Egg White and Eggshell Mortar Reinforcing a Masonry Stone Bridge: Experiments on Mortar and 3D Full-Scale Bridge Discrete Simulations

Abstract

In this study, experimental and numerical investigations were conducted to examine the time-dependent creep and earthquake performance of the historical Plaka stone bridge, which was constructed in 1866 in Arta, Greece. During the original construction of the bridge in 1866, Khorasan mortar with an egg white additive was used between the stone elements. Furthermore, when the bridge underwent restoration in 2015, Khorasan mortar with an eggshell additive was employed between the stone elements. Consequently, two distinct 3D finite-difference models were developed for this study. In the first bridge model, egg white was used in the Khorasan mortar, replacing water at various proportions of 0%, 25%, 50%, 75%, and 100%. In contrast, for the second model, eggshell was incorporated into the Khorasan mixture at percentages of 25%, 50%, 75%, and 100%, relative to the lime amount. Subsequently, the mortars were subjected to curing periods of 1 day, 7 days, and 28 days, and their mechanical properties were determined through unconfined compression strength experiments. Taking into account the determined strengths of the mortars, the kn and ks stiffness values of the interface elements between the stone elements and Khorasan mortar were calculated. In the 3D model, each stone element was individually represented, resulting in a total of 1,849,274 stone elements being utilized. Non-reflecting boundary conditions were applied to the edge boundaries of the bridge model, and the Burger creep and Mohr–Coulomb material models was employed for time-dependent creep and seismic analyses, respectively. Subsequently, time-dependent creep analyses were conducted on the bridge, and seismic events that occurred in the region where the bridge was located were simulated to assess their impact. Based on the results of the time-dependent creep and seismic analyses, we observed that the use of 50% eggshell-mixed Khorasan mortar between the stone elements had a positive influence on the earthquake and creep behaviors of both restored and yet-to-be-restored historical bridges.

1. Introduction

Historical structures play a crucial role in shedding light on the history of mankind. Preserving and transmitting these structures to future generations holds immense significance. Therefore, safeguarding historical buildings under significant external loads such as earthquakes and floods, and implementing earthquake retrofitting measures, are of vital importance. Historical structures were shaped according to the needs of their inhabitants, with each possessing unique architectural characteristics. One of the most important of these structures that shed light on our day is the historical stone arch bridges. These bridges were constructed in regions where communities relied on river transportation. They were frequently used by the local population during the Ottoman Empire and have managed to endure until the present day. Typically, these structures were built using stone elements that were compatible with the geological makeup of the construction area. The length of the stone elements was generally similar, and special mortars were applied between each stone to enable interaction. The Ottoman Empire utilized a specific type of mortar known as Khorasan mortar to facilitate interaction between the stone elements. This mortar, composed of special materials, provided substantial bonding between the stone elements.

Investigating the composition of this mortar and its utilization in the restoration and reinforcement of historical bridges today is of utmost importance. Recently, studies on historical buildings have increased. However, there is not much information and study in the literature about Khorasan mortar, which is used among the stone elements in historical buildings. Bouzas et al. [1] examined the impact of variations in the thickness dimensions of arches on the assessment of a masonry bridge’s load-carrying capacity. The study began with an experimental campaign utilizing geomatic techniques to characterize the bridge’s geometry. Subsequently, a limit analysis model was developed based on the outer dimensions of the arches. The study highlighted the importance of considering both the outer and inner dimensions of the arches in the probabilistic model, as relying solely on the outer dimensions can lead to an overestimation of the load capacity. Majtan et al. [2] examined the structural behavior of masonry arch bridges under extreme flood flows and impact forces from flood-borne debris. The study employed a validated numerical modeling approach, utilizing the smoothed particle hydrodynamics (SPH) method to simulate fluid behavior and determine pressure distributions on a single-span arch bridge. The findings demonstrated that debris impact significantly increased stress levels, particularly when the abutment was fully submerged and the debris had a side-on orientation. Silva et al. [3] introduced two nonlinear finite element modeling approaches for the assessment of damaged masonry arch bridges. These strategies were applied to the Leça railway bridge, serving as a case study, considering different damage scenarios and railway traffic loading. The proposed numerical strategies demonstrated potential in representing localized damage, particularly longitudinal cracks in various opening conditions, in masonry bridges. Furthermore, these calibrated numerical strategies proved to be computationally efficient for nonlinear dynamic analysis under service loading. Ozakgul et al. [4] focused on assessing the structural behavior of a 94-year-old reinforced concrete open-spandrel arch bridge. Dynamic tests, including acceleration measurements, were performed to understand the bridge’s in situ behavior. A 3D finite element model was created based on the original constructional drawings and updated using the Response Surface Method. Structural assessment and evaluation were conducted under UIC-71 live loading, obtaining rating factors and reliability indices for each bridge member using the Load and Resistance Factor Rating approach. The results indicated that the deck of the bridge was the most critical member, suggesting potential strengthening and retrofitting requirements for higher live loads. Chen et al. [5] determined the damage on a historical bridge using different methods. Sonar techniques were used to detect underwater foundation damage in Gongchen Bridge. The results showed that NDT technology improved work efficiency and accurately identified the damage situation of the foundation. Then, it was recommended to reinforce the foundation, seal exposed wooden piles, and remove underwater obstacles for the bridge’s stability. Gönen and Soyöz [6] performed a seismic assessment of masonry arch bridges and highlighted the importance of considering their nonlinear behavior. Initial testing was conducted to gather information on the bridge’s geometry and material properties, and dynamic identification using ambient vibrations was performed to enhance the accuracy of the finite element model. Nonlinear static (NSA), nonlinear dynamic (NDA), and incremental dynamic analyses (IDA) were carried out to assess the bridge’s seismic response. NSA provided conservative displacement results but fell short in capturing the behavior and damage mechanisms associated with higher vibration modes. Moreover, NDA yielded more reliable results but required significant computational resources. Saygılı and Lemos [7] investigated the seismic behavior of two historical stone masonry bridges in Turkey using finite- and discrete-element approaches. The Kazan Bridge exhibited the capacity to withstand seismic excitations, although some damage was predicted due to excessive stresses. Moderate damages, such as slight openings, were observed at the mid-span of the arch. On the other hand, the Şenyuva Bridge was found to be significantly vulnerable and prone to strong damage or collapse during earthquakes with a long return period. It was seen that bridge geometry, including arch span and deck width, influenced the seismic response, with longer spans increasing vulnerability. Papa et al. [8] investigated the arch-backfill interaction problem in masonry bridges using an adaptive NURBS-based limit analysis approach. Comparisons with experimental results and other analysis approaches demonstrated the significance of direct modeling in accurately representing the failure response of the bridge. Zani et al. [9] investigated the response of the Azzone Visconti bridge, a historical masonry arch bridge in northern Italy, to vertical static loads. A detailed 3D nonlinear finite element model was created to assess the effect of soil–structure interaction (SSI) on the bridge’s mechanical behavior. The results showed that the model with SSI and soil nonlinearity accurately reproduces the bridge’s vertical settlements under load tests. Chen et al. [10] examined the weathering of stone ashlars used in the construction of Guyue Bridge in Yiwu, Zhejiang. It was concluded that stone weathering in historic structures is primarily influenced by atmospheric conditions and internal stress caused by loading. Sánchez-Aparicio et al. [11] proposed a non-destructive multidisciplinary approach for the structural diagnosis of masonry arch bridges. The proposed methodology was validated on a case study of a Roman bridge in Spain, showing promising results. Simos et al. [12] focused on earthquake-induced damage on a stone arch bridge (Konitsa Bridge) in an earthquake-prone zone. The bridge, located on an active fault, survived a near-field earthquake in 1996 with no damage. Field studies and laboratory tests were conducted to establish the bridge’s dynamic characteristics and material properties. The study compared the damage caused by near-field and far-field earthquakes and far-field earthquakes were found to be more destructive. Aydin and Özkaya [13] focused on non-linear analyses of arched structures, examining their behavior and the occurrence of cracks and fractures under different loads. It was seen that the load distribution before collapse exhibited a decreasing tendency near the L/2 section, while variations after the L/4 section towards the left and right ends differed, albeit slightly, from the filler and side walls. Conde et al. [14] proposed a multidisciplinary approach to analyze masonry arch bridges, with the Vilanova Bridge in Allariz, Spain, serving as a case study. The results indicated that the cohesion of fill materials and the non-linear properties of masonry significantly influenced the bridge’s structural behavior. Based on numerous numerical simulations, the Vilanova Bridge demonstrated an acceptable safety level, but lower service loads were recommended for preserving the structure’s life expectancy. Karaton et al. [15] investigated the nonlinear seismic responses of the Malabadi Bridge, an Artuqid structure constructed in 1147 on the Batman River in Turkey. Seismic acceleration data were generated for three different levels (D1, D2, and D3) based on the seismic characteristics of the region where the bridge is located. Under D1 and D2 earthquake loadings, no damage was observed in the arches and spandrel walls of the Malabadi Bridge. However, plastic deformations were observed in the backfill material under D2 earthquake loadings. Conde et al. [16] focused on the structural analysis of a masonry arch bridge situated in Galicia, Spain. The bridge’s geometric characterization was conducted using terrestrial laser scanning, providing an accurate and detailed description of the arches. The results indicated that without precise data, caution must be exercised when making idealizations about the geometry of stone arches for numerical calculations, as it often leads to an overestimation of the predicted collapse load. Costa et al. [17] presented an overview of numerical strategies employed to assess the load-carrying capacity of stone masonry arch bridges. Detailed numerical models were developed using the finite element method (FEM), discrete element method (DEM), and rigid block analysis. The results demonstrated a strong correlation between the numerical response of the bridge models and the maximum load, as well as the corresponding hinge mechanism. Bayraktar et al. [18] focused on the operational modal analysis of eight historical masonry arch bridges in Turkey, namely the Aspendos, Pehlivanlı, Mikron, Osmanlı, Şenyuva, Şahruh, Osmanbaba, and Torul bridges. Ambient vibration data were utilized to determine the experimental frequencies, damping ratios, and mode shapes of these bridges. It was seen that the first and second mode shapes of the bridges exhibit bending behavior, while the third mode shape predominantly shows a vertical form. Moreover, the first, second, and third frequencies range between 4–8 Hz, 6–10 Hz, and 8–12 Hz, respectively. Ozmen and Sayin [19] observed the effect of soil structure interaction (SSI) on the seismic behavior of masonry bridges and found significant differences when comparing the model of fixed base and SSI. Shabani and Kioumarsi [20] investigated the effect of three different strengthening models on the bridge. Two methods include using polyparaphenylene benzobisoxazole and carbon fiber-reinforced concrete mortar layers to cover the pier of the bridge and the other one is improving all mechanical properties of indefensible part of the bridges. Effects of the usage of different additives (egg white and eggshell) on Khorasan mortar were not examined in those studies [19,20]. Vuoto et al. [21] explored digital twin concepts for the conservation of cultural heritage buildings. Their study highlights the transformative potential of the digital twin paradigm in preserving Built Cultural Heritage (BCH) assets. Addressing the fragmented nature of current conservation efforts, the utilization of digital twin technology offers a means to enhance management processes through precise asset modeling and advanced analytics. Furthermore, the study emphasizes the importance of collaborative research and interdisciplinary cooperation to fully realize the benefits of this paradigm. Supported by EU and global policies advocating for sustainable BCH utilization, digital twin technology emerges as a crucial tool in safeguarding cultural heritage for future generations. Ferretti and Pascale [22] analyzed the strengthening technics for masonry buildings. The CAM system improved the seismic performance of masonry buildings. Hafner et al. [23] compared the methods for strengthening masonry walls and piers. Angiolilli et al. [24] utilized a reinforced cementitious matrix for strengthening brick and stone masonry walls, reporting a remarkable 420% increase in shear strength. Ferretti et al. [25] examined the combined effect of fiber-reinforced cementitious matrix composite reinforced mortar on structural strength, yielding an average increase of 53.61% in strength. Triantafillou [26] conducted analyses on the impact of textile-reinforced mortar, demonstrating its ability to enhance the strength of unreinforced masonry walls. Kouris and Triantafillou [27] proposed a design methodology for strengthening textile-reinforced mortar, validating it through experiments and establishing a good correlation for masonry structures. Carozzi and Poggi [28] employed fabric-reinforced cementitious matrix to strengthen masonry structures, with polyparaphenylene benzobisoxazole fiber exhibiting the highest strength and increase in performance among the fibers studied. Grande and Milani [29] investigated the fiber-reinforced cementitious matrix strengthening system’s impact on the tensile strength of masonry, proposing a model and validating it through experimental tests. Lemos [30] explored the discrete element method on various masonry structures, affirming its suitability for determining the dynamic behavior of masonry monuments. Gobbin et al. [31] introduced new algorithms to define the seismic behavior of masonry structures using the discrete element method, finding the rigid body method suitable for simple problems. Masi et al. [32] developed a model for testing masonry structures through experiments, with the discrete element model proving successful in modeling the dynamic response of the structures.

