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Floor Bearing Characteristics of Jointed and Layered Rock Foundation
Faculty, IBAT School of Management, KIIT Deemed University; Bhubaneswar and Samir Kumar Das Professor, Department of Mining Engineering, Indian Institute of Technology, Kharagpur, India Email (corresponding author): dheeraj@edumail.nic.in |
ABSTRACT
Analysis of stability (mainly bearing strength and settlement) under a footing on regularly bedded, jointed and layered model rock mass is conducted using non-linear FEM analysis. A simple method to estimate bearing strength of both intact and jointed rock mass under circular, square and rectangular footings is proposed.
In jointed and layered rock mass conditions, the floor bearing characteristics analysis were carried out by considering the parameters like orientations of joint and weak layer with respect to the direction of loading, variations in the joint sets (joint spacing) and the layer thickness (as a function of footing plate width). The location of footing plates with respect to joint and weak layer were also varied.
3D non-linear FEM analyses of floor bearing characteristics (under the similar conditions as in the case of laboratory investigation) were carried out using a commercially available FEM software package. Three dimensional numerical models were developed for different conditions of surface footing foundations using appropriate rock mass properties. The inputs required for the FEM modeling were imported from the laboratory results of the measurements. Rock mass was modeled as elastic-plastic with Drucker- Prager failure criteria for plane strain condition. The joints and the weak layers in the floor rock mass were modeled in order to enable separation of zones in the models. The footing settlements correspond to the maximum applied bearing pressure on floor strata (for different sizes and shapes of footing plates and also under varying anisotropy conditions of floor strata) as obtained from the experimental results and FEM investigations, were compared and the maximum deviation was observed as 32 % whereas the minimum was even less than 1 %.
From FEM analysis it is interpreted that maximum stress concentration occurs at the tip of the footing plate all along the boundary. The stress concentration extends maximum to a distance 2 to 3 times footing plate width in all direction of floor strata. It is further interpreted that maximum vertical settlement occurs near to the vicinity of the footing plate.
From the FEM analysis of jointed rock it is examined that there is a slight movement of the two blocks separated by the joint. The joint plane acts as the rupture plane.
Keywords: FEM, jointed rock, nonlinear, analysis
MODEL STUDY
The present investigation has been carried out using the commercially available finite element program, ANSYS 6.1 (ANSYS 6.1, 2002) on HP_UX Unix workstation running hp_ux11.0 OS. The assumptions for the present analysis are:
Rock has been considered as homogenous, and non-linear in-elastic material.
Loading at the contact plane between the footing plate and the rock surface has been represented as:
Circular area load, of diameter that of footing plate placed centrally over the model floor strata in the case of circular surface footing (fig 1).
Square area load, of length that of footing plate placed centrally over the model floor strata in the case of square surface footing.
The loads were uniformly distributed on the surface area of footing plate located over the model floor strata. The loads were all symmetrically placed with respect to the vertical axis. 3-D 10-Node Tetrahedral Structural Solid elements (solid 92) were considered in this investigation for all types of modeling except joint.
The geo-mechanical properties, mainly Young's modulus and Poisson's ratio, shear strength parameters (called cohesion and angle of internal friction here) of respective floor strata model, as determined earlier in the laboratory, were considered as input for the FEM modeling (Kumar & Das, 2002). The bearing strength as determined from the model plate loading test on simulated floor strata in the laboratory were considered as the surface pressure to be applied on the footing plate resting over the model floor strata (Kumar & Das, 2002). For the purpose of comparison of the results obtained from the ANSYS program and the model plate loading tests for the determination of floor bearing strength in the laboratory, the FEM analysis were carried out on same dimensions and the magnitude of loading corresponding to the plate loading tests as carried out in the laboratory.
The loads were applied in steps with respect to time under ramped loading condition. The non linear analyses with Drucker - Prager criteria for non metal plasticity were followed as failure criteria. The numbers of load sub-steps were varied between 10 – 40 depending on the maximum intensity of the bearing strength and time. The “full Newton Raphson method” was used for the non linear converging solution.
ANALYTICAL PARAMETERS
Three dimensional numerical models were developed for different condition of surface footing foundation using appropriate rock mass properties. Rock mass were modeled as elastic-plastic with Drucker- Prager Criteria for plane strain condition. The following parameters were analyzed:
(i) Prediction of bearing strength of surface footings and settlement of the foundation with regards to size effects:
For this purpose the ground were modeled for different size and shapes of footing plates resting on it under various loading conditions. The geometrical dimensions in the numerical models were employed from the physical modeling tests conducted in the laboratory using model plate loading tests on weak floor simulated strata (Kumar & Das, 2002).
