Simplified study on the evolution of the stored and released energy in the middle rock pillar of a twin-tunnel and its influence on the neighbour tunnel

Simplified study on the evolution of the stored and released energy in the middle rock pillar of a twin-tunnel and its influence on the neighbour tunnel | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Simplified study on the evolution of the stored and released energy in the middle rock pillar of a twin-tunnel and its influence on the neighbour tunnel ALLOUACHE Abdelaziz N This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7586323/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this study, the evolution of energy in the middle rock pillar (MRP) of a soft rock twin tunnel is analysed using dynamic FEA simulation. The most affected region by energy storage is the MRP; however, despite accumulating substantial energy, the energy storage is independent of the in-situ stress according to vertical components output. The energy release is significant in the adjacent tunnel when horizontal components are considered. The plastic radius of both tunnels and their geometry play a significant role; for instance, the overlapping part of both plastic radii (Re) registers the most convergences (horizontal displacements) at the bench level of the tunnel. The energy evolution is also linked to the geometry of the tunnel, where the more arched the tunnel shape becomes, the more energy dissipates in that region of the tunnels, and vice versa at the tunnel arch waist, where the geometry is rather straight. These results can efficiently contribute to the body of knowledge in underground engineering, particularly in twin- tunnel excavation cases. Numerical Analysis Geology Civil Engineering Twin-tunnels middle rock pillar energy evolution FEA dynamic analysis plastic radius overlap Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Figure 18 Figure 19 Figure 20 1. Introduction With the expansion of traffic congestion, underground engineering has emerged as the best alternative to consider in order to satisfy transport needs, as it provides an infinite space for traffic infrastructures. Due to the continuous need for transport development, twin tunnels with small clear spacing have become increasingly common (Chen et al., 2011 ; Li et al., 2012 ; Xia et al., 2013 ; Elwood and Martin, 2016 ; Das et al., 2017 ; Gao et al., 2017 ; Wang et al., 2019 ; Shivaei et al., 2020 ). In terms of numerical simulation, Soliman et al. ( 1993 ) derived that the stress field around the second tunnel was no longer the initial stress field and that the central axis of the tunnel and the stiffness of the stratum were no longer symmetrical. Zheng et al. ( 2021 ) investigated the mechanical properties of the middle soil column between two tunnels with small spacing by using a combination of model tests and numerical calculations (1). In the research of laboratory testing, the uniaxial loading test has been conducted in different orientations and locations to explore the failure mechanisms of twin-tunnels (Dhar et al., 1981 ). Jiang et al. ( 2021 ) exhibited the overall failure process of twin-tunnels through 3D printing-based physical simulations, and they identified the vital part of the structural failure by observation and automated measurements. Jia et al. ( 2021 ) developed a test system for use with the twin-tunnel excavation model to determine the deformation pattern of the surrounding rock and supporting structure. Their research indicated that the failure of twin-tunnels is caused by the concentration of stress that occurs mainly in the middle rock pillar between the two tunnels. However, the stress concentration phenomenon gradually disappears with the increasing clear spacing; when the clear spacing between the twin-tunnels exceeds twice the span of the single tunnel, its influence on the stability of the surrounding rock and supporting structure of the twin-tunnels is negligible. Ng. et al. (2004) presented an interesting numerical investigation on the multiple interactions between large parallel hypothetical twin tunnels constructed in stiff soils using the NATM. This study pays special attention to the influence of the lagging distance between twin tunnel excavated faces (LF), indicating a strong effect of LF on the behaviour of both tunnels. It should be mentioned that the behaviour of tunnels excavated using the NATM method and the mechanized method are very different, caused not only by the tunnel shape but also by the components of the construction loading along the tunnelling direction (Do et al. 2016). Zhu et al ( 2022 ) and Zhou et al ( 2017 ) studied rock specimens containing two rectangular holes and revealed the characteristics of energy evolution and fracture processes of the surrounding rock. The existence of double-hole leads to more degradation of mechanical properties compared to a single-hole. The interactions between twin tunnels complicate the stress field of the surrounding rock, thus affecting the mechanical properties and fracture evolution of the surrounding rock (Wang et al. 2024 ; He et al. 2021 ; Tan et al. 2022 ; Han et al. 2022 , Yang et al. 2024 ). In fact, apart from the circular, square, and elliptical tunnel shapes, the straight-wall-top-arch tunnel shape is more commonly encountered in metal mines and underground tunnel engineering. Therefore, some researchers have begun to investigate the influence of straight-wall-top-arch tunnel shape on the mechanical behaviour of the surrounding rock (Wu et al. 2023 (a), (b); Qiu et al. 2021 ). The obtained results indicate that under the same impact conditions, the stress concentration in the straight-wall-top-arch tunnel is substantially greater than that in the circular tunnels (Wang et al. 2024 ; Wang et al. 2025 ). In this paper, a twin horseshoe- shaped tunnel located in the eastern part of the Algerian east-west freeway, precisely in Djebel Ouahch, Constantine, is taken as a reference for the study. The tunnel geometry influences the effect of the released and accumulated energy at the bench (waist) and the arched sidewall according to the nodal outputs of a dynamic FEA numerical assessment. It is also relevant to mention that the plastic radius of both tunnels plays a prominent role, as the overlap section of both plastic radii registered the highest rate of horizontal displacements, precisely at the middle rock pillar, and the recorded horizontal displacements were fewer on the other side of the bench of both tunnels. The results contribute significantly to the understanding of the behaviour of the MRP of a twin tunnel in the field of tunnelling and underground engineering. 