Damage Creep Analysis and Instability Delay Time of Ultrahigh pillar considering Initial Transverse Deformation

. According to the ultrahigh pillar structure and mechanical characteristics, considering the initial geometric deviation caused by mining disturbance, the simplified model of the pressure bar is established. Based on elastic-plastic theory, the deflection and internal forces of ultrahigh pillar with initial geometric imperfections are analyzed and discussed. Based on the Kachanov theory of damage, the damage of ultrahigh pillar is analyzed, the damage failure time is calculated. The results show that under the influence of the initial deflection, pillar will appear different degree of additional deflection, which grows with the increase of initial deflection and pressure ratio. Analysis of engineering examples show that the start time of damage and instability of delay time of pillar has close relationship with initial deflection, pressure ratio and compressive strength. With the increase of pressure ratio and initial deflection, the damage incubation time and the instability of delay time quickly become small, the greater the compressive strength, the greater the instability of the delay time, which means that the pillar is more stable. The results of this study provide a theoretical basis for stability analysis and control of ultrahigh pillar..


Introduce
Stability analysis of mined out area has always been a hot issue of [1] - [4], and also an extremely complex problem. The structure of mined-out area is an important influence factor of stability [5]. The pillar instability is one of the main forms of mined-out area damage.
At present, a great deal of domestic and foreign literatures has already obtained abundant fruits on the research of pillar stability of mined-out area [6] - [8] and many scholars put forward a variety of theoretical models, such as the rheological model of roof-pillar system [9], nonlinear dynamics model of shell [10]. Li Hong [11] derived the stable model of pillar in asymmetric mining, the theoretical solution is obtained. Exadaktylos G [12] studied the pillar instability phenomenon caused by the axial splitting under the uniaxial pressure based on the fracture mechanics. Yin Shenghua [13] evaluated the sensitive of influence factors on the stability of pillar using the method of range analysis in orthogonal. The current study comprehensively reveals the stability characteristics and instability mode of pillar with large width-height ratio. However, the study of the stability of ultrahigh pillar with small width-height ratio relatively few.
With the increase of mining depth, due to the deterioration of the mining environment, safe and efficient subsequent filling method will become the main mining method [14]. When the surrounding rock conditions are good, the stage room with large height and subsequent filling will become the preferred mining method while a lot of ultrahigh pillars will appear at the same time. The pillar is the key factor to control the stope stability. Because height is far greater than the width, so the overall stability of ultrahigh pillar in the transverse direction is significantly decreased. The deformation characteristics of ultrahigh pillar are similar to that of pressure bar structure. But due to the complexity and the anisotropy of the occurrence environment and rock properties, ultrahigh pillar instability mechanism is more complex. In addition, there has been a certain degree of transverse deformation in the ultrahigh pillar caused by the blasting in the mining process, which has important effect on the stability of ultrahigh pillar. However, the study of the stability of pillar with initial transverse deformation is still less.
In this paper, based on the plastic-elastic theory, we analyzed the stability of ultrahigh pillar with transverse initial deformation firstly and established the computing model. Then, based on the creep damage theory, we analyzed damage process of ultrahigh pillar and obtained the instability time of ultrahigh pillar.

Simplified mechanical
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Elastic-plastic strut model of ultrahigh pillar
Ultrahigh pillar of underground mine is often under high pressure which resulted in a large longitudinal deformation in the axial direction which is fa greater than horizontal deformation that can be ignored. So, the top of the pillar can be simplified as a directional support with only axial deformation. At the bottom of the pillar, because the stiffness of floor is far greater than that of pillar, the angular displacement and linear displacement can be ignored. So the bottom of pillar can be simplified as fixed bearing. The Elastic-plastic strut model of ultrahigh pillar is shown in Fig.1.

Basic assumptions
In order to simplify the analysis process, the following basic assumptions is done in the calculation: (1) The weight of the pillar is not considered； (2) Before entering the stage of crack extension, the deformation of pillar is small and overlying load did not change significantly； (3) In the mining process, rock plastic deformation did not change the constraint type of both ends of pillar; (4) In the process of damage, the elastic modulus and inertia moment of cross section of pillar maintained which meet the plane section assumption.
In order to simplify the analysis, the influence of blasting vibration and horizontal stress are neglected, the effect of vertical load is only considered.

