Analysis Method for Ultimate Load of Instability of Arch Blocks in Anti-arch Water Cushion Pond

Abstract

This invention discloses a method for analyzing the ultimate load of arch block instability in an inverted arch stilling basin. Using structural mechanics methods, a mechanical model of arch block instability is established, dividing the instability process into three stages: lifting of the middle arch block, sliding of the two side arch blocks, and warping of the two side arch blocks. Considering engineering safety and reasonable computational complexity, the model is simplified, and a simplified structural mechanics analysis method that facilitates manual calculation is proposed. This method can calculate the ultimate load of arch block instability corresponding to different anchoring forces in an inverted arch stilling basin, providing a reliable theoretical basis for the structural design of inverted arch stilling basins and possessing significant engineering application value.

Description

Technical Field

[0001] This invention belongs to the technical field of anti-arch water cushion pond engineering, specifically relating to a method for analyzing the ultimate load of instability of the arch block in an anti-arch water cushion pond. Technical Background

[0002] Water cushion ponds are a primary measure for flood discharge and energy dissipation in arch dams. Inverted arch water cushion ponds utilize the shape of the riverbed bedrock to create an inverted arch shape for the bottom slab, leveraging the mechanical properties of the arch structure to fully utilize the compressive strength of concrete and the overload capacity of the arch structure. For arch dam projects in high mountain canyons, the advantages of inverted arch water cushion ponds—such as strong overload capacity, less excavation, and minimal damage to the banks—are widely recognized. However, under high water head and large flow rates, the inverted arch blocks can also experience localized instability. If the construction quality of the water cushion pond arch blocks is poor, especially if there are safety loopholes in the water-stopping facilities between the arch blocks, the water-stopping will fail. The dynamic water pressure generated by the flood discharge will then flood the gaps in the arch blocks, creating a strong radial load that causes the arch blocks to fail anchorage and be pulled out of their anchorage.

[0003] Currently, many mechanical methods for calculating the stability of inverted arch water cushion ponds consider the overall stability of the inverted arch structure. However, due to the existence of structural joints, the thrust at the arch end is often significantly lost when it is transmitted to the center of the inverted arch, resulting in an overestimation of the ultimate radial load and making the project more dangerous. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide a method for analyzing the ultimate load of instability of the arch block of an anti-arch water cushion pond, which can obtain more reliable calculation results and has a better guiding role in the structural design and reinforcement scheme of the anti-arch water cushion pond.

[0005] According to a first aspect of the present invention, the present invention adopts the following technical solution:

[0006] The method for analyzing the ultimate load of instability of the arch block in a counter-arched water cushion pond is based on the following simplification:

[0007] (1) Simplification of structural joints: In order to strengthen the arch effect of the anti-arch type water cushion pond in the project, the width of the structural joint between the arch blocks is generally set to be small or even negligible. Therefore, the influence of the width of the structural joint is no longer considered in the model.

[0008] (2) Simplification of the rotation center: When a certain arch block is locally unstable, the arch blocks on both sides will rotate. However, due to the different magnitudes of the constraint forces on both sides, the rotation center will also be different. When the external constraint on one side is larger, the rotation center is the top. When the external constraint is smaller, the center is selected as the bottom. For safety considerations, it is assumed that the constraint forces on both sides are always insufficient, and the position of the rotation center is unified as the bottom. At the same time, it is assumed that when the arch block rotates, it will only push one arch block outward.

[0009] (3) Simplified arch end constraint: In order to better handle the stability problem of the arch block in the middle of the anti-arch, no arch end constraint was applied.

[0010] The analysis of the ultimate load for the instability of arch block j, which is adjacent to arch blocks i, j, and k from left to right, includes the following steps:

[0011] 1) Solve for the force F exerted by arch block i-1 on arch block i. i ;

[0012] 2) Based on the moment balance of arch block i, the ultimate load F when the arch block is tilted is obtained. ij ;

[0013] 3) Solve for the force F exerted by arch block k+1 on arch block k. k ;

[0014] 4) Based on the moment balance of arch block k, the ultimate load F when the arch block is tilted is obtained. kj ;

[0015] 5) Solve for the ultimate load of arch block j based on the normal force equilibrium of arch block j;

[0016] Arch blocks i, j, and k are adjacent arch blocks from left to right in the inverted arch water cushion pond. Among them, arch block i-1 is the left adjacent arch block of arch block i, and arch block k+1 is the right adjacent arch block of arch block k.

[0017] This invention establishes a mechanical model for arch block instability using structural mechanics methods, dividing the instability process into three stages: lifting of the middle arch block, sliding of the two side arch blocks, and warping of the two side arch blocks. Considering engineering safety and reasonable computational complexity, the model is simplified, and a simplified structural mechanics analysis method that facilitates manual calculation is proposed.