2. Project’s overview The tunnel of "Djebel Ouahch," located in Algeria's Constantine province, serves as the benchmark for this research study. This tunnel is a section of the Maghreb Unity Highway (AUM, acronym in French). It spans around 700 km in length, traversing Algeria with an overall distance of 1,200 km. It crosses the Djebel Ouahch Mountain in northeastern Algeria, situated in the Constantine province, ranging from pk205 + 393 to pk207 + 384.5 ("pk" denotes a French term for kilometric point), totaling a length of 1,909 m. It consists of a twin-tunnel with a radius of 9.57 m, positioned 17 m apart from one another. The typical depth is approximately 120 m, with a local maximum of 140 m, whereas the thinnest coverage is 12 m, featuring an area nearing 190 m² in the form of a lowered vault, illustrated in Fig. 2 . Based on the reconnaissance campaign conducted, both tunnels have consistently intersected the marl-limestone zone linked to the Tellian layer, especially within the Cretaceous argillite unit. Quaternary colluviums seem to appear locally at the surface. Rough colluviums consist of heterogeneous sandstone components, pebbles, and angular boulders with red silty matrix, as illustrated in Fig. 3 . On January 1, 2014, from 16:00 to 16:30, a significant ground subsidence took place in the left tunnel, spanning from mileage pk206 + 150 to pk206 + 280 (with "pk" being a French abbreviation for kilometric point). The collapse significantly harmed the right tube (which was already finished and operational at that time), inflicting serious harm to the lining of the right tube, with the total length of the collapse reaching 140 m. No injuries or fatalities were reported after the incident. In the construction documentation, both tunnels have been dug utilizing the identical procedure. The right tube has been fully excavated and is operational; thereafter, the second tube (left tube) was constructed. The NATM (New Austrian Tunnelling Method) was employed in the construction of the left tube, using the bench-cut excavation approach. A two-tiered support structure was utilized during the construction phase of the left tunnel. The materials for the temporary support scheme included HEB 200 steel arches with a pitch spacing of 0.75 m, 6 m long hollow micro-fissured fiberglass bolts, and a 30 cm thick layer of shotcrete. A 60 cm thick layer of reinforced concrete was utilized for the secondary lining. 3. Methodology Based on a 2D FEA model conducted using Abaqus with a dynamic/explicit solver to identify failure phenomena with a plastic behavior constitutive model, a nodal output is considered to effectively identify the evolution of energy in the MRP and the left tunnel waist. The right tunnel has not been considered in this study since it did not undergo failure despite having been heavily damaged. The model geometry is 200 m long and 200 m deep, with a free mesh using a medial axis algorithm composed of 2806 quadratic elements, as shown in Fig. 5 . Figure 6 represents the expansion of the plastic radius, designated as "Re" of both tunnels with their perfect overlap area. As shown in Fig. 7 , two nodes are selected in the middle rock pillar for the vertical component S22, namely node 296 and node 12. Figure 8 shows the considered nodes for the horizontal and shear components S11 and S12, respectively. A perfect elastic-plastic model represents the behaviour model in this case. An increase in plasticity can clearly reproduce the behaviour in a plastic regime, with the loading accounted for by gravity loading (the tunnel is buried at 120 m, see sect. 2 para 2). The parameters of the behaviour are summarized in Table 1 . The plastic radius "Re" is calculated according to the Hoek & Brown (1980) rock-support analytical model. The calculated value in this case is Re = 16.91 m. Table 1 Behaviour constitutive model used for the simulation. Young modulus (MPa) Poisson’s ratio Yield stress (KPa) Plastic strain (%) Unit weight (KN/m 3 ) 200 0.3 605 0 / / 605 1.25 24 / / 605 2.8 / / 605 4.5 4. Results & discussions 4.1. Results Figure 9 shows the recorded horizontal displacements of the model. According to the output, the maximum displacements are recorded at both tunnels' waists on the MRP side, with the left tunnel reaching 124 mm of convergence and the right tunnel reaching 123 mm, while significantly fewer displacements are observed on the other side of both tunnels' waists, with 41.43 mm in the left tunnel and 41.44 mm in the right tunnel. Figures 10 and 11 show that the recorded displacements are reasonably correlated with those of the modelling, with 90.5% and 88.44% at the critical points in order to validate the proposed model. The Figs. 12 & 13 show the vertical component (S22) of the energy accumulation in the MRP at nodes 296 and node 12 respectively with the energy (in Joules) against time (S). Both nodes show a constant evolution of energy, with a difference in behavior at the post-peak energy level. The lower node registers a peak at almost 700,000 J, exactly at 698,725 J, with a constant post-peak energy evolution. The higher node peaks slightly under the first one, exactly at 693,122 J, with some fluctuation in the post-peak energy behavior. Figure 14 and 15 show the horizontal component (S11) of the energy accumulation in the MRP at nodes 296 and 12, respectively, with energy (in Joules) against time (S). At the deeper point (node 12, Fig. 15 ), there is a constant energy evolution with a peak at 421,192 J, followed by a recovery at 305,163 J, then progressively stabilizing at 342,689 J. At the upper node (node 296, Fig. 14 ), a peak is registered at 346,446 J, followed by a recovery at 142,374 J, then a fluctuating stabilization is witnessed that reaches 302,670 J. Figure 16 and 17 show the shear component (S12) of the energy accumulation in the MRP at nodes 296 and 12, respectively, with energy (in Joules) against time (S). At the higher node (node 296), the two highest peaks are recorded at -58601 J and 56784 J (with two directions for the shear components), followed by fluctuating decreasing peaks toward stabilization. At the lower node (node 12), only two peaks have been recorded: -13197 J and the second at 14762 J, followed by a quasi-constant stabilization. Figures 18 and 19 show the energy evolution at node 246 in the horizontal and shear components (S11, S12), respectively, with energy (in Joules) plotted against time (S). The horizontal component (S11) of node 246 registers a peak at 79001 J, followed by a fluctuating recovery at 29381 J, then stabilizes around 60000 J. The shear component (S12) of the same node registers a first peak at -46438 J and the second at 115440 J, then stabilizes with a fluctuating trend. Figure 20 shows the energy evolution of the horizontal component at "node 242," with the energy (in Joules) plotted against time (s). The horizontal component (S11) of node 246 registers a peak at 225,745 J, followed by a fluctuating recovery at 129,991 J, then stabilizes around 180,000 J. 4.2. Discussions / interpretation In Fig. 9 , the numerical output of the horizontal displacements is shown. It is obvious that both tunnels' waists underwent the most horizontal convergences, while the other side of both tunnels' waists shows significantly lower horizontal convergence values. The output is validated with a fair degree of correlation, as reported in Fig. 10 . The most convergences in the MRP were registered at the straight arch waist of both tunnels and not the arch waist side. The major factor in this phenomenon is the perfect overlap of both tunnels ' plastic radius (Re), as the other sides of both tunnels (the left side of the left tunnel and the right side of the right tunnel) have shown minor convergences compared to the mentioned areas of the MRP. In Figs. 12 and 13 , representing the vertical component (S22) of the energy evolution, both nodes show a constant evolution of energy, with a difference in behaviour at the post-peak energy level. The lower node (node 12) registers a peak of almost 700,000 J, exactly at 698,725 J, with a constant post-peak energy evolution. The higher node (node 296) peaks slightly under the first one, exactly at 693,122 J, with some fluctuations