In fig. 8.2, the load Q forms the rotational moment Mq of the arch beam, and the horizontal thrust T at the arch hinge forms the counter-rotational moment MT, which are balanced. According to the conclusion of the single-layer arch beam model test in Chapter 4, the limit value of MT is related to factors such as rock strength, thickness and initial stress, and it is a comprehensive index. When the limit of MT is greater than Mq, the arch beam remains stable and has a certain safety reserve; When they are equal, the arch beam is in a critical state; When the limit of MT is less than Mq, the arch beam is unstable. Remember, the MT limit is expressed as the above relationship:

Bolt support technology in coal roadway based on rock mass structure analysis

According to the conclusion of single-layer arch beam model test (see section 4.3.5), αM is called bending moment coefficient, and A is the width of rock stratum, which can be used as unit width. Table 4.7 shows the test results of bending moment coefficient of single-layer arch beam, which is a function of initial stress and span-thickness ratio. With the increase of span-thickness ratio, the bending moment coefficient has little change, but it first increases and then decreases with the increase of initial stress. In order to be safe and simplify the calculation, the bending moment coefficient is set as a constant, and it is taken as the value with large initial stress, here it is taken as 0.2 1.

According to the critical state analysis, as shown in Figure 8.2:

Bolt support technology in coal roadway based on rock mass structure analysis

Further written as:

Bolt support technology in coal roadway based on rock mass structure analysis

Derive the above formula:

Bolt support technology in coal roadway based on rock mass structure analysis

By introducing safety factor η and creep factor ξ, the critical span of single-layer arch beam is recorded as Lc, which is obtained by substituting it into formula (8.7):

Bolt support technology in coal roadway based on rock mass structure analysis

Where: η = 4, ξ = 0.5 ~ 0.7. γ is 25kN/m3, and the above formula is further simplified as:

Bolt support technology in coal roadway based on rock mass structure analysis

Where the unit of σc is MPa; The unit of t is m; The Lc unit is m.