## 3 Numerical Verification  
Noting the complexity of the closed form mathematical solutions it is proposed to use a Distinct Element Model (DEM) to model numerically the rocking and overturning response of rigid blocks under regular pulses. A commercial 2D software, the Universal Distinct Element Code, UDEC [78], was selected to model the response of a rigid block to repeated regular pulses. The equations governing two-dimensional rocking response are nonlinear, differential equations that do not admit a closed form analytical solution unless the equations are linearized and the ground motion is idealized. Thus, the need for a numerical solution that adopts an incremental time marching scheme. This can be  
Figure 2: (a) Safe–unsafe boundary for half sine impulse signal and (b) shape of critical half-sine impulses. accomplished using a differential equation solver, or alternatively, can be accomplished using a DEM. The solution scheme is very different from one method to the other. While the equation solver would solve the differential equations of the problem directly, the DEM approach would solve the dynamic equations of motion about the centroids of the blocks using springs at contact points between rigid blocks to control the overlap or inter-block penetration. The time integration in UDEC uses the central difference scheme which is second-order accurate as first-order error terms vanish from the solution. This is an important characteristic that prevents long-term drift in a distinct element simulation. In addition, the scheme is explicit dynamic and does not require any corrective iterations or solution tolerances. Furthermore, the UDEC environment allows for: visualization of results and future work expansion to model multiple blocks subjected to real earthquakes ground motions such as in large historical masonry structures. The proportions of the block adopted matched those of the roman columns found in the temple of [PERSON] in Baalbek, Lebanon with \(\alpha=\) b/h = 0.11. Three sizes of block were numerically modeled of heights 0.5, 1, and 20 m corresponding to available shaking table models for the 0.5 and 1 m [79] and the full size of the column at 20 m. Damping was neglected on the basis that the strong ground motion pulse of a near-fault earthquake signal, which is the cause for the loss of stability and overturning of the structure, is not affected by the actual value of the damping coefficient. Damping may have to be considered when energy is dissipated in the free vibrations of the structure going to rest. For simplicity, the damping coefficient was set equal to zero considering that the numerical analysis presented is focused on identifying the stability threshold at which overturning takes place during strong shaking [80]. The UDEC code models contact between blocks through placing springs along the normal and shear directions at contact points. The contact forces generated by these springs are governed by stress-displacement constitutive relations that are a function of the relative displacements at the ends of each spring. The spring stiffness is an input parameter to the code that should be chosen judiciously: (1) it should be high enough to limit the amount of overlap between the contacting blocks to capture the actual conditions and (2) it should be low enough to obtain tractable execution times, knowing that the stable time increment is inversely proportional to square root of the stiffness. A schematic of the UDEC numerical model is shown in Figure 3a consisting of two rigid blocks: a base block where the ground motion is prescribed and another block that is resting on it. Contact forces between the blocks are determined using: a spring in the direction of contact normal with stiffness = K\({}_{n}\); a spring in the shear direction with stiffness = K\({}_{s}\); and a Coulomb slip model. Figure 3: \(Cont.\)  
In UDEC, the default contact overlap tolerance is one-half the rounding length at the corners. The rounding length in the analysis was set to 0.1% width of the block.