisfying Eurocode 8 recommendations (≥90%).

Identified natural frequencies:

  • Mode 1 (2.62 Hz): Global bending (X-direction)
  • Mode 2 (2.68 Hz): Global bending (Y-direction)
  • Mode 3 (~12.45 Hz): Higher-order bending
  • Mode 4 (~12.45 Hz): Higher-order bending
  • Mode 5 (15.30 Hz): Torsional
  • Mode 6 (22.95 Hz): Vertical
Figure 3. Numerical mode shapes and the effective modal mass of the first six mode shapes of the numerical model. Deformation factor: 3.

The nonlinear dynamic analysis employed a time step of Δt = 0.001 s, consistent with the modal characteristics of the structure.

Analysis Procedures & Results

Nonlinear Static (Pushover) Analysis

Pushover analysis was performed in accordance with Eurocode 8 using two lateral load distributions:

  • Uniform Pattern Analysis (UPA): Forces proportional to mass
  • Modal Pattern Analysis (MPA): Forces proportional to mass and first-mode shape

Material and geometric nonlinearities were included.

Load factor results:

  • UPA: 0.17–0.19
  • MPA: approximately 0.10

Strain contours under monotonic loading show concentration of tensile demand at the transition segment between the octagonal base and cylindrical shaft.

Figure 4. Maximum principal strains for the MPA at maximum displacement (0.40 m): (a-d) PNL (Physical nonlinearity); (e-h) GNL (Geometric nonlinearity). Deformation factor: 2.

The pushover procedure identifies the critical region and provides global capacity curves but does not represent time-dependent inertial interaction.

Nonlinear Dynamic Analysis (NDA)

Time-history analysis was performed using the Bam earthquake record (Iran, 2003):

  • Magnitude: Ms 6.70
  • Horizontal PGA: 0.80 g
Figure 5. Bam earthquake: (a) acceleration time-history in X; (b) acceleration time-history in Y, (c) response spectrum in X, and (d) response spectrum in Y (ab: duration of the effective portion of the earthquake used for the analysis).

Numerical Integration and Damping

  • Time integration: Hilber–Hughes–Taylor (HHT-α) method
  • Time step: 0.001 s
  • Rayleigh damping ratio: 3%
  • Damping calibrated over a frequency range of approximately 2.60–87.27 Hz

Incremental Dynamic Scaling

To evaluate progressive damage development, the ground motion was scaled incrementally to intensity factors of:

0.1, 0.2, 0.3, 0.4, and 0.5

Figure 6. Time history of the relative displacements at the control point on top of the minaret for all dynamic analyses.

This approach enabled identification of the seismic intensity at which cracking initiated and how damage propagated with increasing demand.

Stability Threshold

The structure remains stable up to approximately 0.3Acc, with peak top displacement approaching 15 cm. Beyond this level:

  • At 0.4Acc and higher, significant instability develops
  • Collapse mechanisms initiate near the transition segment

Damage Progression Under Dynamic Loading

Dynamic analysis reveals a sequential damage pattern:

  1. Initial tensile cracking at the transition segment
  2. Vertical crack propagation along the shaft
  3. Increased flexural deformation
  4. Local instability and loss of stiffness
  5. Collapse mechanism formation at higher scaling levels

Crack initiation occurs within the first few seconds of excitation, with amplitude-dependent progression.

Figure 7. Envelopes of hysteresis curves for all acceleration levels and pushover curves: maximum load factor versus lateral displacement in(a) X and (b) Y directions.

Unlike pushover analysis, the dynamic procedure captures the interaction between inertia forces, cyclic demand, and stiffness degradation over time.

Comparative Observations

Within the scope of the study:

  • Pushover analysis identifies the critical region and global load capacity under monotonic loading.
  • Nonlinear dynamic analysis quantifies displacement demand under recorded seismic excitation and captures progressive instability under increasing intensity.

Differences between procedures become more pronounced at higher scaling levels, where inertia effects and modal interaction govern response.

While pushover analysis provides valuable insight into structural capacity, incremental dynamic analysis more directly represents time-dependent collapse mechanisms for slender masonry systems.

Engineering Implications

For slender masonry minarets:

  • Pushover analysis is suitable for preliminary vulnerability assessment and identification of critical regions.
  • Nonlinear dynamic analysis is necessary to evaluate displacement demand under realistic time-varying excitation.

The study demonstrates that dynamic amplification and modal interaction significantly influence response in high aspect-ratio masonry towers.

Within this framework, DIANA enables:

  • Continuum macro-modeling using solid elements for global response simulation
  • Nonlinear constitutive modeling with rotating crack capability
  • Modal extraction satisfying mass participation criteria
  • Stable nonlinear time integration using HHT-α under strong seismic input
  • Incremental dynamic scaling for progressive damage evaluation

Each of these capabilities is exercised directly in the comparative investigation.

Conclusion

The comparative assessment shows that while pushover analysis successfully identifies vulnerable regions in masonry minarets, nonlinear dynamic time-history analysis provides a more comprehensive representation of displacement demand and damage evolution under seismic excitation.

For slender heritage masonry towers, assessment procedures that incorporate inertia effects and time-dependent response are essential for realistic seismic vulnerability evaluation.

Through validated nonlinear constitutive modeling and advanced dynamic solution procedures, DIANA supports rigorous seismic assessment of complex masonry systems consistent with contemporary engineering practice.

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