in the post-peak energy behaviour. The deeper node (node 12) stores more energy than the higher node (node 296) with the increase of in-situ stress; however, the energy release is more prominent at the higher node than at the lower node, which shows a constant evolution (Fig. 13 ). Given that the clear distance between both tunnels is 17 m, and the location of the lower node coincides with a rather straight geometry of the tunnel while the higher node aligns with an arch-shaped geometry, there is an evident influence between the tunnels. The higher node (node 296) undergoes an inertia that makes it sensitive to energy release in the arch-shaped tunnel region, whereas at the lower node (node 12), the peak evolution is constant, knowing that the node is located in a straightforward geometry of the tunnel, which makes the energy dissipation independent of in-situ stress. In Figs. 14 and 15 , representing the horizontal component (S11) of the energy evolution, at the deeper point (node 12, Fig. 15 ), there is a constant energy evolution with a peak at 421,192 J, followed by a recovery at 305,163 J, then progressively stabilizing at 342,689 J. At the upper node (node 296, Fig. 14 ), a peak is recorded at 346,446 J, followed by a recovery at 142,374 J, then a fluctuating stabilization is observed that reaches 302,670 J. The deeper area always records a higher value of energy, with the horizontal component this time, and the energy release is consistently registered at the upper node. The horizontal component shows a more pronounced energy release at the upper node where the 17 m clear spacing and the inertia of the arched side wall, associated with the location of the designated node, have an obvious influence on the energy release. The lower node, on the other hand, registers a peak at 421,192 J, followed by a recovery at 305,163 J, then steadily stabilizes at 342,689 J, knowing that its location is at a straight-shaped geometry of the tunnel, and the in-situ stress only influences the energy peak and not the energy dissipation. In Figs. 16 and 17 , representing the shear component (S12) of the energy evolution, at the higher node (node 296), the two highest peaks are recorded at -58601 J and 56784 J (with two directions for the shear components), followed by fluctuating decreasing peaks toward stabilization. At the lower node (node 12), only two peaks have been registered: -13197 J and the second at 14762 J, then followed by a quasi-constant stabilization. The energy release is still mostly recorded at the upper node, as well as the stored energy; in the shear component, the stored energy is higher at the upper level. In addition to the 17 m clear spacing and two calculated components, the major influence, either in terms of stored or released energy, is mainly governed by the present inertia and the arch-shaped area of the tunnel in the shear component, without an influence from the in-situ stress. In Figs. 18 and 19 , the horizontal and shear components (S11, S12) of energy evolution are represented respectively at node 246, which is located in the straight-shaped region of the tunnel's direct vicinity. The horizontal component (S11) of node 246 registers a peak at 79,001 J, followed by a fluctuating recovery at 29,381 J, then stabilizes around 60,000 J. The shear component (S12) of the same node registers a first peak at − 46,438 J and a second at 115,440 J, then stabilizes with a fluctuating trend. In the immediate vicinity of the tunnel's arch waist, energy evolution is recorded in energy dissipation, with no initial peaks as registered previously with nodal outputs of the MRP in the horizontal component; the same applies to the shear component. In Fig. 20, the horizontal component (S11) of the energy evolution at node 242, located in the arch-shaped region of the tunnel's direct vicinity, is represented. The horizontal component (S11) of node 246 registers a peak at 225,745 J, followed by a fluctuating recovery at 129,991 J, then stabilizes around 180,000 J. This time, the released energy is more significant than in the straight-shaped geometry of the tunnel, confirming that the arched shape plays a role in influencing inertia. It is also relevant to mention that node 242 is higher than node 246 (even if only slightly), which excludes the effect of the in-situ stress during the energy dissipation process. The recorded fluctuations in the calculation output are mainly due to the mechanical reaction of the loading process, where crack propagation is marked by a peak, followed by a recovery of the material subjected to loading. 5. Conclusion This study pointed out the energy evolution in the middle rock pillar (MRP) of a twin tunnel and its vicinity, in surrounding rock with poor mechanical properties. It is important to note that the clear distance and the poor mechanical properties of the surrounding rock are contributory to the degree of interaction between both tunnels due to significant deformations. For instance, this case presents an area of perfect overlap of the plastic radius (Re) of both tunnels, where that region recorded the most horizontal displacements at the level of the tunnel arch waist in the MRP region, and the arch waist on the opposite side recorded many fewer horizontal displacements compared with the MRP side. The energy dissipation in the MRP is completely independent of in-situ stress, as the lower nodes certainly recorded more stored energy, but a more constant energy evolution has also been recorded in the lower nodes, where the tunnel has a rather straight-shaped geometry. The higher nodes are located at a level where the tunnel has an arch-shaped geometry; in addition to the distance from the MRP, inertia also influences the energy evolution behaviour, where the released energy is more pronounced in the arch geometry of the tunnel. In the direct vicinity of the tunnel, the energy dissipation is more prominent, where no initial energy storage is present; only energy dissipation is recorded. In this region as well, the arch-shaped region (higher node) records more energy released than the straight-shaped region (lower node), which confirms the influence of inertia in the energy dissipation process. Declarations Acknowledgment Authors express their deep gratitude for granting access to the construction yard, as well as their accompanying during the internship of the first author. Authors are also thankful to the responsible society of the constructions for their help and providing the necessary data for this work’s fulfilment. Data availability Data will be made available upon reasonable request to the corresponding author. Authors’ contribution Allouache Abdelaziz N : Conceptualization, Data acquisition, Investigation, Data curation, Formal analysis, Methodology, Supervision, Validation, Original draft. Saurabh Kureel : Abaqus software, Supervision, Review & editing. Funding This work did not receive any funding, or any specific grant from any part of any kind. Competing interest Authors of this work declare that there are no competing interests, conflicts, or any related conflicts of interests from any part of any kind AI use declaration During the preparation of this work the authors used “Zero GPT” AI tool (link: https://www.zerogpt.com) in order to check the grammar and spelling. