1. Introduction
The underground water-sealed cavern has changed the traditional way of the surface storage tank [1,2]. In the construction stage of underground caverns, the phenomenon of rock instability and failure is widespread, and the resulting safety problems cannot be ignored. Therefore, the instability and failure of the surrounding rock has always been a hot topic in the research of underground engineering.
Since the 1970s, scholars have carried out a lot of research work. In terms of theoretical research [3,4,5], most scholars regard rock mass as a homogeneous material and obtain analytical solutions according to elastic-plastic theory. However, the deformation of the rock and soil mass has typical nonlinear characteristics [6], and the external loads borne by actual projects are complex and changeable. It is difficult to solve the above problems using analytical methods. Therefore, numerical simulation methods are increasingly favored by scholars [7,8]. The complex deformation characteristics and variable boundary conditions of the rock and soil mass can be fully considered in the analysis of tunnel deformation and failure by using finite element and discrete element methods [9]. For example, [10] obtained the critical spacing of granite double tunnels by using the finite element method. [11] explored the anchoring effect of the anchor bolt and the law of crack propagation by simulating the compression of gypsum standard cylinder specimens with prefabricated cracks. They obtained the conclusion that with the increase of the anchoring angle, the crack development of the tension wing first increased and then decreased, and the failure mode of the prefabricated crack test block changed from shear failure to tension shear composite failure, and then to shear failure. [12] used PFC to study the failure mode of the surrounding rock of a sandy tunnel and reached the conclusion that the local instability of particles will occur in the sandstone tunnel and that the local instability of the particle force chain is more likely to occur than the shear failure when the cohesion is small.
In addition, the physical model test is also a widely used research means. The model test of restoring the field working condition in a certain proportion can reflect the stress characteristics of the rock mass related to underground engineering. [13,14,15] carried out a variety of tunnel model tests to analyse the plate crack failure mechanism of tunnel surrounding rock under different stress states. [16] carried out model tests on the deformation and failure characteristics of circular tunnels in deep composite strata and concluded that the surrounding rock was subjected to tensile failure on the soft rock side in the horizontal direction; shear failure occurred in the top and bottom soft rock areas.
The research of scholars has greatly promoted the development of the tunnel surrounding rock failure theory and support technology, but there are still shortcomings in solving practical engineering problems. For example, multiple types of cross-sections are used synchronously for the main tunnel, according to need [16,17,18,19], but the current research mainly focuses on the analysis of a certain section form and lacks the comparison between different tunnel shapes. In addition, the numerical simulation results of the finite element method are quite different from the actual project, because most of the finite element simulation software are not good at simulating the failure process of rock, a heterogeneous material with a large dispersion. Aiming at this problem, we refer to the previous work: by means of a physical model test and discrete element numerical simulation, the failure modes of a circular sandstone tunnel are compared from the macro and micro perspectives. On this basis, particle flow code (PFC) simulation models of various cross-section shapes are designed to assess the influence of the cross-section shape on rock stability, from the micro scale, to provide a theoretical basis for the selection of the cross-section shape during engineering design.
2. Physical Model Test
2.1. Physical Model Design
Taking a water-sealed cavern in Qingdao as the engineering background, the proposed site of the cave is tectonically located in the north Subei-south Jiaodong block in the southeast of the Jiaodong block of the North China Plate, where the dikes are slightly developed. Geological survey and drilling show that there are mainly granite porphyry, lamprophyre, diabase and other dikes, scattered in the late Yanshanian monzogranite. It passes through the alteration zone during the excavation and the local area is severely altered, which is characterized by kaolinization and chloritization, resulting in the decrease of the rock mass strength.
The field sampling test results show that the compressive strength of this kind of deteriorated rock mass is lower than 30 MPa, and its characteristics of loose structure, weak cementation and strong permeability are similar to the basic characteristics of sandstone. Therefore, we decided to select sandstone, with similar mechanical properties to the field rock mass, as the material for the physical model, and its basic mechanical characteristic parameters (Table 1) were obtained through the uniaxial compression test of standard cylinder specimens.
The circular section is designed with a size similarity ratio of 1:50, and the diameter of the tunnel is 100 mm. The excavation of the tunnel significantly influences the surrounding rocks within the range of three to five times the tunnel span. Thus, specimens measuring 500 mm × 500 mm × 100 mm were prepared (Figure 1).