For the homogeneous and isotropic rock mass the ultimate bearing strength can be expressed by the function:
| qs = f (B, L, D, c, f, E, n, g) | (1) |
where
B, L - dimensions of footing
D - depth of foundation
c, f - shear strength parameters of rock, (a.k.a. "cohesion" and "angle of internal friction")
E - Young's modulus
n - Poisson's ratio
g - unit weight of rock.
The properties of the rock masses described above were used in the analysis; their values were taken from the laboratory results of the measurements. The Table 1 depicts the parameter and the variable used for the FEM analysis purposes.
(ii) Introducing joint (Fig. 2) in the model and to find out the influence of the following parameters on the floor bearing strength:
- Orientation and location of the joint with respect to direction of applied load:
When joint is located at the centre of the footing plate.
When joint is located at the edge of the footing plate.
b = 0o, 20o, 40o, 60o, and 80o.
The joints between the layers and the mass were modeled in order to enable separation of zones in the models (Gens et al., 1995; Lee et al., 1999). The elements for the meshing joint area were taken as target and contact element with the material properties characterized by coulomb sliding and/or tensile separation. Interfaces had the properties of friction, cohesion, normal (kn) and shear (ks) stiffness, and tensile strength (st).
Figure 1. Loading pattern and the boundary conditions adopted for the FEM analysis.
Figure 2. FEM modeling of joint located at the centre of model strata.
Table 1. Parameters and the variables for FEM analysis
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| Physico-mechanical Properties: | |
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| Shape | Square |
| Dimension (LxW) | 17cmX17cm |
| Thickness (T) | 4cm, 8cm, 12cm, 16cm, 20cm |
| Strata condition | Massive, jointed and layered |
| Uniaxial compressive strength | 0.95 MPa |
| Density (g) | 1785 kg/m3 |
| Modulus of Elasticity (Et) | 0.134 GPa |
| Poisson’s ratio (n) | 0.178 |
| Cohesion (c) | 0.82 MPa |
| Angle of internal friction (f) | 20o |
| Uniaxial tensile strength (st) | 0.143 MPa |
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| Footing Plate: Properties of Mild Steel ASTM - 154 were used to model the footing plate of different sizes as surface footing for the FEM analysis. The footings were modeled as an elastic material without weight. | |
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| Footing plate size (B) | 2.5 cm, 5.0 cm, 7.5 cm, 10.0 cm, 12.5 cm |
| Footing plate shape | Circular, square, rectangular |
| Footing plate / strata thickness (ratio) | 0.3125,0.625,0.9375, 1.25, and 1.5625 |
| Modulus of Elasticity (EM) | 199.5 GPa |
| Poisson’s ratio (n) | 0.290 |
| Density (g) | 7861.4 Kg/m3 |
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| Joint properties: The mica sheet with the following properties was used to simulate the joint in the FEM analysis: | |
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| Joint surface | Rough |
| Joint aperture, mm | 0.2 |
| Joint spacing | Remote |
| Joint filling | Soft |
| Friction coefficient | 0.3 |
| Normal Stiffness (Kn) | 200 GPa/m |
| Shear Stiffness (Ks) | 2 GPa/m |
| |
RESULTS
The FEM analysis has been carried out for circular and square footings of various sizes as mentioned in the Table 1. The results have been presented in the form of illustrations showing floor strata deformation behavior, extent of footing settlement (in meter) along the boundary, and stress (Pa) distribution (mainly equivalent stress) in the model strata for circular, square and rectangular footing plates with varying plate sizes (keeping B/T ratio constant to 0.625)
The magnitude of footing settlements in the case of central footing for the maximum bearing pressure (bearing strength) corresponding to the experimental value as applied to floor strata in FEM analysis are mentioned in Tables 2 and 3. The footing settlements as obtained from FEM analysis have been compared with the experimental results and the percentage of deviation is shown in the above mentioned tables. The maximum deviation is observed as 32 % whereas the minimum is near about 1%.