As can be seen from the studies in the literature, there is no study on the use of Khorasan mortar with egg white and eggshell to examine the structural damage behavior of historical buildings. For this reason, this study adds new information to the literature about the structural behavior of historical bridges and the use of Khorasan mortar in historical bridges.

2. General Information about the Content of the Study

The name “Khorasan” is known as a district located in Iran. It is also used instead of concrete in Saudi Arabia. In this study, we evaluated the effects of using Khorasan mortar with different additives as an interface element between the discrete stone elements in historical bridges on time-dependent creep and seismic behavior. The bridge was constructed in 1866 and later destroyed during the 2015 flood. During its original construction in 1866, Khorasan mortar with egg white was employed to ensure the interaction between the stone elements. Therefore, in the initial phase of the study, Khorasan mortars with egg white additives were prepared in the laboratory, and the strengths of the 1-, 7-, and 28-day Khorasan mortars were comprehensively assessed. The mechanical properties obtained from these experiments were used to calculate the properties of the interface elements between the discrete stone elements in the 3D finite-difference model of the bridge. The Burger creep material model was applied to the stone elements and foundation while modeling the bridge, and non-reflecting boundary conditions were defined for the bridge’s boundaries. Each stone element of the bridge was individually modeled. Since the bridge experienced a total of 5 flood disasters, the water level was taken into account, matching the bridge’s height during these flood events. Furthermore, excluding floods, the water level was considered to be 1/6 of the bridge’s height. Taking these water levels into consideration, it was investigated the time-dependent creep behavior of the bridge from 1866 to 2015. As a result of the creep analyses, significant failures were observed around the auxiliary arch parts of the bridge. In the subsequent phase of the study, it was examined the long-term creep and seismic behaviors of the bridge, which was reconstructed following its collapse in the 2015 flood, after using Khorasan mortar with eggshell additives between the stone elements. Khorasan mortars with eggshell additives were prepared in the laboratory, and 1, 7, and 28-day Khorasan mortars were subjected to unconfined compression strength tests. The experiments allowed us to determine the mechanical properties of the eggshell-added Khorasan mortar, which were then employed for the mechanical properties of the interface elements, namely kn and ks stiffness elements, between the discrete stone elements in the bridge model. Subsequently, the bridge model was subjected to the 2023 Kahramanmaraş earthquakes. The seismic analyses revealed that there was no significant seismic damage observed during the earthquake in the historical bridges using Khorasan mortar with eggshell additives between discrete stone elements. Additionally, it was concluded that the use of Khorasan mortar with eggshell additives between stone elements during the reconstruction or restoration of historical bridges provides significant positive contributions to the seismic behavior of the bridge.

3. Burger Creep Material Model

The Burger creep model is a viscoelastic material model that combines elastic and viscous behavior to simulate the time-dependent deformation of materials. It is based on the assumption that materials exhibit both instantaneous elastic strains and time-dependent creep strains when subjected to stress. The model assumes non-linear elasticity, meaning that the stress–strain relationship follows Hooke’s Law [33]. Burger’s model consists of a series connection of a Kelvin model and a Maxwell model.

According to Hooke’s Law, the stress (σ) in a material is proportional to the strain (ε) it experiences, with the proportionality constant being the elastic modulus (E) of the material. The elastic modulus represents the stiffness of the material and determines how it deforms under load. In addition to the instantaneous elastic strain, the Burger creep model incorporates creep, which refers to the time-dependent deformation that occurs in materials under constant stress [33]. The creep strain in the model is described by a creep law, which relates the applied stress to the resulting creep strain over time. The creep behavior is typically characterized by a creep compliance function, which defines the relationship between stress and strain rate. The creep compliance function can take different forms depending on the specific material and its behavior. It can be defined using experimental data or mathematical models that represent the creep behavior of the material accurately. When using the Burger creep model in FLAC3D, the material properties such as Young’s modulus (E) and Poisson’s ratio (ν) are specified to define the elastic behavior of the material [33]. These properties determine the material’s response to instantaneous loading. To capture the time-dependent creep behavior, the parameters provided are related to the creep compliance function. These parameters can include the initial creep strain, the characteristic time constant, and the shape of the creep compliance function. These values are typically determined through laboratory tests or obtained from material property databases. The Burger creep model in FLAC3D allows researchers to simulate the long-term behavior of materials subjected to sustained loading or changing stress conditions over time. It is particularly useful in geotechnical engineering applications, such as analyzing the settlement of soil under a structure or the behavior of rock masses in underground excavations. It is important to note that the specific details and implementation of the Burger creep model may vary depending on the version of FLAC3D and the specific requirements of the analysis to be performed [33]. The Burger creep material model is vital for analyzing historical structures because it accurately simulates how these structures deform over time due to various loads. It helps engineers predict long-term performance issues and plan maintenance accordingly, ensuring the preservation of these valuable assets. This model also assists in assessing risks related to time-dependent effects, enabling the development of strategies to safeguard historical structures.

