# Performance Analysis of Tunnel Boring Machines for
## 1 Introduction  
It is known that hard rock excavation by means of tunnel boring machines (TBM) brings many benefits in terms of productivity and safety, but at the same time the tunnel geometry is limited to a circular shape [1], so the method is more appropriate for projects with constant shapes [2]; moreover, it is not recommended for tunnels shorter than \(1\) km, as the installation times would be of the same order of magnitude as the excavation times [3, 4]. TBMs are classified into large families based on the type of material being excavated. Open TBMs are suitable for hard and compact rocks, single shield machines offer more protection in fractured rock [5, 6] while the double shields generally allow a higher advancement speed compared to the single shield. Regardless of the kind of machine, it is crucial for tunnelling stockholders to predict the performance of the excavation. This issue has been addressed differently by many authors. Empirical and analytical models aimed to predict both the cutting depth of the tools and their wear have been studied by [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Together with the aforementioned studies, three important research schools have drawn the attention of the scientific community to their prediction models, capable of successfully predicting the penetration rate (mm/rev) considering both the parameters of the machine and those of the rock. More in details, pertaining to the machine, total thrust [N], thrust/tool (which depends on cutter diameter), cutterhead diameter, cutter geometry, spacing between tools, and tools numbers are required while, as for the rock parameters,the uniaxial compressive strength (UCS), Brazilian tensile strength (BTS), and drilling rate index (DRI) are needed. The contribution of the Turkish school (Istanbul Technical University) in this field is very relevant [24; 25; 26; 27; 28]; take into account the presence of discontinuities to better describe the properties of the rock mass. The authors investigated in depth the influence of joint orientation, spacing, and persistence on the extent of tool penetration and wear. The Norwegian University of Science and Technology (NTNU) prediction model for hard rock TBMs combines rock boreability properties and TBM parameters in order to determine the main factors influencing the penetration rate predictions [29]. Predictions of advancement rate, cutter wear, and excavation costs are also achieved using the NTNU model. The first version of the model was proposed by [30], and it has been continuously updated with parameters derived from new projects. The analytical model proposed by the Colorado School of Mines (CSM) is able to predict excavation performance based on a \"full-scale\" laboratory test, i.e., the so-called linear cutting test. The conditions of the rock-tool interaction during excavation by TBM are well reconstructed with this test, which allows to estimate the cutting force, the normal force and the tangential force acting on a disk that performs linear paths on a rock specimen with dimensions 1 \(\times\) 1 \(\times\) 0.6 m. For a given case study, the test is commonly repeated several times, evaluating different spacing and penetrations. The first version of the model was developed in 1977 [31; 32] and was later revised [33; 34; 35]. The input parameters are both geotechnical and TBM parameters; based on the physical model coming from the laboratory, a penetration value p (mm/rev) can be assumed and, since the tool radius R (mm) is known, the contact angle is settled, which defines the rock region affected by the stress related to the interaction with the tool. Once this value is known, the tool contact pressure is calculated and the thrust/tool Ft (kN/tool) is assessed.