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication. References Chen RP, Zhu J, Liu W, Tang XW (2011) Ground movement induced by parallel EPB tunnels in silty soils. Tunnelling and Underground Space Technology 26(1):163-171, DOI: 10.1016/j.tust.2010.09.004 Das R, Singh PK, Kainthola A, Panthee S, Singh TN (2017) Numerical analysis of surface subsidence in asymmetric parallel highway tunnels. 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line\u003c/u\u003e.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/df6d8f0b841e0a843e863217.png"},{"id":91198983,"identity":"6fcf62db-2838-42ee-940b-4c0c3d385935","added_by":"auto","created_at":"2025-09-12 15:18:21","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":50654,"visible":true,"origin":"","legend":"\u003cp\u003eTunnel’s cross sections in “meter”.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/76fbad232d63e685b7456fd5.png"},{"id":91198989,"identity":"ed148141-0ebe-4ba9-93b0-f35a51ec2f4f","added_by":"auto","created_at":"2025-09-12 15:18:21","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":344587,"visible":true,"origin":"","legend":"\u003cp\u003eCretaceous Argillite rock, which is the major surrounding rock constitution of the tunnel “T1”.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/e5c015e3448a9c479959b1a1.png"},{"id":91200445,"identity":"8993bd21-501b-4001-bdc1-4eb4ea139b10","added_by":"auto","created_at":"2025-09-12 15:26:21","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":258351,"visible":true,"origin":"","legend":"\u003cp\u003eCollapse phenomenon occasioned damages.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/e57f6816e58eadd70533740a.png"},{"id":91200448,"identity":"7983de6d-245c-4dba-8099-8a2228709c61","added_by":"auto","created_at":"2025-09-12 15:26:21","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":287858,"visible":true,"origin":"","legend":"\u003cp\u003eModel mesh and boundary.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/2b6145d432fcfe1fa77d0f25.png"},{"id":91200795,"identity":"11017fc7-d3cb-4691-90d1-3346f4a3b793","added_by":"auto","created_at":"2025-09-12 15:34:21","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":103413,"visible":true,"origin":"","legend":"\u003cp\u003eExtension of the plastic radius (Re) of both tunnels and their overlap area.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/14716ba2657f63fd5b244068.png"},{"id":91198993,"identity":"081ad212-fd68-4ec7-993b-76985eb0c427","added_by":"auto","created_at":"2025-09-12 15:18:21","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":114270,"visible":true,"origin":"","legend":"\u003cp\u003eNodes considered for the vertical components of the energy evolution in the MRP.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/bc8b4a1fa712a08f7d4e300a.png"},{"id":91199046,"identity":"ac05fdf1-98af-49a2-b60d-8f8863675499","added_by":"auto","created_at":"2025-09-12 15:18:24","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":132296,"visible":true,"origin":"","legend":"\u003cp\u003eNodes considered for the horizontal and shear components of the energy evolution in the MRP.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/b95813b57b259a9e4b93d60c.png"},{"id":91198995,"identity":"31af53d5-2f45-4ca9-87db-626882396d29","added_by":"auto","created_at":"2025-09-12 15:18:21","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":194006,"visible":true,"origin":"","legend":"\u003cp\u003eModel’s horizontal displacements\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/728487f42611baa642bace13.png"},{"id":91200457,"identity":"1a161816-5040-4220-9478-7bd36920b42d","added_by":"auto","created_at":"2025-09-12 15:26:21","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":66087,"visible":true,"origin":"","legend":"\u003cp\u003eRecorded horizontal displacements.\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/4e8a9c7e7f81a645bff65198.png"},{"id":91200452,"identity":"aeeb8999-1a57-49cf-9fb2-0838f3613a07","added_by":"auto","created_at":"2025-09-12 15:26:21","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":12627,"visible":true,"origin":"","legend":"\u003cp\u003eLayout of the monitoring points.\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/cc828825b6637ef6673e360b.png"},{"id":91200459,"identity":"cf35e75b-567b-42d6-a682-59dbcb6591f3","added_by":"auto","created_at":"2025-09-12 15:26:21","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":63701,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the vertical component of “node 296”.\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/4366285df539af4ce8ae8d57.png"},{"id":91201542,"identity":"3e2d2379-8efe-4573-a11e-3f64747b6e3c","added_by":"auto","created_at":"2025-09-12 15:42:21","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":59986,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the vertical component of “node 12”.\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/c53470d73cbc6873db6dce7f.png"},{"id":91201890,"identity":"6a754eb2-b1a6-4a90-bba0-52da5be9315d","added_by":"auto","created_at":"2025-09-12 15:50:21","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":54201,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the horizontal component of “node 296”.\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/c33da9bcbab1b63fcf8cd38d.png"},{"id":91199006,"identity":"e3ea47cf-d30f-43ac-992d-3243b7e193a1","added_by":"auto","created_at":"2025-09-12 15:18:22","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":57351,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the horizontal component of “node 12”.\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/b2d932a93d1c11ea6b19ea18.png"},{"id":91199010,"identity":"86c8c888-c3c7-4ea9-93ef-8480fca70f06","added_by":"auto","created_at":"2025-09-12 15:18:22","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":78640,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the shear component of “node 296”\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/1b3e160ad3ff3d182ade033b.png"},{"id":91200461,"identity":"e63ca78b-62e7-4424-9ab9-be795ede0d01","added_by":"auto","created_at":"2025-09-12 15:26:22","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":61485,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution in the shear component of “node 12”.\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/2bbdf89e47c99580ce03cb4a.png"},{"id":91199007,"identity":"be758563-f269-49fb-9e7c-27aa38652407","added_by":"auto","created_at":"2025-09-12 15:18:22","extension":"png","order_by":18,"title":"Figure 18","display":"","copyAsset":false,"role":"figure","size":61150,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution of the horizontal component of “node 246”.\u003c/p\u003e","description":"","filename":"18.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/833caa1e4eee7f4d971239ce.png"},{"id":91200463,"identity":"579f1672-b0bd-472d-a3a0-9d0280533189","added_by":"auto","created_at":"2025-09-12 15:26:22","extension":"png","order_by":19,"title":"Figure 19","display":"","copyAsset":false,"role":"figure","size":85906,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution of the shear component of node “246”.\u003c/p\u003e","description":"","filename":"19.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/c7c7b73d8ae9e843a49f0e16.png"},{"id":91199008,"identity":"6367a612-f2a9-487d-8017-bc4994814810","added_by":"auto","created_at":"2025-09-12 15:18:22","extension":"png","order_by":20,"title":"Figure 20","display":"","copyAsset":false,"role":"figure","size":56034,"visible":true,"origin":"","legend":"\u003cp\u003eEnergy evolution of the horizontal component at “node 242”.