4. Free-Field and Quiet Non-Reflecting Boundary Conditions

In FLAC3D, non-reflective boundary conditions are employed to model boundaries that allow waves or stresses passing through them to be fully absorbed without reflecting into the model domain. These boundary conditions are particularly crucial when dealing with semi-infinite or unbounded domains, where minimizing the influence of boundary reflections on the results is essential. Non-reflective boundary conditions are commonly used in wave propagation analyses and other dynamic simulations. They are implemented as absorbing boundaries, effectively dissipating incident waves to prevent reflections back into the model domain [33]. These boundaries are especially valuable in modeling wave propagation phenomena, such as seismic waves or acoustic waves, as they ensure a more accurate representation of wave behavior within the model. Two significant non-reflective boundary conditions in FLAC3D are the free-field (FF) boundary condition and the quiet boundary condition. The free-field boundary condition is a fundamental feature in FLAC3D, a powerful numerical modeling tool designed for simulating geo-mechanical problems in three-dimensional space. It represents the behavior of the model’s exterior domain, assuming that the displacements and stresses exerted by the surrounding free-field are negligible. In this approach, the analysis domain is represented with a finite element mesh, and the free-field boundary condition defines the behavior beyond this mesh, assuming uniform and known material properties and displacements [33]. This assumption holds when the model’s boundaries are positioned far enough from the area of interest, ensuring minimal influence on the region of interest. Users specify the external boundaries where the free-field boundary condition is imposed, typically selecting specific nodes or elements at the outer limits of the mesh and designating them as free-field boundary nodes/elements. Material properties of these nodes/elements are set to represent the properties of the surrounding free-field medium, often with zero displacements (both translational and rotational) to reflect the assumption of negligible external influence [33]. The lateral boundaries of the main grid are connected to the free-field grid through the use of viscous dashpots, creating a simulation of a quiet boundary (Figure 1). The unbalanced forces originating from the free-field grid are then imposed on the main-grid boundary. These conditions are mathematically described by the following three equations, which apply to the free-field boundary along one side of the boundary plane, with its normal aligned in the x-axis direction. Similar formulations can be derived for the other side and for the corner boundaries (Equations (1)–(3)) [33]. The density of the material along the vertical model boundary is denoted by the symbol $\rho$. The P-wave speed at the side boundary is represented by the variable $C_p$. The S-wave speed at the side boundary is given by the parameter $C_s$. The area of influence of the free-field grid-point is defined as A. The velocity of the grid-point in the main grid at the side boundary in the x-direction is designated as $V_x^m$. $V_y^m$ signifies the velocity of the grid-point in the main grid at the side boundary in the y-direction. $V_z^m$ corresponds to the velocity of the grid-point in the main grid at the side boundary in the z-direction. $V_x^{ff}$ stands for the velocity of the grid-point in the side free field in the x-direction. Similarly, $V_y^{ff}$ represents the velocity of the gridpoint in the side free field in the y-direction. Lastly, $V_z^{ff}$ denotes the velocity of the gridpoint in the side free field in the z-direction. The free-field grid-point force with contributions from the stresses of the free-field zones around the grid-point in the x-direction is expressed by the variable $F_x^{ff}$. Likewise, the free-field grid-point force with contributions from the stresses of the free-field zones around the grid-point in the y-direction is denoted by $F_y^{ff}$. Lastly, $F_z^{ff}$ stands for the free-field grid-point force with contributions from the stresses of the free-field zones around the grid-point in the z-direction [33].

$$F_x=-\rho C_p\left(V_x^m-V_x^{ff}\right)A+F_x^{ff}$$

(1)

$$F_y=-\rho C_s\left(V_y^m-V_y^{ff}\right)A+F_y^{ff}$$

(2)

$$F_z=-\rho C_s\left(V_z^m-V_z^{ff}\right)A+F_z^{ff}$$

(3)

The quiet boundary condition, on the other hand, is a specialized feature in FLAC3D, designed to simulate boundaries where waves and stresses passing through are entirely absorbed without reflection. It is particularly useful when dealing with semi-infinite or unbounded domains, ensuring that unwanted boundary reflections do not affect the accuracy of the simulation results. Like the free-field boundary condition, it is strategically applied to specific regions on the model’s exterior boundaries, absorbing waves, stress waves, and displacements entirely, to simulate an infinitely distant boundary [33]. Its implementation is crucial for accurately simulating wave propagation and dynamic problems, effectively minimizing artificial reflections that could compromise the accuracy and stability of the simulation. Moreover, the quiet boundary condition excels in accurately analyzing wave propagation, making it particularly valuable in simulations involving seismic waves or similar dynamic phenomena, where precise representation without boundary reflections is essential [33]. Additionally, eliminating reflections significantly contributes to maintaining numerical stability in dynamic simulations, reducing artificial oscillations, and ensuring more reliable and consistent results. A notable advantage of the quiet boundary condition is its ability to simulate unbounded or semi-infinite domains without the need to explicitly model the entire surrounding space, resulting in substantial computational resource savings [33].

5. General Information about Plaka Masonry Bridge

The Plaka Stone Bridge, located in the city of Arta, Greece, is an iconic historical structure with significant cultural and architectural value. Originally constructed during the Ottoman Empire in 1866, this bridge served as a vital transportation link for the local community, facilitating river crossings and trade routes. Over the years, the Plaka Stone Bridge faced numerous challenges, including natural disasters and the test of time. In 2015, a devastating flood caused the collapse of the bridge, necessitating extensive reconstruction and restoration efforts. From the mechanical point of view, the size effect is the reason why the masonry bridge collapses or is heavily damaged during floods [34]. Recognizing the importance of preserving this remarkable piece of history, a comprehensive renovation project was initiated to revive the bridge and ensure its continued existence for future generations. The reconstruction process involved meticulous planning, advanced engineering techniques, and the use of traditional construction methods to retain the bridge’s original character and architectural integrity. Skilled craftsmen and engineers collaborated to painstakingly replicate the intricate stonework and structural elements that defined the bridge’s unique charm. In addition to physical restoration, comprehensive studies and assessments were conducted to enhance the bridge’s resilience and ability to withstand external forces, such as earthquakes and heavy rainfall [35]. The renovated Plaka Stone Bridge stands as a testament to the harmonious blend of historical preservation and modern engineering. It not only re-establishes the vital transportation link it once provided but also symbolizes the cultural heritage and resilience of the local community. The restored bridge serves as a captivating landmark, attracting visitors from around the world, while also honoring the legacy of those who originally built and traversed this remarkable structure. In conclusion, the Plaka Stone Bridge in Arta, Greece, stands as a remarkable example of historical architecture and engineering prowess. The meticulous reconstruction and incorporation of innovative materials ensure its longevity and continued significance as a cherished historical monument. The mechanical properties of the stone elements of the bridge are shown in

Table 1. Material properties of the stone.

Item	Density (kg/m3)	Modulus of Elasticity (Pa)	Poisson Ratio	Cohesion (kPa)	Friction Angle (Degree)	Dilatancy Angle (Degree)
The Lowest Main Arch	2628	7.05 × 107	0.28	3970	47.1	46.2
The Middle Main Arch	2583	6.52 × 107	0.26	3842	47.0	46.1
Supporter Arch (Right)	2280	5.86 × 107	0.21	3370	47.5	47.3
Supporter Arch (Left)	2152	5.24 × 107	0.19	2975	42.6	41.6
Rock Material	2300	4.75 × 107	0.25	1750	36.5	36.2
Foundation	3120	9.00 × 107	0.38	4817	49.6	48.5

. Szabo et al. [36] delineated six distinct patterns for masonry buildings, and the Plaka bridge was characterized as Type D (regular masonry from soft stones) in this study. Furthermore, Szabo et al. [36] classified typologies into two groups: Class 1 and Class 2, with the subject bridge falling under Class 2, denoting “good quality masonry”. Additionally, the construction technique of the Plaka Bridge was meticulously detailed in the study conducted by Giannelos et al. [37]. The examination revealed that the central arch of the bridge, initially perceived as semi-circular, exhibits a more intricate geometry upon closer inspection. A topographic survey unveiled a dual curvature within the arch structure: from its base to a 60° angle, it conforms to an incircle with a radius (R1) of 20.05 m, while transitioning to a smaller incircle with a radius (R2) of 17.41 m towards its apex [37]. The center of the smaller incircle is situated 3.18 m above that of the larger one, resulting in a 0.53-meter elevation of the apex. This curvature variation significantly influences the distribution of structural stress, as elaborated in Section 6. Furthermore, the examination of collapsed fragments and failure surfaces facilitated the identification of construction typology across various bridge segments [37]. The western and eastern approaches feature solid masonry, with approximately 0.40-meter-thick spandrel walls filled with rubble stone masonry and mortar. Conversely, the central 60° segment comprises exclusively arches, while the area between the solid approaches and the central arch segment employs a distinct filling material: tufa stones combined with mortar. This deliberate selection aims to reduce the bridge’s self-weight and alleviate lateral pressure on the spandrel walls [37]. Furthermore, Khorasan mortar has been employed amidst the discrete stones of the Plaka Bridge. The term “Khorasan” was initially coined during the Sassanid era, signifying “the land of the sun” [38]. Throughout the periods of the Ottoman and Roman Empires, mortars composed of tile, lime, and brick were referred to as Khorasan mortar [39]. This study extensively investigated Khorasan mortar and its effects on the seismic behavior of the bridge’s mortar.

