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Fig. 1 Two storeys colonnade with multi-drum columns and multi-blocks segmented trabeations [he structure under investigation (Fig. 1) is a two storey colonnade of the Forum in Pompeii made of white limestone. The colonnade was erected in the main square of the ancient town where political, economic and religious events were taking place. To prevent the passage of the carts, the pave was raised with respect to the height of the adjacent roads py two steps. It is believed that originally (seventh—sixth century B.C.) the Forum had an regular shape. Today, the Forum has a rectangular shape with dimensions 143 m long by 38 m wide. Probably, at the end of the second century B.C., a double row colonnade was puilt which was made of tuff. Some years later in 79 A.D the colonnade was reconstructed and the tuff was replaced with white limestone. The columns of the second storey followed he Ionic order while the columns of the lower storey followed the Doric order. In the second half of the twentieth century, only a small part of the second storey was re-erected or educational and touristic purposes. In 1980, an earthquake produced damages in the solonnades of the Forum and it was decided to remove the beam over the second storey. An “innovative solution” was adopted for the construction of the trabeation. To avoid ong span beams over the columns, short segments were built up providing opposing inclined patterned edges “flat arch” (Fig. 2a). This solution was conceived to simplify

Figure 1 Two storeys colonnade with multi-drum columns and multi-blocks segmented trabeations [he structure under investigation (Fig. 1) is a two storey colonnade of the Forum in Pompeii made of white limestone. The colonnade was erected in the main square of the ancient town where political, economic and religious events were taking place. To prevent the passage of the carts, the pave was raised with respect to the height of the adjacent roads py two steps. It is believed that originally (seventh—sixth century B.C.) the Forum had an regular shape. Today, the Forum has a rectangular shape with dimensions 143 m long by 38 m wide. Probably, at the end of the second century B.C., a double row colonnade was puilt which was made of tuff. Some years later in 79 A.D the colonnade was reconstructed and the tuff was replaced with white limestone. The columns of the second storey followed he Ionic order while the columns of the lower storey followed the Doric order. In the second half of the twentieth century, only a small part of the second storey was re-erected or educational and touristic purposes. In 1980, an earthquake produced damages in the solonnades of the Forum and it was decided to remove the beam over the second storey. An “innovative solution” was adopted for the construction of the trabeation. To avoid ong span beams over the columns, short segments were built up providing opposing inclined patterned edges “flat arch” (Fig. 2a). This solution was conceived to simplify

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Fig. 2 Different construction methods at the trabeation of the Forum in Pompeii (Sarhosis et al. 2015b). a Flat arch (segmented beam), b Solid long beam
Fig. 2 Different construction methods at the trabeation of the Forum in Pompeii (Sarhosis et al. 2015b). a Flat arch (segmented beam), b Solid long beam
Fig. 3. Mechanical representation of the contact between blocks (a); and joint behaviour under normal (b) and shear (c) loads When defining the model in UDEC, a single block covering the region to be analyzed 1 considered. The model features are then introduced by cutting the block into smaller one (e.g. block stones) whose boundaries represent discontinuities (e.g. joints). Blocks can tak any arbitrary geometry and may be different from each other in size and shape. Individua blocks can be considered as rigid or deformable. Rigid blocks are mainly used when th behavior of the system is dominated by the joints. Alternatively, the blocks can be mod elled as deformable and the complexity of the deformation of the blocks depends on th number of zone elements into which they are divided. Zones obey the constitutive mode assigned to them and, for each separate block, the strain can be estimated. In UDEC, th deformable block zones can be assumed to be linear elastic or non-linear according to th Mohr—Coulomb criterion ITASCA 2004). The material parameters for the linear, iso tropic, elastic model are the unit weight of the block (d), the compression modulus (K) anc the shear modulus (G).
Fig. 3. Mechanical representation of the contact between blocks (a); and joint behaviour under normal (b) and shear (c) loads When defining the model in UDEC, a single block covering the region to be analyzed 1 considered. The model features are then introduced by cutting the block into smaller one (e.g. block stones) whose boundaries represent discontinuities (e.g. joints). Blocks can tak any arbitrary geometry and may be different from each other in size and shape. Individua blocks can be considered as rigid or deformable. Rigid blocks are mainly used when th behavior of the system is dominated by the joints. Alternatively, the blocks can be mod elled as deformable and the complexity of the deformation of the blocks depends on th number of zone elements into which they are divided. Zones obey the constitutive mode assigned to them and, for each separate block, the strain can be estimated. In UDEC, th deformable block zones can be assumed to be linear elastic or non-linear according to th Mohr—Coulomb criterion ITASCA 2004). The material parameters for the linear, iso tropic, elastic model are the unit weight of the block (d), the compression modulus (K) anc the shear modulus (G).