\u003c/p\u003e","description":"","filename":"20.png","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/ab31afbf9681fb7d119f2b8e.png"},{"id":91332318,"identity":"6b6c5180-0a02-4e76-9fea-cd4f0d7edd19","added_by":"auto","created_at":"2025-09-15 11:11:11","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2906182,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7586323/v1/c8b76d1f-3a90-4384-861e-4a4dc52a39b5.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eSimplified study on the evolution of the stored and released energy in the middle rock pillar of a twin-tunnel and its influence on the neighbour tunnel\u003c/strong\u003e\u003c/p\u003e","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eWith the expansion of traffic congestion, underground engineering has emerged as the best alternative to consider in order to satisfy transport needs, as it provides an infinite space for traffic infrastructures. Due to the continuous need for transport development, twin tunnels with small clear spacing have become increasingly common (Chen et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Li et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Xia et al., \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Elwood and Martin, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Das et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Gao et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Wang et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Shivaei et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eIn terms of numerical simulation, Soliman et al. (\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1993\u003c/span\u003e) derived that the stress field around the second tunnel was no longer the initial stress field and that the central axis of the tunnel and the stiffness of the stratum were no longer symmetrical. Zheng et al. (\u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) investigated the mechanical properties of the middle soil column between two tunnels with small spacing by using a combination of model tests and numerical calculations (1). In the research of laboratory testing, the uniaxial loading test has been conducted in different orientations and locations to explore the failure mechanisms of twin-tunnels (Dhar et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e1981\u003c/span\u003e). Jiang et al. (\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) exhibited the overall failure process of twin-tunnels through 3D printing-based physical simulations, and they identified the vital part of the structural failure by observation and automated measurements. Jia et al. (\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2021\u003c/span\u003e) developed a test system for use with the twin-tunnel excavation model to determine the deformation pattern of the surrounding rock and supporting structure. Their research indicated that the failure of twin-tunnels is caused by the concentration of stress that occurs mainly in the middle rock pillar between the two tunnels. However, the stress concentration phenomenon gradually disappears with the increasing clear spacing; when the clear spacing between the twin-tunnels exceeds twice the span of the single tunnel, its influence on the stability of the surrounding rock and supporting structure of the twin-tunnels is negligible.\u003c/p\u003e\u003cp\u003eNg. et al. (2004) presented an interesting numerical investigation on the multiple interactions between large parallel hypothetical twin tunnels constructed in stiff soils using the NATM. This study pays special attention to the influence of the lagging distance between twin tunnel excavated faces (LF), indicating a strong effect of LF on the behaviour of both tunnels. It should be mentioned that the behaviour of tunnels excavated using the NATM method and the mechanized method are very different, caused not only by the tunnel shape but also by the components of the construction loading along the tunnelling direction (Do et al. 2016).\u003c/p\u003e\u003cp\u003eZhu et al (\u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) and Zhou et al (\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) studied rock specimens containing two rectangular holes and revealed the characteristics of energy evolution and fracture processes of the surrounding rock. The existence of double-hole leads to more degradation of mechanical properties compared to a single-hole. The interactions between twin tunnels complicate the stress field of the surrounding rock, thus affecting the mechanical properties and fracture evolution of the surrounding rock (Wang et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; He et al. \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Tan et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Han et al. \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e2022\u003c/span\u003e, Yang et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). In fact, apart from the circular, square, and elliptical tunnel shapes, the straight-wall-top-arch tunnel shape is more commonly encountered in metal mines and underground tunnel engineering. Therefore, some researchers have begun to investigate the influence of straight-wall-top-arch tunnel shape on the mechanical behaviour of the surrounding rock (Wu et al. 2023 (a), (b); Qiu et al. \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). The obtained results indicate that under the same impact conditions, the stress concentration in the straight-wall-top-arch tunnel is substantially greater than that in the circular tunnels (Wang et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2024\u003c/span\u003e; Wang et al. \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eIn this paper, a twin horseshoe- shaped tunnel located in the eastern part of the Algerian east-west freeway, precisely in Djebel Ouahch, Constantine, is taken as a reference for the study. The tunnel geometry influences the effect of the released and accumulated energy at the bench (waist) and the arched sidewall according to the nodal outputs of a dynamic FEA numerical assessment. It is also relevant to mention that the plastic radius of both tunnels plays a prominent role, as the overlap section of both plastic radii registered the highest rate of horizontal displacements, precisely at the middle rock pillar, and the recorded horizontal displacements were fewer on the other side of the bench of both tunnels. The results contribute significantly to the understanding of the behaviour of the MRP of a twin tunnel in the field of tunnelling and underground engineering.\u003c/p\u003e"},{"header":"2. Project’s overview","content":"\u003cp\u003eThe tunnel of \"Djebel Ouahch,\" located in Algeria's Constantine province, serves as the benchmark for this research study. This tunnel is a section of the Maghreb Unity Highway (AUM, acronym in French). It spans around 700 km in length, traversing Algeria with an overall distance of 1,200 km. It crosses the Djebel Ouahch Mountain in northeastern Algeria, situated in the Constantine province, ranging from pk205\u0026thinsp;+\u0026thinsp;393 to pk207\u0026thinsp;+\u0026thinsp;384.5 (\"pk\" denotes a French term for kilometric point), totaling a length of 1,909 m.