6. Experimental Background

In this section of the study, we present detailed information about the experiments and test results. Khorasan mortar was prepared by using a mixture of brick powder, hydrated lime, standard sand, eggshell, and egg white to examine the structural failure behavior of historical bridges with more realistic data. The utilization of egg white and eggshell in the production of Khorasan mortar during the Ottoman and Roman Empires was widespread. However, there are no published works documenting the use of eggshell and egg white specifically for producing Khorasan mortar in bridge construction. It is worth noting that the cementing effect of eggshell and egg white surpasses that of fiber, which is why they were chosen for this study. The egg white used in the study was sourced from commercially purchased white eggs. The standard sand was prepared following the TS EN 196-1 [40] standard. Hydrated lime and brick were obtained from a private company. Figure 2 illustrates all the materials used in the experiments. To achieve the maximum Khorasan mortar mixture, egg white and eggshell were used separately as additives, in varying proportions. Nine different mixtures were formulated, including a control sample without any egg white or eggshell. The mixtures containing egg white had ratios of 4:4:3:9 for “water + egg white”, lime, standard sand, and brick powder, respectively. Mixtures prepared with eggshell had the same ratio (4:4:3:9) of materials; only “water + egg white” was replaced by eggshell. While the proportions of water and egg white were altered in these mixtures, the other components remained constant. Specifically, the ratios of “water + egg white” were selected as 0%, 25%, 50%, 75%, and 100%, in the mixtures. The total combined weight of egg white and water was set at 36 g, and the percentage of “water + egg white” in each mixture was determined. A 0% egg white additive ratio means that there is 0 g of egg white and 36 g of water, while a 100% additive is defined as 0 g of water and 36 g of egg white in the mixture. Eggshell additive ratios were used as 25%, 50%, 75%, and 100% of the lime amount. The amount of eggshell was increased while keeping the amounts of water and other materials constant for mixtures involving eggshell. Additionally, the effect of curing time was investigated using different periods (1, 7, and 28 days). The prepared mixtures were assigned codes for easy identification. For instance, AC1 denoted the control sample, C represented curing, and 1 indicated a curing period of 1 day. Similarly, EW25W75C7 stood for a mixture containing egg white (EW) with a 25% percentage, water (W) with a 75% percentage, and subjected to curing (C) for 7 days. Furthermore, mixture ES25C1 was labeled as such, with ES referring to eggshell, 25 representing the eggshell percentage, C indicating curing, and 1 representing the curing time.

Table 2. Mixture ratios and curing time.

Mixture Code	Water (g)	Egg White (g)	Eggshell (g)	Lime (g)	Standard Sand (g)	Brick Powder (g)	Curing Time (Day)
AC1	36	0	0	36	27	81	1
AC7	36	0	0	36	27	81	7
AC28	36	0	0	36	27	81	28
EW25W75C1	27	9	0	36	27	81	1
EW25W75C7	27	9	0	36	27	81	7
EW25W75C28	27	9	0	36	27	81	28
EW50W50C1	18	18	0	36	27	81	1
EW50W50C7	18	18	0	36	27	81	7
EW50W50C28	18	18	0	36	27	81	28
EW75W25C1	9	27	0	36	27	81	1
EW75W25C7	9	27	0	36	27	81	7
EW75W25C28	9	27	0	36	27	81	28
EW100W0C1	0	27	0	36	27	81	1
EW100W0C7	0	36	0	36	27	81	7
EW100W0C28	0	36	0	36	27	81	28
ES25C1	36	0	9	36	27	81	1
ES25C7	36	0	9	36	27	81	7
ES25C28	36	0	9	36	27	81	28
ES50C1	36	0	18	36	27	81	1
ES50C7	36	0	18	36	27	81	7
ES50C28	36	0	18	36	27	81	7
ES75C1	36	0	27	36	27	81	1
ES75C7	36	0	27	36	27	81	7
ES75C28	36	0	27	36	27	81	28
ES100C1	36	0	36	36	27	81	1
ES100C7	36	0	36	36	27	81	7
ES100C28	36	0	36	36	27	81	28

provides comprehensive details regarding the composition of each mixture.

6.1. Preparation of Mortar Samples

Before the mixtures were prepared, the brick ballast was pulverized through 1900 cycles in the Los Angeles abrasion device to obtain brick powder, as illustrated in Figure 3. The Los Angeles abrasion instrument is constructed from steel and has a spacious interior for sample placement. It uses balls to break down the material into finer dimensions. Each full turn of the device is considered one cycle during the experiment. Subsequently, the egg white was vigorously whisked in a porcelain bowl. All the materials were gathered in a container and thoroughly mixed until a uniform blend was achieved. The resulting mixture was then compacted using the Harvard mini compactor mold to create a cylindrical test sample, as shown in Figure 4. The compaction process consisted of 5 layers, with each layer subjected to 10 strokes. An equal amount of the sample was evenly distributed on each layer. Afterward, the samples were carefully removed from the mold using a sample extractor. The diameters and heights of the extracted samples were measured using a caliper, resulting in mean diameter and height values of 33 mm and 71 mm, respectively. To ensure proper preservation, the measured samples were meticulously wrapped with stretch film. Following this, identifying labels were attached, and the samples were placed inside a desiccator, as depicted in Figure 5. The desiccators were stored in a dark and cool environment, away from direct sunlight. The primary goal of using desiccators was to prevent any loss of water content from the samples. Consequently, the samples were kept within the desiccator for the designated curing periods (1, 7, and 28 days).

6.2. Test Setup

The experiments were conducted following the ASTM D2166/D2166M-16 [41] standard. The prepared samples for the test were placed on the bottom table of the automatic triaxial strength test device, as shown in Figure 6. Subsequently, relevant data concerning the sample, such as the mixing code, sample diameter, and height, were input into the program. The test specimens were set at a height of 71 mm, and the test speed was applied at a 10% strain rate (0.71 mm/min) over 10 min. The data recording rate was selected at 1 data point per second during the test. Before starting the experiment, the lower platform was raised to ensure that the sample was adequately loaded. Once the sample made contact with the upper platform and the load value exhibited a change, the load and deformation values were reset in the program, and the test was initiated. Throughout the experiments, real-time monitoring was conducted for both the graphical representation and the physical sample. Upon completion of the experiment, the stress–strain graph was generated. In this graphical representation, the maximum stress value provides the unconfined compression strength of the sample, while half of this value corresponds to the cohesion of the sample.

7. Experimental Results

7.1. UCS Results for Control Mortars

In this section of the study, the results of the unconfined compression strength test of the control sample (mortar) are presented in detail. Stress–strain graphs for the 1-day cured control sample (AC1) are provided in Figure 7a. According to the test results of the AC1 sample, the actual fracture pattern indicated a brittle fracture. When examining the stress (σ)–axial strain (ε) graph obtained from the program, it can be observed that the unconfined compression strength (UCS) of the AC1 sample was 143.22 kPa, and the sample reached the maximum stress value when the strain reached 3.03%. The stress–strain graph of the AC7 sample is presented in Figure 7b. The actual fracture pattern indicated ductile fracture, unlike the AC1 sample. The unconfined compression strength of the AC7 sample was 210.24 kPa. The axial strain value obtained at maximum stress was 2.95% and very close to the ε value of the AC1 sample. The unconfined compression strength value of the AC7 sample was 46.80% larger than AC1, demonstrating the strength-enhancing effect of the lime during the 7-day reaction period. The σ–ε graph for the AC28 sample is provided in Figure 7c. The fracture pattern was similar to AC7 but different from the AC1 sample. The unconfined compression strength of the sample was 338.79 kPa. The ε value obtained at this strength was 4.22%, slightly larger than the ε values of the AC1 and AC7 samples. The unconfined compression strength of the AC28 sample increased by 136.55% compared to the AC1, indicating that the chemical bond between lime and other materials strengthened as a result of the 28-day curing period.

