Tributary Area Method Research Articles

Assessing the stability of high-level pillars in deeply-buried metal mines stabilized using cemented backfill

Obtaining a reasonable assessment of the stability of high-level pillars stabilized using cemented backfill is particularly important when designing deeply-buried metal mines. This research determined the average vertical stress in high-level pillars in deeply-buried metal mines and also their strength when subjected to the confining pressure generated by cemented backfill. An expression was thus proposed for the stability of the cemented-backfill-stabilized high-level pillars in deeply-buried metal mines. The results show that the improved formula for estimating the strength of the high-level pillar gives a better reflection of the impact of on-site size and shape effects on the strength of the high-level pillars. The strength of a high-level pillar subject to on-site size effects was obtained by considering the compressive strength of the rock-mass of the pillar under confining pressure provided by the cemented backfill body in the deeply-buried mine. The shape effect was taken into account by considering that the ratio of the effective width to the height of the high-level pillar modifies its strength. At the same time, the concept of load transfer distance was introduced to determine the average vertical stress in the high-level pillar. This method effectively improves the tributary area method which overestimates the average vertical stress in the high-level pillar. When mining the ore body inside a pillar's zone of influence, the area of the orebody excavated should not exceed 0.80 (otherwise it will cause a significant increase in the average vertical axial stress in the high-level pillars). The new method proposed in this study allows the stability of high-level pillars stabilized using cemented backfill to be assessed when an insufficient amount of large-scale in situ testing has been conducted in a deeply-buried metal mine. It also provides a fast, economical, and reliable method for assessing the stability of mined-out areas in deeply-buried metal mines.

APPLICATION OF THE SAP2000 COMPUTER PROGRAMME IN THE MODELLING OF AN UNDERGROUND QUARRY OF DIMENSION STONE

A numerical analysis of part of the underground quarry of dimension stone with designed maximum dimensions of the rooms was carried out with the SAP2000 software based on the finite element method. The main objective of the analysis was to examine the preliminary stability and the influence of excavation on the redistribution of stresses in the surrounding rock masses, thus confirming the applicability of the tributary area method. The analytical solution with the tributary area method showed satisfactory agreement with the numerical calculations, and the results of the analytical calculations were on the safe side. The obtained results confirmed the predicted behaviour of the rock mass, which confirms the use of SAP2000 as a general-purpose software for civil engineering in the field of mining. The highest concentrations of stresses and strains are found in the edge areas of the pillars, so their correct design is crucial. The safety factor of the point with the highest stresses was determined and a value of 2.21 was obtained, thus fulfilling the stability condition. As for the influence on the surrounding area, stress changes of less than 10% were determined based on the solutions for the vertical stresses at a distance of 20 m from the underground excavation.

An improved pillar design methodology

Empirical pillar design methods are commonly used in the mining industry. The parameters within which these methods are valid are frequently unknown to the user or ignored. In addition, empirical design may not consider essential parameters such as blasting effects, orebody dip and the presence of geological structures, which all adversely affect the stability of the pillars. This can result in potentially serious pillar design strength over-estimates. Although the commonly based tributary area method is generally conservative, as the spans are seldom that large relative to the depth, failing to consider other relevant parameters can result in errors. Problems associated with an under-designed pillar can range from a local pillar collapse to a catastrophic chain reaction collapse (or run). Over-designed pillars are generally safe but reduce the extraction of the orebody, thus adversely affecting the profitability of the mining operation. We used laboratory tests and numerical modelling to understand the effects of pillar orientation, blasting and the presence of discontinuities on pillar strength. Reduction factors were developed with these models to be implemented in conjunction with the existing empirical pillar design methods. For any pillar or mine design, once it is implemented, the actual performance of the system must be checked regularly by observation and monitoring and adjusted if needed. The pillar design approach outlined in this paper can better optimize the pillar mining method by considering other generally ignored but important parameters, thus improving safety, productivity, and economic aspects.

Comparison of Axial Load on Column by Tributary Area Method and 3D Modelling by ETABS

Comparison of Axial Load on Column by Tributary Area Method and 3D Modelling by ETABS

Theoretical Analysis for Stability Evaluation of Rock Mass Engineering Structure under Combined Compression-Shear Loading: A Case Study of Inclined Pillar