[0018] The present invention has the following beneficial effects: Addressing the local instability problem of arch blocks in inverted arch stilling basins, the present invention studies the mechanical mechanism of arch block instability based on a structural mechanics model, establishes an analytical method for the ultimate load of the instable arch block, and obtains the ultimate load of the arch block corresponding to different anchoring forces. The method of the present invention provides a reliable theoretical basis for the structural design and optimization of inverted arch stilling basins, and has significant theoretical significance and practical engineering application value for ensuring the safety of flood discharge at hydropower stations. Attached Figure Description

[0019] Figure 1 This is a schematic diagram of the anti-arch water cushion pond structure described in this invention;

[0020] Figure 2 This is a schematic diagram of the local instability and stress of the intermediate arch block described in this invention;

[0021] Figure 3 This is a schematic diagram of the force exerted on the left arch block when it is about to lift up, as described in this invention;

[0022] Figure 4 This is a schematic diagram of the force exerted on the right arch block as it is about to lift up, as described in this invention. Detailed Implementation

[0023] like Figure 1 As shown, an anti-arch water cushion pond structure includes an anti-arch base 10, arch blocks 1, 2, 3, 4, 5, 6, 7, 8, and 9 arranged on the base 10, with arch block number 5 at the bottom. The central angle of the arch block is α, and the radius of the arc length of the water cushion pond is R.

[0024] First, based on the mechanical mechanism of instability of the water cushion pond arch, the arch structure is appropriately simplified:

[0025] (1) Simplification of structural joints: In order to enhance the arch effect of the anti-arch type water cushion pond in the project, the width of the structural joint between the arch blocks is generally set to be small or even negligible. Therefore, the influence of the width of the structural joint is no longer considered in the model.

[0026] (2) Simplification of the center of rotation: When a certain arch block becomes locally unstable, the arch blocks on both sides will rotate. However, due to the different magnitudes of the constraint forces on both sides, the center of rotation will also be different. When the external constraint on one side is larger, the center of rotation is the top. When the external constraint is smaller, the center is chosen as the bottom. For safety considerations, it is assumed that the constraint forces on both sides are always insufficient, and the position of the center of rotation is unified as the bottom. It is also assumed that when the arch block rotates, it will only push one arch block outward.

[0027] (3) Simplified arch end constraint: In order to better handle the stability problem of the arch block in the middle of the anti-arch, no arch end constraint was applied.

[0028] Therefore, for adjacent arch blocks i, j, and k (1≤j≤5), a unified mechanical model can be constructed for the local instability of the intermediate arch block j, and its forces are as follows: Figure 2 As shown, the calculation steps for the ultimate load of arch block j under stress instability are briefly described below:

[0029] (1) Solve for the force F exerted by arch block i-1 on arch block i. i ;

[0030] (2) Based on the moment balance of arch block i, the ultimate load F when the arch block is tilted is obtained. ij ;

[0031] (3) Solve for the force F exerted by arch block k+1 on arch block k. k ;

[0032] (4) Based on the moment balance of arch block k, the ultimate load F when the arch block is tilted is obtained. kj ;

[0033] (5) Solve for the ultimate load based on the normal force balance of arch block j.

[0034] Since the arch blocks are symmetrical, the simplified structural mechanics algorithm steps will be explained below using the left arch block as an example.

[0035] (1) First step: solve for the force F exerted by arch block i-1 on arch block i. i ;

[0036] When arch block i pushes arch block i-1 on the outer side, arch block i-1 slides upward tangentially along the arc, and the frictional force it experiences is downward tangentially along the arc. According to the force balance of arch block i-1 in the tangential direction, we can conclude...

[0037]

[0038] In the formula:

[0039] μ—coefficient of sliding friction;

[0040] α i-1 —The central angle of the center of arch block i-1 (unit: °). Considering the symmetry of the water cushion pond, when i-1<5, the central angle takes a positive value. Let the central angle of each arch block in the arc be α, then the central angle of the center of the i-th arch block is α(5-i).

[0041] G – Weight of the arch block (unit: N).

[0042] (2) The second step is to solve for the ultimate load F when arch block i is lifted. ij

[0043] Force analysis when the left arch block is about to lift up is as follows: Figure 3 As shown, by referring to point C i To determine the torque, the critical contact force F required to lift arch block i can be obtained based on torque balance. ij As shown in the following formula:

[0044]

[0045] In the formula:

[0046] L m —Line segment C i O i Length (unit: m);

[0047] L CiP —Line segment C i The length of P (unit: m);

[0048] H—Thickness of the arch block (unit: m);

[0049] A max — Anchorage force (unit: kPa) when all anchorage reinforcement bars of the arch block reach their yield strength;

[0050] α i —The central angle of arch block i (unit: °);

[0051] ξ — Effective coefficient of anchoring force.