7.2. UCS Results for Mortars Containing Egg White and Eggshell Stabilized Samples

Unconfined compression strength results of the egg white and eggshell stabilized samples are presented in this section. The effect of the test on the failure type of the samples and the σ–ε graphs of stabilized samples for different curing times (1 day, 7 days, and 28 days) are provided. EW25W75C1 showed the worst strength performance compared to all mixtures for all curing times even though its strength was 28.58% larger than the AC1, as shown in Figure 8a. Axial strain for reaching maximum stress (unconfined compression strength) for egg white-stabilized samples cured for 1 day tended to decrease when egg white in the mixture increased, as shown in Figure 8c,e,g. The maximum value for unconfined compression strength (UCS) of the egg white mixture stabilized and cured for 1 day was EW50W50C1 and the strength of it was 195.76% higher than the AC1. The eggshell stabilized mixtures show a similar stress–strain behavior to that of the egg white stabilized mixtures, for a curing time of 1 day, according to Figure 8b,d,f,h. Although the UCS value of ES25C1 is less than other eggshell stabilized mixtures, it is still 95.93% higher than the AC1. A maximum increase in unconfined compression strength was obtained at ES50C1 when compared to all mixtures cured for 1 day, and it is 282.62%.

The stress–strain graph of mixtures cured for 7 days is shown in Figure 9. UCS values of all mixtures increased significantly. EW25W75C7 was the mixture having the lowest UCS value when compared to all samples cured for 7 days, as presented in Figure 9a. The behavior of the stress–strain indicates that increasing the egg white content causes a sharp reduction in the axial strain value at the point of maximum stress occurring. This value decreased suddenly after adding egg white, by more than 50%, and it is seen in Figure 9c,e,g. However, increasing egg shell content in stabilized samples caused almost no change in the axial stress value reached at maximum stress after 50%, as shown in Figure 9b,d,f,h. ES50C7 was the mixture having the highest UCS value when compared to all mixtures cured for 7 days and the increase in strength was 571.45%, according to the AC1.

Egg white stabilized samples cured for 28 days behave a little bit differently compared to other curing times, since the addition of egg white after 50% has no significant change at axial strain value occurred at maximum stress as observed in Figure 10a,c,e,g. EW25W75C28 is again the worst mixture compared to other mixtures cured for 28 days just as 1 and 7 days. The unconfined compression strength value of ES50C28 is the highest one compared to all mixtures and curing times since it is 1056.44% higher than AC1. Egg shell stabilized samples cured for 28 days behave more brittle compared to other curing times as shown in Figure 10b,d,f,h. Stress-strain behavior of ES50C28 and ES75C28 are similar when compared with lime-stabilized samples as presented in Figure 10f,h. Those results are strong evidence that as the curing time increased, the chemical reaction became stronger, and a lime-like behavior emerged as the amount of CaO in the sample increased.

Figure 11 illustrates the unconfined compression strength (UCS) of all mixtures, including the control sample. The UCS performance of the control sample for all curing times has been depicted on the diagrams to facilitate comparison with the results of all mixtures. In Figure 11a, curing times of 1 day for all mixtures are displayed. The unconfined compression strengths of the EW50WC1, EW75WC1, ES50C1, ES75C1, and ES100C1 samples surpass those of the AC1, AC7, and AC28 samples, even though they were cured for just 1 day. Results for samples cured for 7 days are presented in Figure 11b. With the exception of EW25W75C7, all samples exhibit UCS values higher than the control sample. In Figure 11c, the UCS values of egg white-treated samples after 28 days of curing are compared. The lowest UCS among these samples is 3.45 times, 2.35 times, and 1.46 times that of AC1, AC7, and AC28, respectively, as inferred from the figure.

While creating mortar, slump tests are performed and the consistency of mortar is determined based on these slump tests [42]. Figure 12 illustrates the slump values of both the control sample and the mixtures utilized in the analyses. The slump value serves as a reliable indicator of workability. A notable decrease in slump was observed when a mixture with a 50% addition of eggshell was used as an additive, followed by an increase with higher ratios. A similar trend was observed in mixtures containing egg white + water. The best workability was achieved with the EW50W50 and ES50 mixtures, compared to the control sample.

8. Three-Dimensional Modeling of Plaka Bridge

The finite-difference modeling technique offers a range of advantages that make it a valuable tool in various fields. One of its key strengths is its versatility, as it can be applied to a wide array of problems, including differential equations and partial differential equations [33]. Additionally, it is known for its simplicity, making it relatively easy to understand and implement compared to other numerical methods. Despite its simplicity, finite-difference modeling can yield accurate results when properly utilized, enabling fine-grained resolution and the capture of intricate details within the modeled system [33]. Furthermore, the method is efficient and can be deployed on modern computing architectures, resulting in high-performance simulations and reduced computational time. In this study, the FLAC3D program was employed, which is based on the finite-difference method, to develop the bridge model timeline. Each stone element was individually modeled using a separate FLAC3D code, with brick or wedge elements utilized for each stone. Essentially, our approach utilized the finite-difference method instead of the discrete-element method or finite-element method. This choice was made because the FLAC3D program and the finite-difference method offer a crucial alternative for revealing the time-dependent creep and seismic behavior of stone elements. The Plaka bridge had been destroyed and subsequently rebuilt after a flood disaster in Greece in 2015. Two distinct bridge models were developed. The first model considered the Plaka bridge, which had endured from 1866 to 2015, and utilized Khorasan mortar with egg white additives between the stone elements. On the other hand, the second model accounted for the use of eggshell-added Khorasan mortar between the discrete stone elements. In the creation of both models, each stone element was individually represented. Brick and wedge elements were employed for modeling the stone elements. Brick elements, also known as hexahedral elements, are three-dimensional elements with six faces, eight nodes, and twelve edges. They are widely used in FLAC3D to discretize the geotechnical domain and represent solid materials, such as soil, rock, or concrete [33]. In addition, wedge elements, specialized three-dimensional elements available in FLAC3D, are particularly useful for modeling geological features such as faults, joints, or stratigraphic interfaces. These elements possess a triangular base and taper to a point, resembling the shape of a wedge. Wedge elements are advantageous when modeling inclined or irregular surfaces, as they can conform well to complex geological geometries. Brick and wedge elements used in the bridge modeling process are illustrated in Figure 13.

Khorasan mortar was simulated between the discrete stone elements by considering specialized interface elements. Utilizing the mechanical properties of Khorasan mortar (Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11), the mechanical characteristics (kn and ks stiffness) of the interface elements between Khorasan mortar and stone elements were computed (Equation (4)) [33]. Subsequently, the calculated kn and ks stiffness values, representing Khorasan mortar between the stone elements in the bridge model, were defined to realistically model the interaction between the stones. The most straightforward approach to modeling Khorasan mortar within a three-dimensional model is to introduce interface elements into the bridge model instead of Khorasan mortar. The mechanical properties of the stiffness elements established between the stone elements instead of Khorasan mortar were entirely determined by considering Equation (4). Therefore, in this study, interface elements, as depicted in Figure 14, were defined between distinct surfaces in place of Khorasan mortar. One interface element was computed between two different stones with identical mechanical properties. However, during the bridge modeling process, it was necessary to define interface elements between stones with varying mechanical properties in certain sections of the bridge. As illustrated in Figure 14, two distinct interface elements were computed between stones with different mechanical properties, and the larger interface element between two dissimilar stones with varying mechanical properties was considered in the analyses.

The mechanical properties of the stone elements and Khorasan mortar were utilized in the calculation of interface elements. The ‘Interface’ functionality in FLAC3D is employed to establish and simulate interactions between two distinct sub-grids within a geotechnical model. It represents the boundary or contact surface between different materials or regions. By specifying properties such as bulk and shear moduli, the interface facilitates the transfer of forces and displacements between the interconnected sub-grids, allowing for the modeling of intricate geotechnical phenomena such as soil–structure interaction and fault behavior. The interface feature in FLAC3D offers a versatile approach to simulating the interaction of separate regions, contributing to the comprehension and prediction of the system’s behavior. In this study, the kn (normal) and ks (shear) stiffness coefficients of the interface elements, defined in place of Khorasan mortar between the discrete stone elements, were determined using the following formulation [33].

$${\mathrm{k}}_{\mathrm{n}} \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{k}}_{\mathrm{s}}=\frac{{\mathrm{K}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}+}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\mathrm{G}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}}{{\mathrm{Z}}_{\mathrm{a}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}}}$$

(4)

In Equation (4) [33], K and G represent the bulk modulus and shear modulus, respectively. Additionally, the symbol Z corresponds to the mesh lengths of the stone element and Khorasan mortar. Mesh size plays a crucial role in finite-difference analysis, significantly impacting the results of structural analyses. In this study, the mesh size was determined based on the creep analysis of the bridge. Specifically, 10 different mesh ranges (0.1 m, 0.2 m, 0.3 m, 0.4 m, 0.5 m, 0.6 m, 0.7 m, 0.8 m, 0.9 m, and 1 m) were selected for displacements and stresses and it was observed that no significant changes occurred in the range from 0.1 m to 0.3 m mesh size. Consequently, a mesh interval of 0.3 m was chosen for the bridge model. It is important to note that the mesh intervals in this study were not arbitrarily selected but were based on the results of the creep analysis. For this study, the mesh lengths of the stone elements and Khorasan mortars were set to 0.4 m and 0.03 m, respectively. The mechanical properties of mortars and stones are shown in

Table 3. Mechanical properties of mortars and stones.