Table 2. Comparison among laboratory and FEM result of the footing settlement
(Central circular footing)
Table 2. Comparison among laboratory and FEM result of the footing settlement
(Central circular footing)
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| Plate size (cm) | For B/T ratio = 0.625 | For varying B/T ratio | |||||
| Settlement (mm) | % of deviation | B/T ratio | Settlement (mm) | % of deviation | |||
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| Lab | FEM | Lab | FEM | ||||
| 2.5 | 2.38 | 2.041 | 14.2 | 0.3125 | 2.35 | 2.454 | 4.4 |
| 5.0 | 1.94 | 1.480 | 23.8 | 0.625 | 1.94 | 1.480 | 23.8 |
| 7.5 | 1.56 | 1.410 | 9.7 | 0.9375 | 1.30 | 1.105 | 19.5 |
| 10.0 | 1.04 | 1.024 | 1.9 | 1.25 | 0.95 | 1.000 | 5.3 |
| 12.5 | 0.68 | 0.790 | 16.2 | 1.5625 | 0.72 | 0.930 | 29.2 |
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Table 3. Comparison among laboratory and FEM results of footing settlement
(Central square footing)
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| Plate size (cm) | |||||||
| For B/T ratio = 0.625 | For varying B/T ratio | ||||||
| Settlement (mm) | % deviation | B/T ratio | Settlement (mm) | % deviation | |||
| Lab | FEM | Lab | FEM | ||||
| |||||||
| 2.5 | 1.96 | 1.931 | 1.5 | 0.3125 | 2.29 | 2.666 | 16.2 |
| 5.0 | 1.60 | 2.055 | 28.4 | 0.625 | 1.60 | 2.055 | 28.5 |
| 7.5 | 1.34 | 1.606 | 19.9 | 0.9375 | 1.15 | 1.160 | 0.9 |
| 10.0 | 0.88 | 1.138 | 28.4 | 1.25 | 0.79 | 1.000 | 26.5 |
| 12.5 | 0.62 | 0.800 | 29.1 | 1.5625 | 0.70 | 0.930 | 32.9 |
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Figure 3 shows the deformation characteristics of floor strata in case of circular footing plate of 5 cm with B/T ratio of 0.625. The bearing pressure of magnitude 7.85 MPa was applied in 20 equal sub-steps.
Figure 4 shows the equivalent stress concentration along the X (length) direction of the floor strata in the case of circular central footing of different plate sizes. From this figure it can be interpreted that maximum stress concentration occurs at the tip of the footing plate all along the boundary. The stress concentration extends maximum to a distance 2 to 3 times footing plate width in all direction.
Figure 5 shows the deformation behaviors of the floor strata for different sizes of circular central footing plates with fixed footing size / strata thickness (B/T) ratio. From all these results, it can be interpreted that maximum vertical settlement occurs near the vicinity of the footing plate.
Figure 6 shows the failure behaviors of floor strata in the presence of joint located at an angle 0o from the direction of applied load in the case of circular edge footing. The bearing pressure of magnitude 3.57 MPa was applied in 10 equal sub-steps. From the figure it is seen that there is a slight movement of the two blocks separated by the joint. The joint plane acts as the rupture plane.
Figure 7 shows Equivalent stresses along the entire length of model strata for circular footing of 5.0 cm size with fixed B/T ratio of 0.625 in the presence of single joint of varying orientations. From the graph it is clear that maximum stress concentration exists at the rim of footing plate.
Figure 8 shows the deformation behaviors of the jointed floor strata for varying joint orientation (circular central footing of 5cm sizes). From these figures it can be interpreted that maximum settlement occurs near the edge of joint plane.
Figure 3. Nodal displacements U in vertical direction - Z for footing size of 5.0 cm with B/T ratio of 0.625 (Circular footing).
Figure 4. Equivalent stresses along the entire length of model strata for circular footing of various sizes with fixed B/T ratio of 0.625.
Figure 5. Footing settlements along the entire length of model strata for various sizes of circular footing with fixed B/T ratio of 0.625.
Figure 6. Footing settlement in the presence of joint located at an angle 0o from the direction of applied load i.e. vertical (central circular footing).
Figure 7. Equivalent stresses along the entire length of model strata for circular footing of 5.0 cm size with fixed B/T ratio of 0.625 in the presence of single joint of varying orientations.
Figure 8. Footing settlement along the entire length of model strata for circular footing of 5.0 cm size with fixed B/T ratio of 0.625 in the presence of single joint of varying orientations
CONCLUSIONS
From the above analysis following conclusions may be drawn:
Tensile cracks are initiated at the rim of the footing forcing a depression beneath the footing leading to a stable crack growth as load increases.
With increased loads, the depression may eventually crush the floor with unstable crack growth and with fracture reaching the surface.
The maximum stress concentration occurs at the tip of the footing plate all along the boundary. The stress concentration extends to a distance 2 to 3 times the footing plate width in all direction.
The maximum normalized footing settlement occurs near the vicinity of the footing plate.
The footing settlement correspond to the maximum bearing strength as obtained from the bearing strength tests on simulated floor strata in the laboratory and using FEM analysis shows the minimum and maximum deviation of about 1% and 32 % respectively.
REFERENCES
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