The test was performed on a conventional triaxial testing machine to allow for the real-time monitoring of the changes to various parameters (such as load and deformation). A steel press plate was placed on the left and right flanks of the model and connected with a hydraulic pump, through which the confining pressure was imposed on the left- and right-hand sides. A steel plate was separately set on the top and bottom to ensure the uniform application of stress on the model. Moreover, a layer of Vaseline® was smeared over four flanks of the model to reduce the effects of friction thereon. It was necessary to keep the centre of the vertically loaded column to lie on the same vertical line as the tunnel centre to avoid eccentric loading.
In order to restore the impact of the initial in-situ stress on actual engineering rock mass, according to the geological survey report, the horizontal in-situ stress of the proposed site of the cavern was 2.1–3.0 MPa, so the constant horizontal stress in the design test process is 3 MPa. According to the stress path after the excavation of underground cavern in the actual project, the uniform loading mode should be adopted when applying a vertical load in the model test. Meanwhile, in order to avoid the influence of a too fast loading speed on the overall bearing capacity of the model, the displacement controlled loading speed was finally determined to be 0.02 mm/min.
2.2. Test Results
The change of macroscopic cracks directly reflects the failure process in the surrounding rock. Figure 2 shows the changes in the visible cracking patterns on the surface of specimens recorded by a high-speed camera (In Figure 2, t refers to the test time; σz refers to the vertical load applied by the test machine and the cracks are marked in yellow for the convenience of comparison):
The failure process is summarised as follows: a V-shaped crack with thae length of approximately 10 mm is initiated at the upper left of the entrance under a vertical stress of 19.2 MPa; fine particles drop into the tunnel. The surface rock at the upper left of the entrance is cut and exfoliated by the V-shaped crack under a vertical stress of 20.1 MPa; a vertical crack ① with the length of approximately 70 mm occurs in the upper left (30 mm) of the entrance; a vertical crack ② with a length of approximately 80 mm appears in the lower right of the entrance; rib spalling occurs on two sides of the internal wall of the tunnel, and laminae with a thickness of approximately 1 mm are spalled. As the load increases, the lower end of the crack ① extends to the entrance and finally coalesces with the V-shaped crack thereat. A crack ③ with the length of approximately 70 mm, which develops obliquely (50°) to the upper left, appears at the lower end of the crack ① under the stress of 21.5 MPa; an elliptic lamina with a thickness of approximately 2 mm is spalled from the upper end of the crack ②. Under a stress of 22.2 MPa, the surface rock is cut into blocks by the crack at the upper left of the entrance and the exfoliated rock appears as a triangle, with an area of approximately 700 mm2; the crack ③ extends towards the upper left with an angle of 65°, and rock fragments are frequently lost from the internal wall of the tunnel. A vertical crack ④ some 50 mm long is present at approximately 40 mm above the crack ② under a stress of 23.9 MPa; the crack ③ extends by 150 mm towards the upper left at an angle of 65°; the surface rock breaks into fragments. The specimens are subjected to brittle failure after reaching their peak strength. Furthermore, cracks rapidly develop to cut the rock into blocks and a loud noise is generated during the collapse of the tunnel.
The test results showed that the left and right sides of the entrance are subjected to tension and vertical, or oblique cracks with a large dip angle are mainly formed on the surface of the model. The vicinity of the entrance is under shear and rock spalling from the internal wall of the tunnel persists under loading: fine particles are exfoliated at low load while rock fragments spall under high load. The global failure of the model is brittle.
3. PFC Numerical Simulation
The discrete element method breaks through the limitations of the traditional continuum hypothesis and can analyse the propagation and stress change associated with mesoscopic fractures in the disturbed zone during the failure of the surrounding rock. The model test on a circular cross-section was reproduced by using the numerical simulation software PFC2D using the discrete element method to attain more information pertaining to failure of surrounding rocks at the mesoscopic scale.
3.1. Test Design Using PFC
(1) Particles: set random seeds to randomly generate rigid spheres with a particle size range of 4 to 5 mm inside the wall, with a total number of 13,135.
(2) Boundary condition: the dimension of the PFC model is consistent with that of the physical model, with the surrounding walls of the model as the boundary, and the overall size of the test piece is 50 cm × 50 cm; the wall is set as a translatable rigid body (Figure 3a), that is, the wall can be vertically or horizontally displaced, but not rotated or deformed.
