PSI - Issue 48

Youcef Cheikhaoui et al. / Procedia Structural Integrity 48 (2023) 81–87 Cheikhaoui et al/ Structural Integrity Procedia 00 (2023) 000 – 000

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2.2. The pillars’ loading

Budavari, presented the loading of mine pillars by taking a beam on 2 simple supports and a spring, loaded uniformly across its span, figure2,(Ma et al., 2012). For the purpose of the example, he associates the beam with a horizontally stratified mass, loaded by the effects of gravity, and the spring with a vertical pillar acting as a column located in the center of the span. In this case, the spring is at rest if the deflection of the beam is zero. It will be loaded only if the beam is deformed by deflection, figure 2.a. In order to find out qualitatively the load on the spring when the beam is loaded (uniformly distributed load in the case of figure 2.b, it is necessary to consider the deformability of the beam figure 2.c and of the spring, figure 2.d and to analyze the partial diagrams of free bodies of this hyper static system). By analyzing this problem, we notice that the load on the spring is a function of the stiffness of the two elements of the structure, the spring and the beam. Thus, for a given loading of the beam, an increase in the stiffness of the spring will result in a greater proportion of the load being taken by it. The previous example could be generalized by using several springs in parallel, of various stiffnesses. Conceptually, the conclusions would remain similar. The load on each spring would be a function of the deformation characteristics of the beam, its own stiffness, the stiffness of the other springs and the position of the springs along the beam.

Fig. 2. Load transmission by deformation of the beam-spring system (Ma et al., 2012).

2.3. Tributary area theory TAT This approach allows, from only a few data, to determine the load on a pillar. The tributary area theory (TAT) assumes that after excavation, the pillar supports the weight of the ground contained in an imaginary prism, extending from the excavation level to the free surface, Figure 3. The tributary area theory accepts that the average load of the pillar can be obtained by the following equations (Brown and Brady, 1985): For a pillar, the average load at the center of the pillar is defined by: = ( 0 + ) 2 2 (1) : The load on the pillars calculated by the tributary area method, 0 : Room width, : Pillar width If our ground consists of different layers of thickness ℎ i the vertical stress , of the initial state is defined by the following equation: =∑ ℎ 1 (2) where σ v is Blank vertical stress, γ i Specific gravity for each layer I, h i Layer thickness.