An inclined pillar is a typical support structure under compression and shear loads for underground mining. The shear load caused by the inclination of the ore-body affects the bearing capacity of the pillar. At present, there is no systematic investigation on the influence of shear load on the stress state evolution and bearing capacity of the inclined pillar. Additionally, there is still a lack of effective evaluation of the bearing capacity of the inclined pillar in the presence of additional shear load. In this research, the theoretical analysis method is used to solve these problems. First, the compressive and shear load components on the inclined pillar were calculated by the tributary area method, and the average stress state of the inclined pillar, considering the influence of the shear load, was characterized by a series of generalized stress circles. The factors that affect the shear load, such as the area extraction ratio, the inclination of the ore-body, and the in-situ stress ratio, were analyzed, and it reveals that there are three kinds of stress paths of the inclined pillar and their trajectories are straight line, circle, and curve, respectively. Then, a shear strength model was proposed to evaluate the bearing capacity of inclined pillars. The expression of this model is multiplied by a vertical pillar strength model and a dimensionless coefficient that is named the contribution factor of shear load (CFSL). Some cases of inclined pillars were employed to verify the rationality of this model. Finally, the factors that affect the bearing capacity of pillars were analyzed. This investigation presents that the shear load affects the stress path and determines the bearing capacity of the pillar. Therefore, the shear load should not be neglected in pillar design and stability analysis.

Semi-analytical solution for gable roofs under uplift wind loads

A common type of failure observed during hurricane events is the detachment of the roofs of light-frame wood houses because of the inability of the roof-to-wall connection (RTWC) to transfer uplift wind loads acting on the roofs. This paper introduces a computationally efficient procedure for conducting nonlinear analysis of the roofs to determine the connections forces using a semi-analytical solution. The new procedure simulates an entire roof as a continuous beam resting on elastic supports through the use of statically indeterminate slope deflection equations that include shear deformations. It accounts for the two-dimensional spatial variation of the uplift wind pressure. The new model was validated against a three-dimensional finite element model and also through a comparison with the results of experimental testing of a gable roof house previously conducted at the University of Western Ontario, Canada. The model was used to determine the load sharing among the trusses under a realistic load. It was also used to assess the adequacy of using the tributary area method under a code load. Based on the validation results, the semi-analytical solution model demonstrated an ability to predict the response of a roof under uplift wind loads that vary both in time and space. The main advantage of the new model is the efficient computational time compared to three-dimensional finite element modeling. As such this model can be used in the future for conducting probabilistic analysis of the roof taking into account the variation of the connection properties.

Effect of roof to wall connection stiffness variations on the load sharing and hold-down forces of Australian timber-framed houses

Roof to wall connections (RWCs) play a major role in wind load sharing and load transfer of timber-framed houses. The lack of coherence in the load sharing and load transfer can cause premature failure of timber-framed houses. Thus, it is essential to provide RWCs with adequate strength and stiffness, to ensure the dynamic roof wind loads are adequately shared and transferred through the structural system to the foundations. Typical construction defects in RWCs (i.e. missing nails) reduce the stiffness of the RWCs, introducing discontinuities in load paths, contributing to total roof failure under windstorms. This study combines realistic spatially and temporally varying wind loads, captured from model-scale wind tunnel-studies, with a validated finite element model of the roof structure, that accurately describes the load sharing through the structural system and evaluate the effect of RWC stiffness variations and realistic wind loading variation on the load sharing and hold-down forces of Australian contemporary timber-framed houses. The combined effect of RWC stiffness variations and realistic wind loading variation are compared to the traditional design method approach (i.e. tributary area method and use of simplified design roof pressures from wind loading standard AS/NZS 1170.2 (2011)). Results show that the maximum RWC hold-down force from AS/NZS 1170.2 (2011) is about 10% greater relative than the maximum RWC hold-down force from realistic dynamic wind loads, and overestimates wind loads for most wind angles. The maximum RWC hold-down force derived by the load sharing method with defective connections (i.e. single and double nails missed on the RWCs), induced hold-down forces 10% higher than that of the traditional method at neighbouring RWCs. Further, hold-down force variation between traditional and load sharing methods can increase up to 50% in a real house roof structure, as the stiffness of their RWCs vary throughout the system due to the material non-linearity and construction defects/human errors.

Comparison of Seismic Responses of Updated Lumped-Mass Stick Model and Shaking Table Test Results

A conventional lumped-mass stick model is based on the tributary area method to determine the masses lumped at each node and used in earthquake engineering due to its simplicity in the modeling of structures. However the natural frequencies of the conventional model are normally not identical to tho...