(3) Cavern: a wall with a specified shape is generated at the centre of the model and the particles in the wall are deleted to simulate the excavation of the cavern. After the excavation of the cavern, the wall of the cavern is deleted to restore the free state of the surrounding rock (Figure 3b). The walls on left- and right-hand sides of the model were subject to a constant confining pressure of 3 MPa through a servo mechanism in the loading stage. A displacement constraint was applied to the upper and lower sides and the load was uniformly applied at a displacement rate of 0.002 mm/min (Figure 3c).
(4) Contact: the contact between particles is a parallel-bonding model to simulate rock-like materials. The results obtained through the calculation model approximate to those attained through the physical model test in terms of failure mode and applied stress (after multiple adjustments). The final calibrated parameters are displayed in Table 2:
3.2. Changes in Mesoscopic Fractures
The calculation function was compiled by using FISH language to monitor the changes in the mesoscopic fractures in the model in the loading-induced failure process. The red and dark-blue separately represent the tensile and shear fractures, as shown in Figure 4:
The tensile fractures in two sides of the entrance start to aggregate under a stress of 15.4 MPa. At 16.2 MPa, shear fractures occur on two sides; the fractures to the right of the entrance increase and gradually accumulate, appearing in a V-shaped pattern. At 21.2 MPa, the fractures on both the left and right sides of the entrance are distributed in a strip shape and extend approximately along the vertical direction; a small number of particles burst into the tunnel. As the stress increases to 23.2 MPa, fractures to the left of the entrance grow and particles are accumulated, then exfoliated; fractures to the right of the entrance aggregate to form an elliptical fracture zone with a width of 20 mm, causing rib spalling to occur on the right wall. At 24.2 MPa, fractures on the left side are accumulated to form a coalescent fracture zone with a width of 20 mm and a length of 120 mm. When the applied stress reaches 24.3 MPa, the fracture zone to the left of the entrance widens to 27 mm; a long, narrow coalescent-fracture zone with a width of 20 mm and a length of 200 mm is generated on the right. In this case, the specimens are under the peak load: thereafter, the rate of change in the number of fractures increases and the specimens become unstable.
The mesoscopic fractures mainly appear as tensile fractures, and a small number of shear fractures are found in the vicinity of the entrance during the PFC test. The fractures are distributed in an approximately V-shaped pattern on both sides of the entrance and a few particles are exfoliated in the tunnel at a low load. The quasi-vertically distributed fracture concentration zone is separately formed on both sides of the entrance with an increasing load. When approaching the peak load, mesoscopic fractures develop and evolve into a long, narrow coalescent-fracture zone on both sides of the entrance. Under these conditions, the global failure of the model is brittle. By comparing the results obtained through the model test and the PFC test, it can be found that the results match in terms of the bearing capacity, failure mode and fracture distribution. Therefore, the physical model test can be favourably reproduced by utilising the PFC.
4. PFC Tests Based on Different Tunnel Shapes
4.1. Test Design
Four types of cross-sections commonly used in practical engineering were selected based on their mesoscopic parameters. The following four types of PFC simulation models (Table 3) were designed according to the area equivalence principle.
4.2. Test Results
The Figure 5 shows the fracture evolution process during the loading of models of different cross-sections.
The pre-peak fracture evolution process of the models at different stress levels is described as follows:
(1) When the vertical stress reaches 70% of σzmax in the pre-peak stage, a small number of tensile fractures start to accumulate at the end of the long axis on the left-hand side of the elliptic cross-section model. Fractures with a length of approximately 60 mm are distributed in thin strips and extend obliquely (80°) towards the upper right from the lower right of the ovoid cross-section. Moreover, the rock is cut by some fractures in the left wall, thus leading to particle exfoliation. The fractures on the horseshoe-shaped cross-section mainly concentrate on the spandrel to the right-hand side (albeit they are few in number). As for the vertical-wall arched cross-section, a V-shaped fracture zone is formed in the spandrel on the right-hand side, and some particles are exfoliated from the tunnel walls.
(2) When the vertical stress reaches 80% of σzmax in the pre-peak stage, the fractures at the long axis in the left-hand side of the elliptical cross-section begin to develop vertically upwards, and a small number of particles drop from the tunnel walls. Several fractures are accumulated within the area some 12 mm from the lower left of the ovoid cross-section, thus resulting in rib spalling from the tunnel walls. More fractures occur at the spandrel in the left-hand side of the horseshoe-shaped cross-section and a few particles are exfoliated within the tunnel. A V-shaped fracture zone is also formed in the left-hand side of the spandrel of the vertical-wall arched cross-section and the number of exfoliated particles is the largest.