[0052] (3) The third step is to solve for the force F exerted by arch block k+1 on arch block k. k

[0053] According to the force balance of arch block k+1, we can conclude that:

[0054]

[0055] (4) Fourth step, solve for the ultimate load F when arch block k is lifted. jk

[0056] When the right arch block tilts up, the force is as follows Figure 4 As shown, by referring to point C k To determine the torque, we can use torque balance to find the critical contact force F that causes the arch block k to lift. jk As shown in the following formula:

[0057]

[0058] (5) Fifth step, solve for the ultimate load of arch block j.

[0059] Based on the normal force balance of arch block j, the ultimate load can be obtained as follows:

[0060]

[0061] Example

[0062] right Figure 1 The ultimate load at which the arch blocks of the anti-arch water cushion pond structure become unstable is calculated. The effective anchorage force coefficient ξ is taken as 0.2. Using the analysis method proposed in this invention, the ultimate load at anchorage level A for each arch block is calculated. max The ultimate loads at 0, 10 kPa, 20 kPa, 50 kPa, 100 kPa, 150 kPa, and 200 kPa are summarized in the table below.

[0063] Table 1. Calculation results of the ultimate radial load for arch block instability under different anchoring force levels.

[0064]

[0065]

[0066] To verify the reliability of the calculation method of this invention, Figure 1The local pull-out resistance of the arch blocks in the anti-arch water cushion pond was analyzed using the discrete element method, and the results of the structural mechanics analysis method of this invention and the discrete element method were compared. As shown in Table 2, it can be found that under the same anchorage level, the ultimate load between the arch blocks obtained by the structural mechanics method of this invention is not much different from that obtained by the discrete element method. This is because for large-span anti-archs, the arch action is usually insufficient to provide enough constraint. From a safety perspective, the formula only considers the action of the four surrounding arch blocks for each arch block, without considering the superposition of the axial forces of each arch block under the arch action (i.e., the arch action).

[0067] Table 2 Comparison of Calculation Results with Discrete Element Method

[0068]

Claims

1. A method for analyzing the ultimate load of instability of arch blocks in a counter-arch water cushion pond, characterized by: A mechanical model for the instability of the arch blocks in a counter-arch water cushion pond is established using structural mechanics methods. The instability process is generally divided into three stages: lifting of the middle arch block, sliding of the two side arch blocks, and tilting of the two side arch blocks. When a certain arch block experiences local instability, the two side arch blocks will rotate. The rotation center of the two side arch blocks is uniformly located at the bottom, and no arch end constraints are applied. The method includes the following steps: 1) Solve for the force F of arch block i-1 on arch block i i ; 2) According to the moment balance of i-th voussoir, the limit load F when the voussoir is lifted up is obtained ij ; 3) Solve for the force F that the k+1 arch block exerts on the k arch block k ; 4) According to the moment balance of the kth arch block, the ultimate load F is obtained when the arch block is lifted up kj ; 5) Solve for the ultimate load of arch block j based on the normal force equilibrium of arch block j; Arch blocks i, j, and k are adjacent arch blocks from left to right in the anti-arch water cushion pond. Among them, arch block i-1 is the arch block to the left of arch block i, and arch block k+1 is the arch block to the right of arch block k. In step 1), when arch block i pushes arch block i-1 on the outer side, arch block i-1 slides upward tangentially along the arc, and the frictional force it experiences is downward tangentially along the arc. The force F exerted by arch block i-1 on arch block i is... i The equation is to be solved as follows: Where μ is the sliding friction coefficient of the arch block, α i-1 Let G be the central angle of the center of arch block i-1, G be the weight of the arch block, z be a natural number, and arch block z be the bottom arch block of the water cushion pond. The ultimate load F when arch block i is lifted in step 2) ij The equation is to be solved as follows: Where H is the thickness of the arch block; A max α represents the anchorage force when all anchorage reinforcements of the arch block reach their yield strength, in kPa. i Let L be the central angle of arch block i, ξ be the effective anchoring force coefficient, and L be the angle of the center of the arch block i. m For line segment C i O i The length of L CiP For line segment C i The length of P; j is a natural number, z is a natural number, and arch block z is the bottom arch block of the water cushion pond; In step 3), the critical contact force F that causes arch block k to tilt can be obtained based on the torque balance. jk The equation can be solved as follows, where z is a natural number, the z-th arch is the bottommost arch of the water cushion pond, and k is a natural number:

2. The method for analyzing the ultimate load of instability of the arch block of the anti-arch water cushion pond according to claim 1, characterized in that: In step 4), the critical contact force F when arch block k tilts up jk The equation is to be solved as follows: In step 5), the instability and pullout of arch block j will cause the adjacent arch blocks on both sides to tilt upwards. The equation for the ultimate normal load of arch block j is:

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