Assessing the effects of rock mass gradual deterioration on the long-term stability of abandoned mine workings and the mechanisms of post-mining subsidence – A case study of Castle Fields mine

Post-mining subsidence is a serious problem in urban areas constructed above abandoned mine workings. Subsidence may severely damage the environment and threaten the stability of surface structures and infrastructure. This study aims at analysing the role of gradual weakening of rocks on chimney collapse and sinkhole subsidence formation, in shallow mine workings. The factors leading to the degradation of the rock masses mechanical parameters are explained. The long-term stability of mine workings at Castle Fields mine, in West Midland, UK, is investigated as a case study. Progressive chimney collapse resulted in a large sinkhole in this area, and damaged the Dudley cricket ground, in 1987. The modified Tributary Area Method, TAM, is used to assess the effects of deterioration on the long-term loading of pillars. The long-term strength of pillars is also assessed based on empirical and theoretical approaches. Then, three-dimensional (3D) numerical modelling is used to assess the role of deterioration on the mechanisms of chimney failure and sinkhole development. A deteriorating model is proposed to incorporate the effects of gradual deterioration of rock mass mechanical parameters. Results of this study indicate that gradual deterioration due to weathering plays a significant role in the propagation of cavity and the formation of sinkholes above abandoned mine workings. Ignoring the gradual softening and the effects of deterioration, in the stability analysis of mine workings, may result in catastrophic consequences.

Effect of Columns Stiffness's and Number of Floors on the Accuracy of the Tributary Area Method

Effect of Columns Stiffness's and Number of Floors on the Accuracy of the Tributary Area Method

Proposal of 2D finite element model for square pillar stability analysis

It is suggested that 2D finite element method models are used for square pillar stability analysis. Equivalent width of strip pillar that is represented by FE models is determined from tributary area method by introducing equivalence of square and strip pillar loads in tributary area method analysis. With FE analysis it is shown that pillar loads are close to expected, but tributary area method is limited to pillars of small heights and with horizontal deposits. Proposed methodology provides opportunity to model much complex conditions and sequencing of excavation with respect of sequential stress redistribution.

Aerodynamic profiling of terminal building using computational fluid dynamics approach

A case study of isolated building is studied using ANSYS CFX and SAP2000. The plan idea of 30m by 60m is chosen for terminal building. The model is subjected to different wind incidence from 0° to 90° and 45° with 30° interval for 55m/s wind speed. By using tributary area method, the forces at the each mesh node are summed up to get corresponding wind force at that joint within that area. The best effective structural system is determined by designing the structure for each wind incidence. Wind analysis and design is carried out for increasing wind speed above 55m/s to identify the collapse pattern of structure. External supporting members are suggested to withstand that maximum wind speed.

A new approach for estimating internal stability of reinforced soil structures

The current practice for designing reinforced soil structures involves the use of empirical or semi-empirical methods to estimate the maximum reinforcement loads using different variants of tributary area method. As the soil–reinforcement interaction is not explicitly captured in these methods, a number of limitations exist in estimating the maximum reinforcement load and pullout resistance. This paper presents a new method called the soil–reinforcement interaction method to assess the internal stability of reinforced soil structures. As a part of its derivation, a more refined interface friction model was developed that accounts for the increased friction from soil dilation and subsequent reduction in friction with displacement. By combining the interface friction model and reinforcement stiffness, an analytical model was developed to relate the displacement, strain, force, and mobilized frictional length for a given reinforcement. Using this model, an approach was developed to assess the internal stability of reinforced soil structures. The method allows estimation of the factor of safety against internal failure by considering the soil–reinforcement interaction and boundary conditions. To demonstrate the applicability of this method, several reported pullout tests and three full-scale reinforced walls are modeled.

The effect of rock mass gradual deterioration on the mechanism of post-mining subsidence over shallow abandoned coal mines

In this study, the effect of gradual deterioration on the long-term stability of shallow abandoned room and pillar coal mines is investigated. The conventional Tributary Area Method is adopted and modified to take into account the rate of deterioration on the lifetime of pillars. Moreover, a simple quantitative approach is proposed to incorporate the effects of deterioration in numerical modelling. The suggested methods are then used to assess the stability of the shallow abandoned room and pillar coal mines around Dolphingstone village, UK. Results of this study indicate that the gradual deterioration plays a significant role in the long-term stability of shallow abandoned coal mines. The stand-up time of a room and pillar mining panel is also estimated based on the modified Q-system. A good agreement is observed between the results of numerical modelling and the rock mass classification system.