\u003c/p\u003e\u003cp\u003eIt consists of a twin-tunnel with a radius of 9.57 m, positioned 17 m apart from one another. The typical depth is approximately 120 m, with a local maximum of 140 m, whereas the thinnest coverage is 12 m, featuring an area nearing 190 m\u0026sup2; in the form of a lowered vault, illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eBased on the reconnaissance campaign conducted, both tunnels have consistently intersected the marl-limestone zone linked to the Tellian layer, especially within the Cretaceous argillite unit. Quaternary colluviums seem to appear locally at the surface. Rough colluviums consist of heterogeneous sandstone components, pebbles, and angular boulders with red silty matrix, as illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eOn January 1, 2014, from 16:00 to 16:30, a significant ground subsidence took place in the left tunnel, spanning from mileage pk206\u0026thinsp;+\u0026thinsp;150 to pk206\u0026thinsp;+\u0026thinsp;280 (with \"pk\" being a French abbreviation for kilometric point). The collapse significantly harmed the right tube (which was already finished and operational at that time), inflicting serious harm to the lining of the right tube, with the total length of the collapse reaching 140 m. No injuries or fatalities were reported after the incident.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eIn the construction documentation, both tunnels have been dug utilizing the identical procedure. The right tube has been fully excavated and is operational; thereafter, the second tube (left tube) was constructed. The NATM (New Austrian Tunnelling Method) was employed in the construction of the left tube, using the bench-cut excavation approach. A two-tiered support structure was utilized during the construction phase of the left tunnel. The materials for the temporary support scheme included HEB 200 steel arches with a pitch spacing of 0.75 m, 6 m long hollow micro-fissured fiberglass bolts, and a 30 cm thick layer of shotcrete. A 60 cm thick layer of reinforced concrete was utilized for the secondary lining.\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cp\u003eBased on a 2D FEA model conducted using Abaqus with a dynamic/explicit solver to identify failure phenomena with a plastic behavior constitutive model, a nodal output is considered to effectively identify the evolution of energy in the MRP and the left tunnel waist. The right tunnel has not been considered in this study since it did not undergo failure despite having been heavily damaged. The model geometry is 200 m long and 200 m deep, with a free mesh using a medial axis algorithm composed of 2806 quadratic elements, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Figure\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e represents the expansion of the plastic radius, designated as \"Re\" of both tunnels with their perfect overlap area.\u003c/p\u003e\u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e, two nodes are selected in the middle rock pillar for the vertical component S22, namely node 296 and node 12. Figure\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e shows the considered nodes for the horizontal and shear components S11 and S12, respectively. A perfect elastic-plastic model represents the behaviour model in this case. An increase in plasticity can clearly reproduce the behaviour in a plastic regime, with the loading accounted for by gravity loading (the tunnel is buried at 120 m, see sect. 2 para 2).\u003c/p\u003e\u003cp\u003eThe parameters of the behaviour are summarized in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. The plastic radius \"Re\" is calculated according to the Hoek \u0026amp; Brown (1980) rock-support analytical model. The calculated value in this case is Re\u0026thinsp;=\u0026thinsp;16.91 m.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003eBehaviour constitutive model used for the simulation.\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"5\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eYoung modulus (MPa)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003ePoisson\u0026rsquo;s ratio\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eYield stress (KPa)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c4\"\u003e\u003cp\u003ePlastic strain (%)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c5\"\u003e\u003cp\u003eUnit weight (KN/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e200\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e605\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e605\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e1.25\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e\u003cp\u003e24\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e605\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e2.8\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e/\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e605\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c4\"\u003e\u003cp\u003e4.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c5\"\u003e\u0026nbsp;\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e"},{"header":"4. Results \u0026 discussions","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1. Results\u003c/h2\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e shows the recorded horizontal displacements of the model. According to the output, the maximum displacements are recorded at both tunnels\u0026apos; waists on the MRP side, with the left tunnel reaching 124 mm of convergence and the right tunnel reaching 123 mm, while significantly fewer displacements are observed on the other side of both tunnels\u0026apos; waists, with 41.43 mm in the left tunnel and 41.44 mm in the right tunnel. Figures \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e show that the recorded displacements are reasonably correlated with those of the modelling, with 90.5% and 88.44% at the critical points in order to validate the proposed model.\u003c/p\u003e\n \u003cp\u003eThe Figs. \u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e \u0026amp; \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e show the vertical component (S22) of the energy accumulation in the MRP at nodes 296 and node 12 respectively with the energy (in Joules) against time (S).\u003c/p\u003e\n \u003cp\u003eBoth nodes show a constant evolution of energy, with a difference in behavior at the post-peak energy level. The lower node registers a peak at almost 700,000 J, exactly at 698,725 J, with a constant post-peak energy evolution. The higher node peaks slightly under the first one, exactly at 693,122 J, with some fluctuation in the post-peak energy behavior.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e15\u003c/span\u003e show the horizontal component (S11) of the energy accumulation in the MRP at nodes 296 and 12, respectively, with energy (in Joules) against time (S). At the deeper point (node 12, Fig. \u003cspan class=\"InternalRef\"\u003e15\u003c/span\u003e), there is a constant energy evolution with a peak at 421,192 J, followed by a recovery at 305,163 J, then progressively stabilizing at 342,689 J. At the upper node (node 296, Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e), a peak is registered at 346,446 J, followed by a recovery at 142,374 J, then a fluctuating stabilization is witnessed that reaches 302,670 J.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e16\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e17\u003c/span\u003e show the shear component (S12) of the energy accumulation in the MRP at nodes 296 and 12, respectively, with energy (in Joules) against time (S).