Mixture Code	E (Pa)	Poisson Ratio	G (Pa)	K (Pa)
EW50W50C28 (Mortar)	3.47 × 107	0.34	1.29 × 107	3.61 × 107
ES50C28 (Mortar)	6.01 × 107	0.28	2.35 × 107	4.55 × 107
The Lowest Main Arch	7.05 × 107	0.28	2.75 × 107	5.34 × 107
The Middle Main Arch	6.52 × 107	0.26	2.59 × 107	4.53 × 107
Supporter Arch Right	5.86 × 107	0.21	2.42 × 107	3.37 × 107
Supporter Arch Left	5.24 × 107	0.19	2.20 × 107	2.82 × 107
Rock Material	4.75 × 107	0.25	1.90 × 107	3.17 × 107
Foundation	9.00 × 107	0.38	3.26 × 107	1.25 × 108

. When computing the kn and ks values for the interface elements, the averages of the K and G values of the stone elements and Khorasan mortar are considered, as depicted in

Table 4. Stiffness parameters between the discrete surfaces.

Location of Interface Element	G (Average) (Pa)	K (Average) (Pa)	kn and ks Stiffness (Pa/mm)
Between The Lowest Main Arch and EW50W50C28	2.02 × 107	4.48 × 107	4.35 × 105
Between The Middle Main Arch and EW50W50C28	1.94 × 107	4.07 × 107	4.03 × 105
Between Supporter Arch Right and EW50W50C28	1.86 × 107	3.49 × 107	3.62 × 105
Between Supporter Arch Left and EW50W50C28	1.75 × 107	3.22 × 107	3.37 × 105
Between Rock Material and EW50W50C28	1.60 × 107	3.39 × 107	3.35 × 105
Between Foundation and EW50W50C28	2.28 × 107	8.06 × 107	6.73 × 105
Between The Lowest Main Arch and ES50C28	2.55 × 107	4.95 × 107	5.06 × 105
Between The Middle Main Arch and ES50C28	2.47 × 107	4.54 × 107	4.75 × 105
Between Supporter Arch Right and ES50C28	2.39 × 107	3.96 × 107	4.33 × 105
Between Supporter Arch Left and ES50C28	2.28 × 107	3.69 × 107	4.08 × 105
Between Rock Material and ES50C28	2.13 × 107	3.86 × 107	4.06 × 105
Between Foundation and ES50C28	2.81 × 107	8.53 × 107	7.44 × 105

, to obtain the kn and ks interface coefficients. The calculated kn and ks values are presented in Table 4. The variable “K” denotes the bulk modulus, while “kn” and “ks” represent stiffness values in the normal and shear directions, respectively. In the context provided, “EW50W50C28” signifies that the mixture contains a 50% ratio of egg white to water and was cured for 28 days. When referencing “between the lowest main arch and EW50W50C28”, this indicates that Khorasan mortar with the composition EW50W50C28 was utilized in the Lowest Main Arch section of the bridge.

While examining historical bridges in the literature, the influence of Khorasan mortars is often disregarded. However, it is crucial to accurately model Khorasan mortars within the bridge model to yield more realistic results. Therefore, these values contribute significantly to the literature regarding the modeling of historical stone arch bridges. Subsequently, the Burger creep material model was employed for the bridge’s rockfill, arch, and foundation materials. This material model allowed for an accurate representation of the time-dependent failure behavior of the bridge material. When using this material model, the following material properties were incorporated: shear modulus, bulk modulus, density, cohesion, friction angle, and dilatancy angle values. The bridge model was constructed with a non-linear approach, employing the Burger model to characterize the non-linear behavior of materials. Additionally, free-field and quiet-boundary conditions were applied to the lateral boundaries of the three-dimensional bridge model. These boundary conditions serve to minimize the reflection of earthquake waves within the model. Furthermore, a reflecting boundary condition was specified for the base boundaries of the bridge model. The bridge model comprised a total of 1,849,274 individual brick elements. Each brick element was individually modeled within the overall structure. An overview of the three-dimensional bridge model is presented in detail in Figure 15. The height of the primary arch of the bridge was modeled as 18.7 m. Subsequently, two auxiliary arches were created, measuring 7.8 m and 8.9 m, respectively. The thickness of the primary arch section in the model was established at 0.57 m. To incorporate structure–soil interaction into the analysis of significant structures like bridges, soil was modeled beneath the bridge. Furthermore, soil was included in the bridge model to simulate a more realistic effect of earthquake waves on the bridge. Additionally, the foundation section extended downward by four times the height of the bridge (Figure 15).

9. Three-Dimensional Numerical Analysis Results

9.1. Long-Term Creep Analysis Results beween 1866 and 2015 for Bridge Using Khorasan Mortar with Egg White Additive

Historical bridges represent one of the most significant ways of illuminating our history. These bridges provide insights into the lifestyles of people who resided in the past. Therefore, it is essential to scrutinize the prospects of these bridges, considering the genuine structural behavior of historical bridges that have managed to endure to this day. Restoration efforts should be carried out without compromising the historical fabric. This section of the study gives a comprehensive examination of the time-dependent creep analyses of the Plaka bridge. This bridge, constructed in 1866, has withstood a total of five floods over its history. The specific dates of these flood disasters are detailed in

Table 5. Dates of the flood disasters.

Symbol	Date of Flood Disaster	Explanation
FD1	1919	53 years after the bridge was built
FD2	1930	64 years after the bridge was built
FD3	1948	82 years after the bridge was built
FD4	1975	109 years after the bridge was built
FD5	2015	149 years after the bridge was built

.

The water level at the bridge was simulated using hydrostatic water forces and a water table. During the examination of the time-dependent creep analyses, it was considered that Khorasan mortar mixed with egg white should be used between the stone elements. The primary reason for this choice is that when the bridge was constructed in 1866, Khorasan mortar with an egg white additive was employed to bond the discrete stone elements together. As egg white-mixed Khorasan mortar enhances the interaction between these stone elements, the stiffness values (kn and ks) of the interface elements were calculated, taking into account the mechanical properties of Khorasan mortar with egg white additive. The computed kn and ks values are provided in detail in Table 3. Stiffness values were established between individual stone elements, and the bridge was prepared for realistic creep analyses. In the execution of time-dependent creep analyses, specific time intervals were defined within the FLAC3D program. The definition of precise time steps is crucial for conducting more realistic analyses. To carry out the time-dependent creep analyses, we initially commenced with 1-second analyses. After each 1-second creep analysis, all stresses and displacements on the bridge were reset. Subsequently, the next 1-second analysis was conducted, and this analysis cycle was repeated 4,698,864,000 times, covering 149 years of creep analyses. The stages of analysis for the creep analyses are comprehensively presented in Figure 16.

