3.1.1. Working Principle of Single-Point Loading Device

Reasonable and accurate counterweights directly affect the accuracy of the test data, and an efficient and feasible counterweight method is an important part of the test design. To solve the drawbacks of traditional loading methods, a pulley-group loading scheme was designed [28–30], and its working principle is shown in Figure 5. In the figure, pulleys X1, X2, and X3 form a fixed pulley group, which is fixed on the ground, and pulleys S1, S2, and

S3 form a movable pulley group, which is connected to the upper structure using elastic supports to simulate the vertical displacement of the main arch ring. The pulley groups are connected by a wire, starting from the axis of pulley S3, winding counterclockwise around pulleys X1, S3, X2, S2, X3, and S1 in sequence, and a counterweight block is suspended at the end of the wire.

**Figure 5.** Diagram of the working principle of a loading system with a single set of pulleys.

In the aforementioned loading scheme, the ratio of the actual load (*F*) borne by the loading point to the self-weight load (*G*) of the counterweight block is denoted as the load magnification factor, represented by *FS*. The schematic calculation diagram is shown in Figure 6. In Figure 6, taking pulley S2 as the research object, the load magnification factor *F*<sup>2</sup> = *F*<sup>6</sup> can be determined according to the principle of moment balance. By analogy, *Fi* = *G*(*i* = 1, 2, ··· , 7) can be obtained; thus, the theoretical value of the load magnification factor for the single-point loading device under this design scheme can be derived as follows:

$$F\_S = \frac{F}{G} = \frac{\sum\_{i=1}^{7} F\_i}{G} = \mathcal{T} \tag{1}$$

**Figure 6.** Simple diagram of single pulley force.

For the moving pulley system, the load magnification factor, *FS*, alters along with the changes in the number of pulleys, and the relationship between the two is expressed as follows:

$$F\_S = 2X\_m + 1,\\
\left(X\_m \in N\_+\right) \tag{2}$$

where *Xm* represents the number of moving pulleys.

It is worth noting that as the number and the diameter of the pulleys increase, the total frictional force between the pulleys and the rope will also escalate, causing a reduction in the mechanical efficiency of the single pulley system's loading apparatus. Simultaneously, a rise in frictional force extends the time required for the tension of the outer wire rope to reach the inner wire rope. Therefore, when designing an experiment, comprehensive consideration should be given to various factors, such as the load magnification factor, mechanical efficiency, and the time required for the system to stabilize.