\u003c/p\u003e\n \u003cp\u003eAt the higher node (node 296), the two highest peaks are recorded at -58601 J and 56784 J (with two directions for the shear components), followed by fluctuating decreasing peaks toward stabilization. At the lower node (node 12), only two peaks have been recorded: -13197 J and the second at 14762 J, followed by a quasi-constant stabilization.\u003c/p\u003e\n \u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e18\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e19\u003c/span\u003e show the energy evolution at node 246 in the horizontal and shear components (S11, S12), respectively, with energy (in Joules) plotted against time (S). The horizontal component (S11) of node 246 registers a peak at 79001 J, followed by a fluctuating recovery at 29381 J, then stabilizes around 60000 J. The shear component (S12) of the same node registers a first peak at -46438 J and the second at 115440 J, then stabilizes with a fluctuating trend.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;20 shows the energy evolution of the horizontal component at \u0026quot;node 242,\u0026quot; with the energy (in Joules) plotted against time (s).\u003c/p\u003e\n \u003cp\u003eThe horizontal component (S11) of node 246 registers a peak at 225,745 J, followed by a fluctuating recovery at 129,991 J, then stabilizes around 180,000 J.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2. Discussions / interpretation\u003c/h2\u003e\n \u003cp\u003eIn Fig. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e, the numerical output of the horizontal displacements is shown. It is obvious that both tunnels\u0026apos; waists underwent the most horizontal convergences, while the other side of both tunnels\u0026apos; waists shows significantly lower horizontal convergence values. The output is validated with a fair degree of correlation, as reported in Fig. \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e. The most convergences in the MRP were registered at the straight arch waist of both tunnels and not the arch waist side. The major factor in this phenomenon is the perfect overlap of both tunnels \u0026apos; plastic radius (Re), as the other sides of both tunnels (the left side of the left tunnel and the right side of the right tunnel) have shown minor convergences compared to the mentioned areas of the MRP.\u003c/p\u003e\n \u003cp\u003eIn Figs. \u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e, representing the vertical component (S22) of the energy evolution, both nodes show a constant evolution of energy, with a difference in behaviour at the post-peak energy level. The lower node (node 12) registers a peak of almost 700,000 J, exactly at 698,725 J, with a constant post-peak energy evolution. The higher node (node 296) peaks slightly under the first one, exactly at 693,122 J, with some fluctuations in the post-peak energy behaviour. The deeper node (node 12) stores more energy than the higher node (node 296) with the increase of in-situ stress; however, the energy release is more prominent at the higher node than at the lower node, which shows a constant evolution (Fig. \u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e). Given that the clear distance between both tunnels is 17 m, and the location of the lower node coincides with a rather straight geometry of the tunnel while the higher node aligns with an arch-shaped geometry, there is an evident influence between the tunnels. The higher node (node 296) undergoes an inertia that makes it sensitive to energy release in the arch-shaped tunnel region, whereas at the lower node (node 12), the peak evolution is constant, knowing that the node is located in a straightforward geometry of the tunnel, which makes the energy dissipation independent of in-situ stress.\u003c/p\u003e\n \u003cp\u003eIn Figs. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e15\u003c/span\u003e, representing the horizontal component (S11) of the energy evolution, at the deeper point (node 12, Fig. \u003cspan class=\"InternalRef\"\u003e15\u003c/span\u003e), there is a constant energy evolution with a peak at 421,192 J, followed by a recovery at 305,163 J, then progressively stabilizing at 342,689 J. At the upper node (node 296, Fig. \u003cspan class=\"InternalRef\"\u003e14\u003c/span\u003e), a peak is recorded at 346,446 J, followed by a recovery at 142,374 J, then a fluctuating stabilization is observed that reaches 302,670 J. The deeper area always records a higher value of energy, with the horizontal component this time, and the energy release is consistently registered at the upper node. The horizontal component shows a more pronounced energy release at the upper node where the 17 m clear spacing and the inertia of the arched side wall, associated with the location of the designated node, have an obvious influence on the energy release. The lower node, on the other hand, registers a peak at 421,192 J, followed by a recovery at 305,163 J, then steadily stabilizes at 342,689 J, knowing that its location is at a straight-shaped geometry of the tunnel, and the in-situ stress only influences the energy peak and not the energy dissipation.\u003c/p\u003e\n \u003cp\u003eIn Figs. \u003cspan class=\"InternalRef\"\u003e16\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e17\u003c/span\u003e, representing the shear component (S12) of the energy evolution, at the higher node (node 296), the two highest peaks are recorded at -58601 J and 56784 J (with two directions for the shear components), followed by fluctuating decreasing peaks toward stabilization. At the lower node (node 12), only two peaks have been registered: -13197 J and the second at 14762 J, then followed by a quasi-constant stabilization. The energy release is still mostly recorded at the upper node, as well as the stored energy; in the shear component, the stored energy is higher at the upper level. In addition to the 17 m clear spacing and two calculated components, the major influence, either in terms of stored or released energy, is mainly governed by the present inertia and the arch-shaped area of the tunnel in the shear component, without an influence from the in-situ stress.\u003c/p\u003e\n \u003cp\u003eIn Figs. \u003cspan class=\"InternalRef\"\u003e18\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e19\u003c/span\u003e, the horizontal and shear components (S11, S12) of energy evolution are represented respectively at node 246, which is located in the straight-shaped region of the tunnel\u0026apos;s direct vicinity. The horizontal component (S11) of node 246 registers a peak at 79,001 J, followed by a fluctuating recovery at 29,381 J, then stabilizes around 60,000 J. The shear component (S12) of the same node registers a first peak at \u0026minus;\u0026thinsp;46,438 J and a second at 115,440 J, then stabilizes with a fluctuating trend. In the immediate vicinity of the tunnel\u0026apos;s arch waist, energy evolution is recorded in energy dissipation, with no initial peaks as registered previously with nodal outputs of the MRP in the horizontal component; the same applies to the shear component.\u003c/p\u003e\n \u003cp\u003eIn Fig.