(3) When the vertical stress reaches 90% of σzmax in the pre-peak stage, fractures in the left-hand side of the elliptical cross-section occur in an increasing number and rib spalling occurs in the left wall. A triangular fracture zone with a width of approximately 40 mm is found in the left-hand side of the ovoid cross-section and more rocks are exfoliated in blocks; a vertically distributed strip-shaped fracture zone with a width of approximately 20 mm and a length of approximately 100 mm is generated in the right-hand side. The rock is cut by fractures at the spandrel on the left of the horseshoe-shaped cross-section, causing particle exfoliation in blocks and the increasing fracturing of the arch foot to the right. Fractures at the spandrel and arch foot on the right-hand side of the vertical-wall arched cross-section coalesce to form a semi-circular fracture zone, and rib spalling occurs within the tunnel.
(4) At the peak load, a vertical fracture zone with a width of approximately 30 mm and a length of approximately 190 mm is formed in the left-hand side of the elliptical cross-section and particles drop, en bloc, into the tunnel. The symmetric semi-circular fracture zones are formed on both sides of the ovoid cross-section, each with a maximum width of 50 mm. An obliquely coalescent fracture zone with a width of approximately 20 mm and a length of approximately 170 mm is formed at the arch foot to the lower right of the horseshoe-shaped cross-section. Symmetric semi-circular fracture zones are found in both sides of the vertical-wall arched cross-section, with a maximum width of approximately 50 mm, and the rock on the tunnel walls is exfoliated in blocks.
By comparing the failure processes of the various models, it can be found that there are a small number of fractures on the elliptical and horseshoe-shaped cross-sections at the same applied load; moreover, rock failure in the tunnel is mainly manifest as particle accumulation and exfoliation, and the global stress structure of rocks is finally damaged by the long coalescent-fracture zones. To be specific, the fractures on the elliptic cross-section are mainly distributed at the end of the long axis and those on the horseshoe-shaped cross-section mainly occur in the spandrel and arch foot. There are more fractures on the ovoid and vertical-wall arched cross-sections and significant rib spalling occurs within the tunnel. In addition, a large fracture zone is separately formed on both sides of the tunnel wall and the rock on the tunnel wall is exfoliated en bloc.
5. Analysis on a Mesoscopic Scale
5.1. Comparison of Force Chains
The Figure 6 compares the force chains of various cross-sections when the applied stress reaches 70% of σzmax in the pre-peak stage. The stress on the four types of cross-sections shows a weak effect on the upper and lower sides of the entrance while significant stress concentration appears in both sides.
The stress concentration on the elliptic cross-section is significant at the end of the long axis, where the force chain is approximately vertical. According to Figure 5b, mesoscopic fractures are first concentrated in that position and eventually evolve into a vertical fracture zone with the direction consistent with the force chain. The force chain on the horseshoe-shaped cross-section is distributed along the tunnel wall, and thus, the arch foot and spandrel bear a high stress. As shown in Figure 5l, the rock particles at the spandrel on the left are accumulated and burst under high stress. Fractures at the arch foot on the right are developed due to stress concentration and finally evolve into a long, narrow coalescent-fracture zone. The ovoid and vertical-wall arched cross-sections show similar failure modes and distributions of force chains. Both cross-sections present a height-span ratio of approximately 1.5 and are under significant tensile stress due to the large included angle between the tunnel wall and force chains. Owing to the high stress on the spandrel, fractures are first accumulated thereat, develop along the direction of the force chains and eventually coalesce with those at the arch foot, causing rock to spall from the sidewalls en bloc.
5.2. Bearing Capacity and the Number of Fractures
The following Figure 7 separately compares the changes of the peak stress level and the number of fractures in the models of different cross-sections:
By taking the circular cross-section as the reference, the peak loads on the elliptic, ovoid, horseshoe-shaped and vertical-wall arched cross-sections separately account for 99.55%, 98.31%, 97.12% and 90.16% of that on the circular section, respectively. The ultimate bearing capacity of the elliptic cross-section approximates to that of the circular section and the bearing capacity of the vertical-wall arched cross-section is the lowest.
