## 2 Numerical Simulation with FLAC3D
### The Iterative Search Method for Assessing the Strength Requirement of Backfill Roof  
In the stability analysis of the backfill roof, some studies [24,25] have traditionally relied on the examination of plastic zone expansion to evaluate the roof state. When a massive plastic zone occurs in the backfill roof, it can be confidently concluded that the roof has become unstable. In contrast, if minor plastic zones are localized and only appear at specific locations, estimation of the roof's stability becomes challenging. Due to the absence of a quantitative metric linking the extent of plastic zones to the stability of the backfill roof, judgments based on plastic zone analysis remain highly subjective.  
Some studies [11, 12, 13, 22] reported that displacement can provide valuable insights into the state of side-exposed backfill, which correlates the displacement at critical locations of the backfill with its stability, thereby reducing the subjectivity in stability assessment and evaluation of the strength requirement. It is intuitive that the exposed backfill with sufficiently high strength can remain stable without significant displacement under its self-weight or external loads. Conversely, significant deformation occurs when the exposed backfill possesses low strength. Therefore, theoretically, there is a critical strength of the backfill roof at which it is in a state of limit equilibrium. When the strength of the backfill decreases from this critical value, the backfill roof changes from a stable state to an unstable state. This critical value represents the strength requirement for the backfill roof to maintain limit equilibrium.  
Building on this concept, an iterative search method for assessing the strength requirements through numerical simulation is introduced. In this method, the cohesion \(c\) is employed to represent the backfill strength, as the MC strength criterion is applied to  
Figure 3: Comparison between initial ground stress generated in numerical models and those obtained from on-site data.  
model the backfill in numerical simulations. This approach has been widely accepted in previous studies that utilized numerical modeling with the MC model to evaluate the strength requirement for backfill with side or base exposure [11; 12; 13; 14]. Therefore, the critical cohesion (_c_) can be estimated through the following steps:  
(1) Choose the initial search range. The critical cohesion should fall within the initial search range. In most cases, an initial search range of 200-2000 kPa is suitable for backfill roofs [23; 26].  
(2) Update the search range and approximate the critical cohesion. The midpoint of the current search range is assigned with the cohesion of the backfill roof, and the mining and filling processes for each layer are simulated. The maximum vertical displacement of the backfill roof under the current cohesion value is extracted, and any abrupt changes in displacement are examined. If a sudden change in displacement is observed, the current cohesion value is set as the lower bound of the search range. If no abrupt change occurs, the current cohesion value is set as the upper bound. This process is repeated to progressively narrow the search range and converge toward critical cohesion.  
(3) Determine the critical cohesion from the cohesion-displacement curve. After a number of iterations, the figure for variations of the maximum displacement of the backfill roof with different cohesion can be plotted. From the cohesion-displacement curve, the cohesion corresponding to the point of abrupt displacement change is identified as the critical cohesion (i.e., strength requirement) for the backfill roof.