\u0026nbsp;20, the horizontal component (S11) of the energy evolution at node 242, located in the arch-shaped region of the tunnel\u0026apos;s direct vicinity, is represented. The horizontal component (S11) of node 246 registers a peak at 225,745 J, followed by a fluctuating recovery at 129,991 J, then stabilizes around 180,000 J. This time, the released energy is more significant than in the straight-shaped geometry of the tunnel, confirming that the arched shape plays a role in influencing inertia. It is also relevant to mention that node 242 is higher than node 246 (even if only slightly), which excludes the effect of the in-situ stress during the energy dissipation process.\u003c/p\u003e\n \u003cp\u003eThe recorded fluctuations in the calculation output are mainly due to the mechanical reaction of the loading process, where crack propagation is marked by a peak, followed by a recovery of the material subjected to loading.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study pointed out the energy evolution in the middle rock pillar (MRP) of a twin tunnel and its vicinity, in surrounding rock with poor mechanical properties. It is important to note that the clear distance and the poor mechanical properties of the surrounding rock are contributory to the degree of interaction between both tunnels due to significant deformations. For instance, this case presents an area of perfect overlap of the plastic radius (Re) of both tunnels, where that region recorded the most horizontal displacements at the level of the tunnel arch waist in the MRP region, and the arch waist on the opposite side recorded many fewer horizontal displacements compared with the MRP side.\u003c/p\u003e\u003cp\u003eThe energy dissipation in the MRP is completely independent of in-situ stress, as the lower nodes certainly recorded more stored energy, but a more constant energy evolution has also been recorded in the lower nodes, where the tunnel has a rather straight-shaped geometry. The higher nodes are located at a level where the tunnel has an arch-shaped geometry; in addition to the distance from the MRP, inertia also influences the energy evolution behaviour, where the released energy is more pronounced in the arch geometry of the tunnel.\u003c/p\u003e\u003cp\u003eIn the direct vicinity of the tunnel, the energy dissipation is more prominent, where no initial energy storage is present; only energy dissipation is recorded. In this region as well, the arch-shaped region (higher node) records more energy released than the straight-shaped region (lower node), which confirms the influence of inertia in the energy dissipation process.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eAcknowledgment\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAuthors express their deep gratitude for granting access to the construction yard, as well as their accompanying during the internship of the first author. Authors are also thankful to the responsible society of the constructions for their help and providing the necessary data for this work’s fulfilment.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eData will be made available upon reasonable request to the corresponding author.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors’ contribution\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eAllouache Abdelaziz N\u003c/u\u003e: Conceptualization, Data acquisition, Investigation, Data curation, Formal analysis, Methodology, Supervision, Validation, Original draft.\u003c/p\u003e\n\u003cp\u003e\u003cu\u003eSaurabh Kureel\u003c/u\u003e: Abaqus software, Supervision, Review \u0026amp; editing.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work did not receive any funding, or any specific grant from any part of any kind.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interest\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAuthors of this work declare that there are no competing interests, conflicts, or any related conflicts of interests from any part of any kind\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAI use declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eDuring the preparation of this work the authors used “Zero GPT” AI tool (link: https://www.zerogpt.com) in order to check the grammar and spelling. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eChen RP, Zhu J, Liu W, Tang XW (2011) Ground movement induced by parallel EPB tunnels in silty soils. Tunnelling and Underground Space Technology 26(1):163-171, DOI: 10.1016/j.tust.2010.09.004\u003c/li\u003e\n \u003cli\u003eDas R, Singh PK, Kainthola A, Panthee S, Singh TN (2017) Numerical analysis of surface subsidence in asymmetric parallel highway tunnels. Journal of Rock Mechanics and Geotechnical Engineering 9(1):170-179, DOI: 10.1016/j.jrmge.2016.11.009\u003c/li\u003e\n \u003cli\u003eDhar BB, Ratan S, Sharma DK, Rao PM (1981) Model study of fracture around underground excavations in weak rocks. 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Mech. 184, 183\u0026ndash;201, https://doi.org/10.1016/j.engfracmech.2017.08.029\u003c/li\u003e\n \u003cli\u003eZhu Q, Li D, Han Z, Xiao P, Li B (2022) Failure characteristics of brittle rock containing two rectangular holes under uniaxial compression and coupled static-dynamic loads, Acta Geotech. 17 131\u0026ndash;152, https://doi.org/10.1007/s11440-021-01196-8.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"National higher school of technology and engineering","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Twin-tunnels, middle rock pillar, energy evolution, FEA, dynamic analysis, plastic radius overlap","lastPublishedDoi":"10.21203/rs.3.rs-7586323/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7586323/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIn this study, the evolution of energy in the middle rock pillar (MRP) of a soft rock twin tunnel is analysed using dynamic FEA simulation. The most affected region by energy storage is the MRP; however, despite accumulating substantial energy, the energy storage is independent of the in-situ stress according to vertical components output. The energy release is significant in the adjacent tunnel when horizontal components are considered. The plastic radius of both tunnels and their geometry play a significant role; for instance, the overlapping part of both plastic radii (Re) registers the most convergences (horizontal displacements) at the bench level of the tunnel. The energy evolution is also linked to the geometry of the tunnel, where the more arched the tunnel shape becomes, the more energy dissipates in that region of the tunnels, and vice versa at the tunnel arch waist, where the geometry is rather straight. These results can efficiently contribute to the body of knowledge in underground engineering, particularly in twin- tunnel excavation cases.\u003c/p\u003e","manuscriptTitle":"Simplified study on the evolution of the stored and released energy in the middle rock pillar of a twin-tunnel and its influence on the neighbour tunnel","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-12 15:18:16","doi":"10.21203/rs.3.rs-7586323/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"bdee882b-1bee-4f66-9078-1a38b158c40c","owner":[],"postedDate":"September 12th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":54533702,"name":"Numerical Analysis"},{"id":54533703,"name":"Geology"},{"id":54533704,"name":"Civil Engineering"}],"tags":[],"updatedAt":"2025-09-12T15:18:17+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-12 15:18:16","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7586323","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7586323","identity":"rs-7586323","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}