Figure 8 shows that the stresses on the ovoid and vertical-wall arched cross-section models, when more fractures appear, are lower than those on the other three cross-sections. The rate of change in the number of fractures in various models increases once the stress reaches 17.3 MPa. In terms of the number of fractures at a given applied load, various cross-section models are shown (in descending order) as: ovoid, vertical-wall arched, horseshoe-shaped, elliptical and circular cross-sections. The number of fractures in the ovoid cross-section model suddenly rises to approximately 480 at an applied stress of 21.5 MPa, which is twice that in the horseshoe-shaped cross-section model and 2.5 times those in the elliptic and circular models.
Above all, the cross-section with a large height-span ratio has a lower bearing capacity. Relative to the vertical-wall arched cross-section, the height-span ratio of the ovoid cross-section model is approximately 1.5; however, the symmetric rock fracture zones have been formed in two sides of the tunnel wall after the applied stress reaches 21.9 MPa, that is, 90% of σzmax. The rock which bears the load decreases with the height-span ratio and the force chain is transferred to a greater depth to generate a temporarily stable stress structure, thus improving the global bearing capacity.
5.3. Rock Characteristics in Various Stages of Failure
The failure process of rocks is divided into the fracture initiation, development and coalescence stages according to the fracture distribution characteristics, applied stress and the number of fractures in the rock. The fracture initiation stage starts from the occurrence of fracture accumulation; the fracture development stage begins with particle exfoliation; the fracture coalescence stage begins from the occurrence of a coalescent fracture zone or a sudden growth in the number of fractures. The fracture and stress characteristics in different stages are described in Table 4.
In the initiation stage, the stress on the horseshoe-shaped cross-section model, at which fractures occur, is less than 30% of the peak stress in the pre-peak stage. The stress levels on the ovoid and vertical-wall arched cross-sections are 30% to 40% of that in the pre-peak stage. Both the stress levels on the circular and elliptic cross-sections are higher than 40% of the peak stress in the pre-peak stage. The corresponding stress response occurs earliest on the horseshoe-shaped cross-section. In the fracture development stage, the stresses on the circular, elliptic and horseshoe-shaped cross-sections are all higher than 70% of the peak stress in the pre-peak stage while those on the ovoid and vertical-wall arched cross-sections account for approximately 65% thereof. The stresses on the ovoid and vertical-wall arched cross-sections, when particles are ejected, are lower than those on the other three cross-sections. This indicates that the stress concentration is more significant in both cross-sections. As for the percentage of peak stress in the pre-peak stress on various cross-sections in the fracture–coalescence stage, the cross-sections are shown (in descending order) as: circular, elliptical, horseshoe-shaped, vertical-wall arched and ovoid cross-sections. In terms of the applied stress, the cross-sections are (in descending order): circular, elliptical, horseshoe-shaped, ovoid and vertical-wall arched cross-sections. The circular and elliptic cross-sections are subject to a higher stress when fractures coalesce, which more approximates to the peak load, and the failure of both cross-sections is brittle. The fracture zone where fractures coalesce in large areas is formed earliest in the ovoid cross-section and the tunnel still retains a certain bearing capacity after the rock is spalled from the sidewalls.
6. Conclusions
Based on the engineering background of an underground water-sealed gas storage cavern, a physical model test of a circular cavern is designed in this paper. The failure process of the model is reproduced by the discrete element analysis software, PFC, and the instability and failure mechanism of the surrounding rock are discussed from both the macroscopic and microscopic perspectives. On this basis, the PFC simulation tests of several different tunnel shapes are designed. By comparing the test results, the following conclusions are drawn:
(1). In the model test, the surrounding rocks on the left and right sides of the tunnel are significantly tensioned, mainly forming vertical cracks or oblique cracks with large angles. The vicinity of the entrance is subjected to shear and rock if the internal wall of the tunnel bursts under load. The overall failure of the model is brittle. The PFC test results of circular cross-section model are consistent with the results obtained through the physical model test in terms of the bearing capacity, failure mode and fracture distribution.
(2). The cracks of the elliptical section model are mainly distributed at the end points of the long axis, and rock blocks on sidewalls are exfoliated. The main reason for the failure of the cavern is the formation of a through fracture zone at the right arch corner of the horseshoe-shaped model. The failure mode and force chain distribution of ovoid shaped and vertical-wall arched cross-sections are similar: the cracks in both develop along the direction of force chain, stripping the side wall rock mass into blocks, resulting in the instability and failure of the cavern.