Analysis of deformation and stress of pillar
(1) Transverse deformation analysis For the mechanical model shown in Figure 1, the boundary conditions are expressed as： The transverse deformation caused by the overlying load on the top of pillar is 1 ( ) w x ， then the transverse deformation of pillar can be obtained as According to the theory of strut, Eq. (3) satisfies the differential equation as shown Eq. (4) ： Where, E is elastic modulus, I is inertia moment, P is overlying load. Taking Eq. (2) and Eq. (3) into Eq. (4), then we can be obtained as The transverse deformation 1 ( ) w x can be obtained as： Where，a，b，c，d are constants. Taking Eq. (1) into Eq. (6), we can obtain as Where ， cr P is the limit load of pillar. For the strut model in this paper, The total transverse deformation can be calculated as When α >1, the overlying load exceeds the limit load of pillar, the overall instability will occur. When 0<α <1, the creep failure will occur under the effect of overlying load. The analysis on the transverse deformation of pillar under different conditions was carried out. The threedimensional distribution map of pillar transverse deformation under overlying load, α and pillar height when 0 500 f = mm is shown in Fig.2. It can be seen from the Fig.2 that transverse deformation distributes symmetrically along the height of pillar, which is the largest in the middle part and the value is 5mm. With the increasing of α , the transverse deformation of the pillar increases nonlinearly.   (2) Analysis of stress Because of the existence of initial transverse deformation, axial and bending deformation will appear in ultrahigh pillar. Therefore, there are two kinds of stress in pillar internal which are separatly axial stress and bending stress. During the deformation of pillar, the internal stress of pillar can be expressed as 1 0 Where, 0 σ is the axial stress， 1 σ is the bending stress.
Based on the geometric equation, bending strain in pillar can be obtained as Then the 1 σ can be calculated as The stress of any point in the pillar can be expressed as

Analysis of damage process
The rock in the concave of the section of the middle of pillar begin to yield, which is shown in Fig.5. Due to the axial compressive stress is larger than bending stress, this paper only considered the plastic yield zone shown in Figure 5. Assuming that axial compressive stress caused by the overlying load maintained in the analysis process. Based on the Kachanov creep theory [15], the creep damage model for rock can be expressed as： is Change rate of rock damage factor. Taking t: 0~t，D: 0~D，based on Eq. (16) using integral, the damage variable evolution equation is obtained as When I t t = , the concave of the section of the middle of pillar begin to damage.

Delay time of failure
After I t t > , the damage zone will be gradually extended to the interior. Assuming that the plane sections" hypothesis is still applicable in the process of damage. Setting distance of neutral axial moving to internal is δ Δ , the thickness of fracture layer is 2 δ Δ , which is shown in Fig.6.
Based on the Eq. (15), the maximum compressive stress of time t can be obtained as Where, II t is the delay time of failure.

Engineering case analysis 4.1 Engineering background overview
Dong guashan is a famouse copper mine in China，the mining depth has been greater than 1000m. In order to improve the efficiency of mining, large-diameter longhole blasting-stage room and subsequent filling mining method is the main mining method，the stope height is always greater than 80m. Therefore, heightwidth ratio is usually very large, buckling deformation and instability are easy to occur in the pillar under the overlying load.

Calculation of damage and instability time
Limit load of pillar can be obtained as The influence of various factors on the I the greater the t  is, the more stable is, but the less the growth rate is. Fig.9 Effect of compressive strength on the damage

Conclusion
Based on the elastoplastic theory and strut method, this paper established the calculation model of ultrahigh pillar with initial transverse deformation. The transverse deformation of the pillar is analyzed. Then, the damage process and instability time of ultrahigh pillar are analyzed based on the creep theory. The main conclusions are as follows.
(1) For the ideal pillar, when the overlying load is less than the critical load of the pillar, there is only compression deformation but no transverse deformation in the pillar. As for the ultrahigh pillar with initial transverse deformation, the additional transverse deformation will be produced under the action of the overlying load. The research shows that the additional transverse deformation of pillar is closely related with the ratio of overlying load and limit load (α ) and initial transverse deformation ( 0 f ). Additional transverse deformation increases linearly with the increasing of initial transverse deformation. With the increasing of ratio of overlying load and limit load, additional transverse deformation increases nonlinearly. The greater the ultimate compressive strength is, the greater the t  is.
(3) The initial transverse deformation of pillar, which is caused by the blasting in the mining process, has an important influence on the stability of pillar. Therefore, in order to ensure the stability of the pillar, the disturbance to the pillar should be reduced so as to reduce the 0 f . In addition, with the increase of the mining depth, the overlying load increases and α becomes larger, which results in shorter stability time of pillar. At this time, we should increase the width of the pillar and reduce the height of the pillar to strengthen the stability of pillar. (4) The results of this study has an important guiding role in the improvement of stability control and management.