Fig. 4 Geometry of the two storey colonnade of the Forum of Pompeii (Sarhosis et al. 2015b) where At, is the change in shear stress and Au, is the change in shear displacement. Stresses calculated at grid points along contacts are submitted to the Mohr—Coulomb failure criterion which limits shear stresses along joints. The following parameters are used to define the mechanical behaviour of the contacts: the normal stiffness (JKn), the shear stiffness (JKs), the friction angle (Jfric), the cohesion (Jcoh), the tensile strength (Jten) and the dilation angle (Jdil).
Fig. 4 Geometry of the two storey colonnade of the Forum of Pompeii (Sarhosis et al. 2015b) where At, is the change in shear stress and Au, is the change in shear displacement. Stresses calculated at grid points along contacts are submitted to the Mohr—Coulomb failure criterion which limits shear stresses along joints. The following parameters are used to define the mechanical behaviour of the contacts: the normal stiffness (JKn), the shear stiffness (JKs), the friction angle (Jfric), the cohesion (Jcoh), the tensile strength (Jten) and the dilation angle (Jdil).
Table 1 Properties of the limestone blocks (Kastemeier et al. 2010; Angrisania et al. 2010)
Table 1 Properties of the limestone blocks (Kastemeier et al. 2010; Angrisania et al. 2010)
Table 2 Properties of the joint interfaces (Kastemeier et al. 2010; Angrisania et al. 2010) 5 Response to static loading conditions Since the original columns that were placed adjacent to the structure have collapsed and thus the continuity of the “flat arch” beam above the columns has been lost, mechanisms can develop between inclined faces of the blocks which results in thrust forces and hence sliding of the blocks and opening of the joints. The response of the structure under gravity load has been examined.
Table 2 Properties of the joint interfaces (Kastemeier et al. 2010; Angrisania et al. 2010) 5 Response to static loading conditions Since the original columns that were placed adjacent to the structure have collapsed and thus the continuity of the “flat arch” beam above the columns has been lost, mechanisms can develop between inclined faces of the blocks which results in thrust forces and hence sliding of the blocks and opening of the joints. The response of the structure under gravity load has been examined.
Fig. 5 Displacement vectors (Jfric = 15°)
Fig. 5 Displacement vectors (Jfric = 15°)
ylocks in the trabeation is also 14°. Figure 7a, b show the principal stress distribution when joint friction angle is 36.8° and [5° respectively. From Fig. 7, stress intensities are reducing as the joint friction angle ncreases. When the joint friction angle is 36.8°, the maximum principal compressive stress n the structure is 0.27 MPa and the minimum principal tensile stress is 0.11 MPa. Sim- larly, when the joint friction angle is 15°, the maximum principal compressive stress in the structure is 0.54 MPa and the minimum principal tensile stress is 0.24 MPa. These values -orroborate the idea that blocks are rigid and no failure can be expected to occur in them. n fact, the strength in the blocks is expected to be at least one order of magnitude higher, yoth in tension and in compression (Angrisania et al. 2010). Also, for the continuous beam left part of the structure), high tensile stresses occurred at the bottom of the beam while ‘ompression occurred at the top of the overlapped elements. Conversely, the left layout Fig. 6 Joints opened in the structure
ylocks in the trabeation is also 14°. Figure 7a, b show the principal stress distribution when joint friction angle is 36.8° and [5° respectively. From Fig. 7, stress intensities are reducing as the joint friction angle ncreases. When the joint friction angle is 36.8°, the maximum principal compressive stress n the structure is 0.27 MPa and the minimum principal tensile stress is 0.11 MPa. Sim- larly, when the joint friction angle is 15°, the maximum principal compressive stress in the structure is 0.54 MPa and the minimum principal tensile stress is 0.24 MPa. These values -orroborate the idea that blocks are rigid and no failure can be expected to occur in them. n fact, the strength in the blocks is expected to be at least one order of magnitude higher, yoth in tension and in compression (Angrisania et al. 2010). Also, for the continuous beam left part of the structure), high tensile stresses occurred at the bottom of the beam while ‘ompression occurred at the top of the overlapped elements. Conversely, the left layout Fig. 6 Joints opened in the structure
Fig. 7 Principal stress distribution for different values of joint friction (tensile: red, compression: blue)
Fig. 7 Principal stress distribution for different values of joint friction (tensile: red, compression: blue)