(3). The peak loads on the elliptic, ovoid, horseshoe-shaped and vertical-wall arched cross-sections separately account for 99.55%, 98.31%, 97.12% and 90.16% that on the circular cross-section.
(4). The bearing capacity of the circular and elliptical cross-section models is relatively high when they are destroyed, but the remaining load reserve is low, and the failure of rock mass presents obvious suddenness. From the perspective of the failure mode, when ovoid-shaped and vertical-wall arched cross-sections models are destroyed, the side wall of the cavern forms a fracture zone, which cannot give full play to the overall bearing capacity of the rock mass. In general, under the stress path shown in the test, while retaining a certain bearing capacity, the horseshoe-shaped section will not have the phenomenon of large rock mass collapse, which is more suitable for use in the corresponding practical engineering.
Conceptualization, Y.C.; Formal analysis, H.L. and X.W.; Investigation, D.G., Y.Z. and L.Z.; Resources, D.G. and L.H.; Data curation, H.L.; Writing–original draft, H.L.; Writing–review & editing, X.W.; Supervision, L.H., Y.Z. and L.Z.; Project administration, Y.C. and X.W.; Funding acquisition, Y.C. All authors have read and agreed to the published version of the manuscript.
Not applicable.
Not applicable.
All data: models, or code that support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare that there is no conflicts of interest that may affect the research reported in this paper.
Footnotes
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Enlarge this image.Figure 1. Physical test model of sandstone.
Enlarge this image.Figure 2. Development process of surface cracks.
Enlarge this image.Figure 3. PFC test model: (a) Boundary condition; (b) Cavern size; (c) Loading condition
Enlarge this image.Figure 4. Evolution process of meso cracks.
Enlarge this image.Figure 5. Evolution process of meso cracks in different cross-section models.
Enlarge this image.Figure 6. Force chains of different cross-section models: (a) Force chain of elliptic cross-section model; (b) Force chain of ovoid shaped cross-section model; (c) Force chain of horseshoe-shaped cross-section model; (d) Force chain of vertical-wall arched cross-section model.
Enlarge this image.Figure 7. Peak stress levels for each model.
Enlarge this image.Figure 8. Crack number-vertical stress curve.
Strength parameters of specimen materials.
| Material | Density |
Uniaxial Compressive Strength(MPa) | Elastic Modulus (Gpa) | Water Saturated Bending Strength (Mpa) | Wave Velocity (m/s) |
|---|---|---|---|---|---|
| Sandstone | 2119 | 21.2 | 10.1 | 7.7 | 2940 |
Mesoscopic parameters of PFC test.
| Elastic Modulus (GPa) | Porosity | Density (kg.m−3) | K-Ratio | Tenpbond |
Copbond |
Fapbond |
|---|---|---|---|---|---|---|
| 35.0 | 0.16 | 2119.0 | 2.0 | 6.0 | 7.5 | 30.0 |
Dimensions of different cross-sections.
| Cross-Section Shape | Ellipse | Ovoid Shape | Horseshoe Shape | Vertical-Wall Arch |
|---|---|---|---|---|
| Height-span ratio | 0.83 | 1.45 | 1.0 | 1.5 |
| Area | 78.36 mm2 | 78.83 mm2 | 78.71 mm2 | 78.76 mm2 |
| Size | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] |
Rock mass characteristics at the different failure stages of each model.
| Circular | Ellipse | Ovoid Shape | Horseshoe Shape | Vertical-Wall Arch | ||
|---|---|---|---|---|---|---|
|
Fracture
|
Distribution | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] |
| Stress level | 10.82 MPa |
10.25 MPa |
7.73 MPa |
6.18 MPa |
7.72 MPa |
|
| Number | 4 | 1 | 2 | 4 | 2 | |
|
Fracture
|
Distribution | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] |
| Stress level | 17.55 MPa |
17.21 MPa |
15.73 MPa |
16.69 MPa |
14.13 MPa |
|
| Number | 50 | 47 | 43 | 26 | 21 | |
|
Fracture
|
Distribution | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] | [Image omitted. Please see PDF.] |
| Stress level | 22.83 MPa |
22.53 MPa |
19.61 MPa |
21.2 MPa |
18.32 MPa |
|
| Number | 319 | 281 | 221 | 246 | 179 | |
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