Since the bridge was subjected to five flood disasters in 149 years, the water level of the bridge fluctuated during these flood events. Under normal conditions, except for flood disasters, the water level at the bridge was considered to be one-sixth of the bridge’s height. However, during flood events, the water level at the bridge was assumed to be at the same level as the bridge height. Following the analyses, an evaluation was made regarding the principal stress values that occurred on the bridge over 149 years. The flood events that the bridge encountered during this period are indicated on the graphs with the symbol FD (Flood Disaster). Following the results of the time-dependent creep analysis, the time-dependent principal stress values at Point 1, Point 2, and Point 3 are elaborated upon in Figure 17. When examining the time-dependent principal stress values at Point 1, it becomes evident that there were minimal principal stresses during the initial 53 years. After the first flood disaster (FD1), an increase in principal stresses at Point 1 was observed. This rise can be attributed to the water level reaching Point 1, subjecting it to significant hydrostatic forces. During the bridge’s second flood disaster (FD2), there was an approximate 0.2 MPa increase in principal stresses at Point 1 (Figure 17a). Notably, significant stress increases occurred at Point 1 between the second and third flood disasters. Furthermore, after the third flood disaster, stress values at Point 1 increased by approximately 0.3 MPa. This result demonstrated the fact that flood disasters profoundly alter the principal stress behavior of historical bridges. The maximum principal stress value observed at Point 1 during the fourth flood disaster was 4.11 MPa. In this study, the maximum capacity value for Point 1 was determined as 6.3 MPa. In Figure 17b, the principal stress values that evolve at Point 2 are detailed. Significant principal stress values were not observed at Point 2 until the first flood disaster. However, the principal stress change at Point 2 after the first flood disaster was 0.23 MPa, significantly altering the time-dependent failure behavior of Point 2. No significant principal stress changes were detected at Point 2 between the first and second flood disasters. The principal stress value just before the second flood disaster at Point 2 was 0.76 MPa. The principal stress value after the second flood disaster at Point 2 was 1.38 MPa. It is evident that the second flood disaster also led to substantial principal stress changes at Point 2. The third flood disaster caused more significant principal stress changes at Point 2 compared to the first and second flood disasters. After the third flood disaster, an approximate principal stress change of 0.24 MPa was observed at Point 2. Similar to other flood disasters, the fourth flood disaster significantly increased the principal stress values at Point 2. It was concluded that the maximum capacity value for Point 2 was 4.8 MPa. In Figure 17c, detailed time-dependent principal stress values occurring at Point 3 over 149 years are presented. It was found that no significant principal stress values occurred at Point 3 during the first 53 years. However, 53 years later, after the first flood, significant changes were observed in the principal stress values at Point 3. Following the first flood disaster, principal stress increased by approximately 0.39 MPa at Point 3. Furthermore, the flood disaster accelerated the rate of principal stress increase at Point 3. This result provides valuable insights into how flood disasters alter the structural behavior of historical stone bridges. After the second flood disaster, there was an approximate 0.17 MPa increase in principal stress at Point 3. Significant principal stress changes were observed at Point 3 between the second and third flood disasters. Following the third flood disaster, critical increases in principal stress values were noted at Point 3, and it was concluded that the maximum principal stress value at Point 3 was 3.98 MPa between the third and fourth floods. Substantial increases in principal stress values at Point 3 were observed between the third and fourth flood disasters. This outcome suggests that the bridge experienced structural fatigue after the third flood. The rate of principal stress increase at Point 3 accelerated significantly after the fourth flood disaster. After the fifth flood, it was observed that structural damage occurred at Point 3 when the maximum principal stress value reached 5.4 MPa.

In Figure 18, the stress-displacement behavior of the historical bridge is presented in detail. According to Figure 18, it was concluded that the displacement values over time at Point 1 were significantly greater than those at Point 2 and Point 3. Additionally, when comparing these three different points, it was observed that the smallest displacement values occurred at Point 2. The largest displacement value recorded at Point 1 over 149 years was approximately 29.7 mm. Furthermore, the largest displacement values observed over 149 years at Point 2 and Point 3 were 15.8 mm and 19.7 mm, respectively. Point 1, Point 2, and Point 3 reached their maximum displacement values at 6.3 MPa, 4.8 MPa, and 5.4 MPa, respectively. This result indicates that the maximum principal stress value for the structural behavior of Point 1, Point 2, and Point 3 on historical stone bridges is 6.3 MPa, 4.8 MPa, and 5.4 MPa, respectively.

9.2. Long-Term Creep Analysis Results between 2015 and 2164 for Bridge Using Khorasan Mortar with Eggshell Additive

In this section of the study, a comprehensive examination of the time-dependent creep behavior of the bridge following its restoration was conducted. During the 2015 restoration of the bridge, Khorasan mortar with eggshell additives was employed between the individual stone elements. The mechanical properties of these mortars are extensively detailed in Table 2. Furthermore, when eggshell-added Khorasan mortar was used between separate stone elements, the calculated kn and ks stiffness parameters between the stones were calculated, and are presented in Table 3. Once the calculated kn and ks stiffness values were defined for the individual stone elements, the bridge underwent static analysis from 2015 to 2164. These analyses took into consideration the flood disasters the bridge had previously experienced. In essence, time-dependent creep analyses were conducted, assuming that the bridge would encounter floods in 2068, 2079, 2097, 2124, and 2164 (

Table 6. General information about flood disasters that are assumed to occur in the future.

Symbol	Year of Flood Disaster	Explanation
FD1	2068	It is thought that a flood disaster would occur 53 years after the bridge was restored.
FD2	2079	It is thought that a flood disaster would occur 64 years after the bridge was restored.
FD3	2097	It is thought that a flood disaster would occur 82 years after the bridge was restored.
FD4	2124	It is thought that a flood disaster will occur 109 years after the bridge was restored.
FD5	2164	It is thought that a flood disaster would occur 149 years after the bridge was restored.

). This approach allowed us to compare how Khorasan mortar with egg white additive and Khorasan mortar with eggshell additive influenced the time-dependent creep behavior of the bridge. During flood disasters, it was assumed that the water level would rise to match the height of the bridge, and the water level was modeled using water loads and a water table. These analyses provided us with detailed insights into the future structural behavior of the restored Plaka bridge. The time-dependent creep behavior of the bridge from 2015 to 2164 is comprehensively presented in Figure 19, with the ‘FD’ symbol denoting flood disasters. Based on the principal stress values evolving at Point 1, Point 2, and Point 3, it was observed that the largest principal stress values occurred at Point 1, while the smallest principal stress values were recorded at Point 2. According to the analyses conducted between 1866 and 2015, it was deduced that the optimum capacity values for Point 1, Point 2, and Point 3 were 6.3 MPa, 4.8 MPa, and 5.4 MPa, respectively. However, as per the results of the creep analysis conducted from 2015 to 2164, the maximum principal stress values at Point 1, Point 2, and Point 3 were determined to be 1.92 MPa, 0.91 MPa, and 1.48 MPa, respectively (Figure 19). These findings suggest that Point 1, Point 2, and Point 3 would not experience any structural damage in the event of five flood disasters occurring between 2015 and 2164. Moreover, they indicate that the historical bridge constructed with Khorasan mortar containing eggshells exhibited greater durability than the bridge constructed using egg white, displaying enhanced resistance to floods without collapsing. Consequently, this study strongly recommends the use of eggshell-containing Khorasan mortar between individual stone elements in the restoration of historical bridges.

9.3. Modal and Seismic Analysis Results for Bridge Using Khorasan Mortar with Eggshell Additive

In this section of the study, detailed modal and seismic analysis results for the bridge that has been restored using Khorasan mortar containing eggshells are presented in detail. In this study, while the Burger creep material model was used for creep analyses, the material model was changed for seismic analyses. Specifically, the Mohr–Coulomb material model was used for the seismic analyses. The material parameters for the Mohr–Coulomb model were obtained from the literature (Table 1) [28]. All assumptions of this material model are included in the FLAC3D program [33]. Figure 20 illustrates the modal behavior of the restored Plaka bridge. According to Figure 20, the first mode of the restored bridge has a frequency of 12.071 Hz. Additionally, the second and third modes of the bridge have frequencies of 14.623 Hz and 21.136 Hz, respectively. The first mode (with a frequency of 12.071 Hz) is characterized by ‘bending’ deformation dominance, while the second mode (with a frequency of 14.623 Hz) is dominated by ‘axial’ deformation. These findings offer valuable insights into the modal behavior of historical bridges that have undergone restoration with Khorasan mortar containing eggshells.

In Figure 21, Figure 22, Figure 23 and Figure 24, the seismic response of the restored bridge to 10 different earthquakes is investigated in detail. When conducting seismic analyses of the bridge, non-reflective boundary conditions were implemented. For lateral boundaries, free-field and quiet-boundary conditions were considered to minimize wave reflection within the model. Hysteresis damping was incorporated into the earthquake analyses, and G/Gmax values of the materials were utilized for calculating the dampings. Earthquake accelerations were defined in the program using the acceleration-time format. Although the seismic events used in these analyses had durations exceeding 40 s, only the most critical 40 s of each earthquake were considered due to the computational length of the analyses. Detailed mechanical properties of the earthquakes employed in these analyses are provided in

Table 7. Earthquake characteristics [43].

Symbol	Earthquake	Mw	Distance (km)	PGA (cm2/s)	PGV (cm/s)	PGD (s)
EQ1	Pazarcık (Kahramanmaraş)	5.5	6.87	49.84	2.84	0.55
EQ2	İslahiye (Gaziantep)	5.7	10.46	363.52	13.85	1.02
EQ3	Ekinözü (Kahramanmaraş)	5.5	10.93	79.35	4.26	0.42
EQ4	Pazarcık (Kahramanmaraş)	7.6	8.6	1966.74	186.78	661.9
EQ5	Elbistan (Kahramanmaraş)	7.6	7	635.45	170.79	614.52
EQ6	Yeşilyurt (Malatya)	5.6	6.15	25.23	2.36	12.77
EQ7	Nurdağı (Gaziantep)	6.6	6.2	445.29	40.49	9.27
EQ8	Doğanşehir (Malatya)	5.6	10.23	47.28	2.90	0.40
EQ9	Nurdağı (Gaziantep)	5.6	6.98	44.15	2.91	0.73
EQ10	Defne (Hatay)	6.4	21.7	445.38	75.78	44.90

.