Numerical analysis of longwall mining layout for a Wyoming Trona mine

At Solvay Mine, located in southwestern Wyoming, a subhorizontal trona seam is mined at depths of between 460 and 490m using mechanized room-and-pillar and longwall mining methods. The stratigraphy at the mine generally consists of horizontally laminated (i.e., bedded) sedimentary rocks comprised mostly of shales and sandstones with significantly contrasting mechanical properties. Most notably, a 43–82m-thick massive, brittle sandstone unit (Tower Sandstone), located approximately 100m above the mining level, has a tendency to promote stress arching within the overburden rock that can bridge over panel-scale mine instabilities and can lead to violent multi-panel collapse failure. One such violent collapse is the 5.1 magnitude seismic event due to a 1×2km multi-panel failure on February 3, 1995. It has proven difficult to account for this arching behavior with conventional mine design methods, such as the tributary area method. Therefore, over the past two decades or more, Solvay Mine has been utilizing numerical modeling techniques along with field instrumentation/monitoring as part of an integrated program to gain an enhanced understanding of the complex response of the overlying stratigraphy (i.e., arching) to mining. In 2005 and 2006, several longwall panels in the northwest and southeast areas of the mine were instrumented and monitored during mining. Two- and three-dimensional numerical models were developed and calibrated on the basis of the instrumentation data, and these models were then used for mine design verification (e.g., pillar and panel dimensions). This mining case study illustrates the complex excavation response due to the contrasts in stratigraphy at Solvay Mine and presents a numerical modeling study that captures the dominant aspects of these conditions.

Practical Analysis for Horizontal Diaphragm Design of Wood-Frame Single-Family Dwellings

AbstractSeismic design of wood-frame single-family dwellings’ (WFSFD) lateral force–resisting systems requires determination of the stiffness of horizontal diaphragms and shearwalls. During design, sizes and locations of shearwall openings are often changed, altering shearwall stiffness and loads and requiring a significant redesign effort. Rigid and tributary area method analyses are examined for different geometries of L-shaped WFSFD and include stiffness reductions for roof geometry and pitch. These methods are applied to historic earthquake damage reports and compared using rigid, semirigid, or flexible horizontal diaphragm analyses useful in design practice. Most WFSFD should be designed using an envelope method because they contain a mix of horizontal diaphragm types owing to the effects of roof pitch and geometry on diaphragm stiffness. Cases occur where determination of semirigid or flexible diaphragm behavior is difficult because the analysis results are contradictory or unclear. This suggests th...

PARAMETRIC STABILITY ANALYSIS OF ROOM AND PILLAR METHOD IN DEEP COAL MINES

The goal of this paper is, to examine the validity of tributary area method, which is used as an empirical mean to estimate pillar dimensions, in room and pillar mines, taking into consideration a safety factor. A two-dimensional, linear elastic, finite element model (Phase 2D) is created for a simple case of a uniform pattern of room and pillar mine layout, that is lying flat 1200 m below ground surface. Sensitivity analysis has been done to study the effect of horizontal-to-vertical stress ratio, K, on the maximum stress that pillar can sustain without failure. Four cases have been presented in this investigation (K = 0.5, 1, 1.5 and 2). The results reveal that, the maximum stress on the pillar increases as K increases. Alternatively, factor of safety deteriorates as mining depth (i.e., increase of K-value) increases. The results also show that displacements/convergence expand as mining depth goes down.

Finite-element modeling of a light-framed wood roof structure

Past high speed wind events have exposed the vulnerability of the roof systems of existing light-framed wood structures to uplift loading, contributing greatly to economic and human loss. This paper further investigates the behaviour of light-framed wood structures under the uplift loading of a realistic pressure distribution. A three-dimensional finite-element model is first developed to capture the behaviour of a recently completed full-scale experiment. After describing the components used to develop the numerical model, a comparison between the numerical prediction and experimental results in terms of the deflected shape at the roof-to-wall connections is presented to gain confidence in the numerical model. The model is then used to analyze the behaviour of the truss system under realistic and equivalent uniform pressure distributions and to perform an assessment of the use of the tributary area method to calculate the withdrawal force acting on the roof-to-wall connections.

Calculation of Axial Loads on Columns by Tributary Area Method and Finite Element Method

Basic concept of structural design is to transmit the loading from superstructure to substructure. This idea normally required sound knowledge of structural design and rational engineering judgments. Recently, there have several techniques that can be utilized to determine the superstructure loading, such as finite element method and tributary area method. However, the compatibility of both methods in order to determine the loading from superstructure is prime important and has been investigated in this research framework. Axial loading, represented as products from dead load and service load, which are imposed on the top of slab is directly transmit to the column nearby and modelled through computer simulation. Models of slab were then varies and studies through comparison with broad dimensions of slab thickness, ranging in 100 mm to 600 mm. Results has shown the increasing of slab thickness will indirectly increases the rigidity characteristic of slab and potential to distribute the axial load equally for all column members. Axial load against slab thickness on corner, edge, center, outer and inner column demonstrated the incompatibility for both methods, finite element method and tributary area method in determining the axial loading from superstructure.

The Effect of Number of Storey's and Columns Dimensions on the Accuracy of Tributary Area Method

The Effect of Number of Storey's and Columns Dimensions on the Accuracy of Tributary Area Method