Fig. 8 Geometry of the colonnade when the upper columns disestablished The role of the excitation frequency in the collapse mechanism was examined using sinusoidal excitations. The frequency of the excitation varied from 1 to 7 Hz and the amplitude of the base acceleration ranged from 1.7 to 80 m/s”. From the initial results analysis, it became apparent that high frequency motion requires large base acceleration amplitude to cause rocking motion and deformability between the individual blocks of the colonnade leading to collapse. Alternatively, a low frequency of excitation resulted in a monolithic response of the structure leading to collapse. Also, a parametric investigation has been carried out and the amplitude of the base acceleration was increased by small
Fig. 8 Geometry of the colonnade when the upper columns disestablished The role of the excitation frequency in the collapse mechanism was examined using sinusoidal excitations. The frequency of the excitation varied from 1 to 7 Hz and the amplitude of the base acceleration ranged from 1.7 to 80 m/s”. From the initial results analysis, it became apparent that high frequency motion requires large base acceleration amplitude to cause rocking motion and deformability between the individual blocks of the colonnade leading to collapse. Alternatively, a low frequency of excitation resulted in a monolithic response of the structure leading to collapse. Also, a parametric investigation has been carried out and the amplitude of the base acceleration was increased by small
Fig. 9 Geometry of the colonnade when the upper columns and the upper part of the beam disestablished
Fig. 9 Geometry of the colonnade when the upper columns and the upper part of the beam disestablished
Fig. 11 Joints opened (in red) at the structure; Jfric = 26° (collapse state) Fig. 10 Joints opened (in red) at the structure; Jfric = 16° (collapse state)
Fig. 11 Joints opened (in red) at the structure; Jfric = 26° (collapse state) Fig. 10 Joints opened (in red) at the structure; Jfric = 16° (collapse state)
Fig. 14 Safe-unsafe boundary of segmental beam model Fig. 15 Location of the blocks with large displacement
Fig. 14 Safe-unsafe boundary of segmental beam model Fig. 15 Location of the blocks with large displacement
6.2 Response to seismic excitations Four earthquake records with different characteristics such as the magnitude, the pre- dominant period, the frequency and the type of seismic wave (sinusoidal or not) were used in order to investigate how they affect the stability of the two storey colonnade. The records selected are shown in Figs. 18 and 19 and based on the same acceleration and time scales. The four records selected for this study are: the earthquake of Bucharest 1977, Irpinia 1980, Kalamata 1986, and L’ Aquila 2009. The 2007 Bucharest earthquake took
6.2 Response to seismic excitations Four earthquake records with different characteristics such as the magnitude, the pre- dominant period, the frequency and the type of seismic wave (sinusoidal or not) were used in order to investigate how they affect the stability of the two storey colonnade. The records selected are shown in Figs. 18 and 19 and based on the same acceleration and time scales. The four records selected for this study are: the earthquake of Bucharest 1977, Irpinia 1980, Kalamata 1986, and L’ Aquila 2009. The 2007 Bucharest earthquake took
Fig. 16 Safe-unsafe boundary of solid beam model
Fig. 16 Safe-unsafe boundary of solid beam model
Fig. 17 Comparison of segmental beam model and solid beam model
Fig. 17 Comparison of segmental beam model and solid beam model
Fig. 18 Acceleration Time History for selected earthquakes
Fig. 18 Acceleration Time History for selected earthquakes
Fig. 19 Acceleration response spectrum for 5 % damping ratio
Fig. 19 Acceleration response spectrum for 5 % damping ratio
Fig. 20 Maximum displacement of top left corner of system under scaled earthquake motion records with segmental beam
Fig. 20 Maximum displacement of top left corner of system under scaled earthquake motion records with segmental beam
Fig. 21 Maximum displacement of top left corner of system under scaled earthquake records with solid beam
Fig. 21 Maximum displacement of top left corner of system under scaled earthquake records with solid beam
Fig. 22 Maximum displacement of top left corner of system under scaled earthquake records with solid and segmental beam
Fig. 22 Maximum displacement of top left corner of system under scaled earthquake records with solid and segmental beam
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Historical ArchaeologyHistoric PreservationMasonry structures
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