The seismic analyses in this study are based on important ground motions that occurred in Turkey in 2023, resulting in the loss of thousands of lives. Given the proximity of the Plaka bridge to Turkey, utilizing these earthquakes in three-dimensional seismic analyses holds significant importance for assessing the bridge’s earthquake response. In Figure 21, the principal stress behavior of the Plaka bridge was investigated under five different earthquakes. During the EQ 1 earthquake, substantial principal stresses were observed in the middle of the main arch section of the bridge and at the foot parts of the auxiliary arches (Figure 21a). At Point 1 on the main arch, a maximum principal stress value of 5.4 MPa was recorded during the EQ 1 earthquake. During the EQ 2 earthquake, significant principal stress values manifested in the middle and along the edges of the main arch of the Plaka bridge. Furthermore, substantial principal stress values were evident at the peripheries of the auxiliary arches (Figure 21b). Under the EQ 4 earthquake, significant principal stress values emerged at the edges of the main arch of the bridge and in the lower sections of the secondary arches. Additionally, substantial principal stresses were observed in the lower regions of the main arch. Nevertheless, no critical principal stress values were noted in the upper portions of the auxiliary arches. The maximum principal stress value recorded at Point 4 during the EQ 4 earthquake was 4.5 MPa (Figure 21d). In Figure 22, the principal stress behavior of the Plaka bridge was analyzed under five distinct earthquakes. The highest principal stress values on the bridge during the EQ 6 earthquake manifested in the upper segments of the main arch (Figure 22a). Furthermore, substantial principal stress values were observed in the lower sections of the auxiliary arches. Conversely, minimal stress values were detected at the far-right and -left extremities of the bridge. The maximum principal stress value recorded at Point 6 during the EQ 6 earthquake was 5.05 MPa. In Figure 22b, the seismic behavior of the Plaka bridge during the EQ 7 earthquake is comprehensively presented. As a result of the earthquake analysis, significant principal stress values were observed in the auxiliary arch segments of the bridge (Figure 22b). During the EQ 9 earthquake, the highest principal stress values manifested in the upper segments of the main arch section of the Plaka bridge. Additionally, notable stress values were evident in the base sections of the auxiliary arches. The maximum principal stress value recorded at Point 9 during the EQ 9 earthquake was 4.7 MPa (Figure 22d). These findings indicate that the principal stress values in the body of the Plaka bridge, restored using Horasan mortar mixed with eggshell, are lower compared to those restored using Horasan mortar mixed with egg white. This suggests that such mortars significantly enhance the seismic resilience of the Plaka bridge.

Figure 23 illustrates the displacement patterns of the Plaka Bridge in response to five different earthquakes. During the EQ 1 earthquake, the most extensive horizontal displacement values were recorded in the main arch section of the Plaka bridge. The maximum horizontal displacement value recorded at Point 1 during the EQ 1 earthquake was 17.8 mm. Additionally, the maximum horizontal displacement in the auxiliary arches was approximately 11 mm (Figure 23a). In Figure 23b, the displacement behavior of the Plaka bridge was assessed during the EQ 2 earthquake. Following the EQ 2 earthquake, the most significant displacements occurred in the central section of the main arch. Conversely, the smallest displacements were observed at the bridge’s base. The maximum horizontal displacement recorded at Point 2 during the EQ 2 earthquake was 15.1 mm (Figure 23b). The EQ 3 earthquake resulted in maximum horizontal displacement values within the central regions of the main arch. Additionally, substantial displacements occurred in the lower segments of the secondary arches, while minimal displacements were observed in the upper sections of the auxiliary arches. The maximum horizontal displacement recorded at Point 3 during the EQ 3 earthquake was 16.2 mm (Figure 23c). These findings underscore the criticality of the main arch sections in historical bridges. In Figure 24, the detailed horizontal displacement behavior of the bridge during five different earthquakes is examined. The most significant horizontal displacements observed on the bridge when considering the EQ 6 earthquake analysis were at the edges of the main arch section. During the earthquake, minimal displacement values were registered at the base of the auxiliary arches. Furthermore, the upper parts of the auxiliary arches exhibited maximum displacement values of approximately 10 mm. The largest recorded displacement value at Point 6 during the earthquake was 15.4 mm (Figure 24a). Under the EQ 8 earthquake, maximum displacements were observed at the edges of the main arch section of the bridge. Significant displacements also occurred at the edges of the auxiliary arch components. This finding underscores the primary importance of the main arches in the seismic behavior of masonry arch bridges (Figure 24c). Therefore, when constructing or restoring masonry arch bridges, careful consideration should be given to the middle of the main arch sections. Under the EQ 10 earthquake, significant horizontal displacements occurred both in the middle and at the edges of the main arch section of the Plaka bridge. The maximum displacement value recorded during the earthquake analysis at Point 10 was 13.3 mm. These results highlight the crucial role played by both the main arch and auxiliary arch sections in the seismic behavior of masonry arch bridges. Moreover, it was observed that the incorporation of eggshell-containing Khorasan mortar in masonry arch bridges positively contributes to their seismic displacement behavior. In light of these analysis results, it is recommended to use eggshell-containing Khorasan mortar between the stone elements during the reconstruction and restoration processes of historical bridges.

10. Conclusions

In this study, the long-term creep and seismic behaviors of a historical bridge were investigated using Khorasan mortar with additives of both egg white and eggshell. The mechanical properties of Khorasan mortar were derived from experiments, and the stiffness parameters between the stone elements were computed based on these mechanical properties. Subsequently, three-dimensional creep and seismic analyses of the bridge were conducted to assess how the behavior of the bridge is influenced when either egg white- or eggshell-added Khorasan mortar is used between the stone elements. In the first part of the study, experiments were carried out to obtain the mechanical properties of Khorasan mortar. Based on the experimental findings, it was observed that the unconfined compression strength of the control soil and all other mixtures increased with longer curing times. Notably, the maximum mixture was found to contain 50% additives of either egg white or eggshell. To optimize the use of egg white, water content was reduced, contributing to efficient water usage, a critical global resource. This reduction in water content led to increased unconfined compression strength in the samples. Furthermore, the eggshell additive exhibited superior performance compared to egg white, due to its stronger and faster chemical reaction, attributed to its lime content. Importantly, even the weakest strength value obtained with additives remained higher than that of the control sample, affirming the positive impact of adding Khorasan mortar with these additives. However, it was determined that exceeding a 50% additive ratio reduced the strength compared to samples stabilized at this percentage. Nevertheless, the strength values remained significantly higher than those of the control soil. In the second part of the study, three-dimensional numerical analyses were performed. Historically, during the construction of the bridge in 1866, Khorasan mortar with egg white additives was used between the stone elements. Static analysis of the bridge from 1866 to 2015 (149 years) revealed that significant principal stresses were concentrated in the main-arch and auxiliary-arch sections following five different flood disasters. This analysis suggested that one of the primary causes of the bridge’s collapse during the 2015 flood was the use of Khorasan mortar with egg white additives in these critical sections. Analyzing the bridge constructed with Khorasan mortar containing egg white additives from 1866 to 2015 revealed maximum principal stress values of 6.3 MPa, 4.8 MPa, and 5.4 MPa at Point 1, Point 2, and Point 3, respectively. When the bridge was being restored in 2015, Khorasan mortar with eggshell additives was employed between the stone elements. For this reason, the bridge was statically analyzed from 2015 to 2164 considering Khorasan mortar with eggshell additives between stone elements. The results indicated that no structural damage would occur in any part of the bridge, even when subjected to five different hypothetical flood disasters. For the bridge constructed with eggshell-added Khorasan mortar from 2015 to 2164, maximum principal stress values of 1.92 MPa, 0.92 MPa, and 1.48 MPa were observed at Point 1, Point 2, and Point 3, respectively. These results demonstrated that the use of eggshell-mixed Khorasan mortar between stone elements provided more favorable contributions to the long-term creep behavior of the bridge. Moreover, according to modal analysis results of the bridge using eggshell-added Khorasan mortar, the natural frequencies of Mode 1, Mode 2, and Mode 3 are 12.071 Hz, 14.623 Hz, and 21.136 Hz, respectively. Additionally, the maximum horizontal displacement value observed in the bridge body during these seismic analyses was 17.8 mm. The seismic principal-stress and displacement values obtained from the seismic analyses are very similar to those found in the literature. This result validates the accuracy of the seismic analyses of the Plaka bridge and demonstrates the reliability of using Khorasan mortar for assessing the seismic behavior of masonry bridges. Considering these comprehensive results, it is highly recommended to use eggshell-mixed Khorasan mortar, rather than egg white-mixed Khorasan mortar, for restoring or rebuilding historical bridges.